Upload
poonamgupta
View
14.215
Download
2.586
Embed Size (px)
DESCRIPTION
statistics for economics for class 11
Citation preview
PREFACE TO THE THIRD REVISED EDITION
Periodic updating and review accommodate new knowledge as well as adds freshness even as it allows for continuity. This third revised edition is an effort to that end.
Statistics for Economics for class XI in its revised format brings forth the changed mode by the Central Board of Secondary Education, New Delhi in 200506. The revision includes sufficient exercises keeping in mind learning tools of Statistics in the context of the study of Economics.
This volume incorporates extensive and colourful diagrams and illustrations to enhance a better and friendlier understanding of concepts of Statistics. Answer to the numerical questions in the exercise of unit 3 are also provided so that the students can verify the solutions.
It is hoped that this revised edition will be of great help to both teachers and students.
— N.M. SHAH
PREFACE
Stamtics has become an anchor for social, economic and scientific studies Stat,st,cal methods are widely used in several disciplines, be i, planning bul ' management, psephology (study of voting patterns), psycholo^ oT adve Sn;
steps. A hst of formulae has been provided at the end of each chapter of unit 3 and rJr rTT —• ^ong years of teLh ng thrsubta
I have to acknowledge that in the wriring of this volume I have got immense help rom my frtends and relation. The pubUshers have been very cooperative aTrelp ll
1 Trr r "tT derstanding w'fe bes d s tsp rin '
me and looktng after the household, has done ind.spensable work for the volume Z
M.M. Shah, himself a scholar and teacher of economics, and who retired as dean of acuity of Commerce, Nagpur University and Prmc.pal, G.S. College of clt rct
Shri Ram College of Commerce, Delhi April, 2002
—N.M. SHAH
SYLLABUS
STATISTICS FOR ECONOMICSXI
One Paper
3 Hrs.
100 marks
104 Periods/50 xMarks 5 Periods/3 Marks 25 Periods/12 Marks 64 Periods/30 Marks 10 Periods/5 Marks
PARTA
STATISTICS FOR ECONOMICS
1. Introduction
2. Collection, Organisation and Presentation of Data
3. Statistical Tools and Interpretation
4. Developing Projects in Economics
Unit 1: Introduction
What is Economics? ^^
Meanmg scope and importance of statistics m Economics
Unit 2: Collection, Organisation and Presentation of Data p,,,
SoToflot^tm^^^^^^^ W imrr^  «
and National Sam;:;! sT^.y^lZZT ^^^^ ^^
Organisation of Data: Meaning and types of variables; Frequency Distribution.
Unit 3: Statistical Tools and Interpretation
interpretation for the rite d^ived) P'd'
deviat.o„,_Lore„z curvLMeaJ^aX ^^^ 
Karl Pearson's method
Oirrelationmeaning, scatter diagramMeasures of correlation(two variables ungrouped data), Spearman's rank correlation
"n^ inL'"on typeswholesale price index, consumer price
"umLrs production, uses of index numbers; inflation and iLx
Unit 4: Developing Projects in Economics in n • j
Tu . J ■ Periods
latlo^ both TaL tT^rV" data, secondary
of L eimnl. f I ' organisations outlets may also be encouraged. Some
of the examples of the projects are as follows (they are not mandatory but suggestivT) (t) A report on demographic amongst households; suggestive).
(«) Consumer awareness amongst households
(»/) Changing prices of a few vegetables in your market
(w) Study of a cooperative institution—milk cooperatives
^eJ^t to enable the students to develop the ways and
u 1
CONTENTS
UNIT 1 : Introduction
1. What is Economics
2. Introduction—Meaning and Scope .
UNIT 2 : Collection and Organisation of Data
3. Collection of Primary and Secondary Data
4. Organisation of Data
Presentation of Data
5. Tabular Presentation
6. Diagrammatic Presentation
7. Graphic Presentation
UNIT 3 : Statistical Tools and Interpretation
8. Measures of Central Tendency
9. Positional Average and Partition Values
10. Measures of Dispersion
11. Measures of Correlation
12. Introduction to Index Numbers
UNIT 4 : Developing Projects in Economics
13. Preparation of a Project Report
1 10
22 52
76 87 108
138 178 232 313 354
394
WHAT IS STATISTICS?
S : T : A : T : I : S : T : I : C : S :
Scientific Methodology
Theory of Figures
Aggregate of Facts
Tables and Calculation for Analysis
Investigation
Systematic Collection
Tabulation and Organisation
Interpretation
Comparison
Systematic Presentation
tc fc ea dc in lif wl his
in (
UNIT 1 —
HmODUCTIOr^
m Whui is ficonoinifs? ■ Jlieaiiiiig, Scope and Imporianct^ of l§)tatislics in Economms
■ ■
Chapter 1
what is economics?
1 Introduction
2. Activity
3. Definition of Economics
4. Nature of Economics
Economics as a Science (i^ Economics as an Art
introduction
If each of us possessed 'Aladdin's magic lamp, which we had merely to rub in order to get our desires fulfilled immediately, there would be no economic problem and no need for a science of economics. In real life we are not lucky as Aladdin, we have to work to earn our livelihood. All people in this world work to satisfy their unlimited wants and desires. Every one requires food to eat, clothes to wear and house to live in. Besides these in daily life. They need television, mobile phone, motor bike, car etc., to lead a comfortable life. The person visits the market and enquires about the varieties and prices of the item which he wants to purchase. Thinking about his source and alternative choices, he uses his sense of economy and decides to buy that item. This is economics.
So,
A customer is a person who buys goods to satisfy his wants.
A Producer is a person who produces or manufactures goods.
A Service holder is a person who is in a job to earn either wages or salary to buy goods.
A Service provider is a person who provides services to society to earn money, e g doctors, scooter drivers, lawyers, bankers, transporters, etc.
All above persons are busy in different activities to earn, called economic activity in ordinary business of life. They face in their life the problem of scarcity of income
z
Statistics for EconomicsXI »purcha.
Thus,
and^J^Ttu' "" ^r^'^ff'^^l^dge with economic activities relating to earning ar^ spending the wealth and tncome. Economics is the study of how human beinJZa^
tZ^ tn":^' T""" unlimited wLts in sZZ Z^LTZ
^omm maxnntse thetr satisfaction, producers can maximise their profits and society can maxtmtse its social welfare'. ^^
infn^W mT publication of Adam Smith's "An Inquiry
into the Nature and Causes of Wealth of Nations", in the year 1776 At its Mxth Z
name of economics was 'PoHtical Economy'. Some of the suggested names '
— Catallactics or the science of exchange.
— Plutology or the science of wealth.
— Chrematistics or the science of moneymaking
~ "y't to^om s''.'^'  ^ '^o'^tical
(HoItVoTr E'^gli^h has its origin into two Greek words : Oikos
(Household) and nomos (to manage). Thus, the word economics was used to mean home management with limited funds available in the most possible economLal mTn2
activity
Iife.?herlf " ^ day
1. Noneconomic Activities
2. Economic Activities
or Activities. These activities are those which have no economic aspect
or are not concerned with money or wealth, viz.
~ bkll^S '' 8«together, attending
Dirtnday parties or marriages etc. o , &
" wSrCo?'''!'''^'"'^ '' ^"^"dwara, mosque or church to
worship God, attending mass prayer (Satsang), etc.
 Political activities such as various activities performed by different political parties namely by Bhartiya Janata Party (BJP), Congress Party etc
~ ^'ds or helping
 Parental activities, such as love and affection towards their children
—^ ^l^ey involve any
nf Activities. Different types of activities are performed bV different types
of people (doctors, teachers, businessmen, industrialists, lawyers etc.) so as taelT^
What is Economics? 3
living. Every one is concerned with one or the other type of activity to earn money or wealth to meet their wants. An economic activity means that activity which is based on or related to the use of scarce resources for the satisfaction of human wants. Economic activities are classified as under :
the
ECONOMIC ACTIVITIES
i
nt
>ution
kos ome
I day
ispect nding to lies ilping
any
[types I their
Production : Production is that economic activity which is concerned with increasing the utility or value of goods and services. Manufacturing shirt with the help of cloth (raw material) and tailoring (labour) etc. is an act of production. Transporting sand from river bank to a town, where it is needed, is also an act of production. Here utility is created through transportation of goods to the person who needs it.
Consumption : Consumption is that economic activity which is concerned with the use of goods and services for the direct satisfaction of individual and collective wants. Consumption activity is the base of all production activities. There would have been no production if there would have been no consumption. For example, eating bread, drinking water or milk, wearing shirt, services of lawyer or doctor etc. are consumption activities.
Investment : Investment is that economic activity which is concerned with production of capital goods for further production of goods and services. Investment indirectly satisfies human wants. For example, the production of printing press machines to print newspapers, books, magazines etc. or investment in computers to provide Internet, banking and related services.
Exchange : Exchange is that economic activity which is concerned with sale and purchase of commodities. This buying and seUing is mostly done in terms of money or price. So, it is also called ""Product Pricing'' which relates to determination of the price of the product under different conditions of the market, viz., perfect competition, imperfect competition, monopoly etc.
Distribution : Distribution is that economic activity which deals with determination of price of factors of production (land, labour, capital and enterprise). This is known as the 'Factor Pricing', e.g., price of land is rent, that of labour is wage, that of capital is interest and price of entrepreneur is profit. Distribution is the study to know how the national income or total income arising from what has been produced in the country (called Gross Domestic Product or GDP) is distributed through salaries, wages, profits and interest.
"Economics is that branch of knowledge that studies consumption, production, exchange and distribution of wealth".
—Chapman
10 Statistics for EconomicsXI
definition of economics
Economics has been defined by many economists m different ways The set of cat"go^s^'"'" '' ^^mto the folwL; W
1. Wealth definition—Adam Smith
2. Material welfare definition—Alfred Marshall
3. Scarcity definition—Lionel Robbins
4. GroAvth definition—Paul A. Samuelson
1. Wealth Definition
(/) Adam Sm^th the father of modem economics, m his book 'An Inquiry mto the Nature and Causes of Wealth of Nations' in 1976 defined that
production and expansion of wealth as the subject matter of
(«) According to J.B. Say, Economics as "the science which deals wtth wealth"
ofTaTth! " ^^^ consumption
{in) Ricardo shifted the emphasis from production of wealth to distribution of wealth
Criticism : This definition is not a precise definition. It gives importance to wealth rather than production of human and social welfare. importance to wealth
The wealth definition of economics was discarded towards the end of the 19th century. 2. Material Welfare Definition
"Economics is a sUuiy of mankind i„ the ordinary business of life it examines that
What is Economics?
5
Criticism :
{a) In economics, we study immaterial things also.
(b) Welfare cannot be measured in terms of money.
(c) Welfare definition makes economics a purely social science.
II h "Tf " " "  d^ffnt times.
Then basic difference between Adam Smith's and Marshall's definition is that Ad.m
riti* sr. ™ —hiri^
3. Scarcity Definition
There are three important aspects in this definition. They are •
Icf of Tir m" human wants which is the
tact of Me. When one warn gets satisfied, another want crops up.
fr" '' are scarce in relation
coal IS used m factories, m running railways and in thermal stations for electric generation and by households, etc. electric
In short, according to Robbins, Economics is a science of choice It deals with how Crit^sm : Sfet™s
Scarcity definition of economics has been criticised on the following grounds ■ (0 The defimtion is impractical and difficult. It is narrow and restricted in scope It
^ development. It has notS^^^
Hi) The definition makes economics a human science.
4. Growth Definition
Paul A. Samuelson defines—
to p^u^ vanous commod,Ues overtime and distribute them for consumptionZ^ or
12 Statistics for EconomicsXI
The definition combines the essential elements of the definitions by Marshall and Robbms. Accordmgly, economics is concerned with the efficient allocation and use of scarce means as a result of which economic growth is increased and social welfare is promoted. The definition has been accepted universally. In short, the growth definition of economics is most comprehensive of all the earUer definitions.
iture of economics ^
Nature of economics—as a science or art. It is science and art as well.
NATURE OF ECONOMICS
ics as Art
A. ECONOMICS AS A SCIENCE
Science can be divided into : (a) Natural science, and (b) Social science : Sciences like Physics Biology and Chemistry are natural or physical sciences, where experiments can be conducted in the laboratory under controlled conditions. Relationships can be decided between cause and effect, which are based on facts. Observations can be made and used to prove or disprove theories. The results apply universally.
Economics is a social science because it is systematic study of economic activities of human beings. Economics is a science as it is a branch of knowledge where various facts have been systematically collected, classified and analysed.
The following arguments are given in favour of economics as a science.
(/) Systematised Study : The study of economics is systematically divided into
consumption, production, exchange and distribution of wealth and finance which
have their own laws and theories. Economics as social science which is a systematic
study of human behaviour concentrating on maximum satisfaction to households
maximum profit to producers and maximum social welfare to the society as a whole. ^
Hi) Scientific Laws : Economics is a science because its laws are universally true Different laws m economics namely, law of demand, law of supply, law of dimimshing marginal utility, law of returns,
Gresham's law etc. are applicable to all types of economies, whether capitaKstic, socialistic or mixed economy
y
ae. of ; to
What is Economics? y
(m) Cause and Effect Relationship : Economic laws establish cause and effect relationship like the laws in other sciences. For example, the law of demand shows the relationship between change in price and change in demand..It shows that mcrease in price of a commodity (the cause) will decrease its demand (the effect) establishing the negative or inverse relationship between price and quantity demanded. The law of supply shows that the increase in price of a commodity (cause) will increase its supply (the effect) establishing the positive relationship between price and supply of quantity of commodity.
(iv) Verification of Laws : Like other sciences economic laws are also open to verification. These economic laws can be verified through any empirical investigation.
On the basis of the arguments given above, we can say economics is a science—not exacriy natural or physical science but social science that studies economic problems and policies in a scientific manner.
Economics—A Positive or Normative Science
(a) Economics as a Positive Science
A positive science is one which makes a real description of an activity. It only answers what ts} what was! It has nothing to suggest about facts, positive economics deals with what IS or how the economic problems facing a society are actually solved. Prof. Robbins held that economics was purely a positive science. According to him, economics should be neutral or silent between ends; /.e., there should be no desire to learn about ethics of economic decisions. Thus, in positive economics we study human decisions as facts which can be verified with actual data.
Some exampi es of Economics as a positive science are : {i) India is second largest populated country of the world. (k) Prices have been rising in India.
(m) Increase in real per capita income increases the standard of living of people. (iv) The targeted growth rate of the tenth fiveyear plan is 8 per cent per annum. {v) Fall in the price of commodity leads to rise in its quantity demanded.
(vi) Minimum wage law increases unemployment.
(vii) The share of the primary sectors in the national income of India has been declining.
{viii) Ordinary business of life is affected enormously by tsunami, earthquakes, the bird flue, droughts, etc.
(b) Economics as a Normative Science
A normative science is that science which refers to what ought to be} what ought to have happened} Normative economics deals with what ought to be or how the economic problems should be solved. Alfred Marshall and Pigou have considered the normative aspect of economics, as it prescribes that cause of action which is desirable and necessary to achieve social goals. It makes an assessment of an activity and offers suggestions for that. The statements which make assessment of activity and offer suggestions are called
14 Statistics for EconomicsXI
normative statements. The normative statements, in fact, are the opinions of different
persons relating to tightness or wrongness of a particular thing or policy. Normative
statements cannot be empirically verified. That part of economics which deals with
normative statements is called Normative Economics. Thus, economics is both positive and normative science.F^smvc
Some examples of Economics as a normative science are :
(/) Minimum wages should be guaranteed by the government in all economic activities. (//) India should not take loans from foreign countries. [Hi) Rich people should be taxed more. [iv) Free education should be given to the poors.
{v) Effective steps should be taken to reduce incomeinequalities in India. (vi) India should spend more money on defence. {vii) Government should stop minimum support price to the farmers.
(vtii) Our education system should produce sufficient qualified and trained persons to the economy.
Economics as positive science and normative science is inseparable. In reality economics has developed along, both positive and normative lines. The role of economist is not only to explain and explore as positive aspect but also to admire and condemn as negative aspect which is essential for healthy and rapid growth of economy.
In the followii^ examples first part of statement is positive giving facts and second i part IS normative based on value judgements.
H) Indian economy is a developing economy, the government should make development through correct and proper planning.
(ii) A rise in the price of a commodity leads to a fall in demand of quantity of commodity, therefore government should check rise in prices.
(iti) Rent Control Act provides accommodation to the needy peoples, therefore, the act should be honestly implemented.
B. ECONOMICS AS AN ART
Art IS practical application of knowledge for achieving some definite aim. It helps in solution of practical problems Art is the practical application of scientific principles. Sc ence lays down princip es while art puts these principles into practice. Economics is an art as it gives us practical guidance in solution to various economic problems. '
We all know that there is oil shortage in India. The information given by economics .sposmve sconce We also know the govermnent aims at removing' oil shortage X information supplied by economics is normative science. In order to achieve the objective of full availability of oil m India, the govermnent has followed the path of oil plaLng The path of planmng is an art as it implies practical application of knowledge with a view to achieve some specific objectives. So, we can say that economics is an art. Economics is, thus, a science as well as an art.
What is Economics? y
exercises
i Explain the origin of word 'Economics', i What is economic activity?
Distinguish between noneconomic and economic activities.
Make a Ust of economic activities that constitute the ordinary business of life.
What are your reasons for studying Economics?
How will you choose the wants to be satisfied?
Give Adam Smith's definition of economics.
Define economics in the words of Alfred Marshall.
"Economics is the science of choice." Explain.
Which is the most accepted definition of economics? Give the definition. Explain welfare definition of economics.
"Economics is about making choices in the presence of scarcity." Explain. How scarcity and choice go together? What is meant by economics? Economics is a science? Give reasons.
Discuss the nature of economics as a science. Give argimient in favour of economics as a science.
Is economics an art? Give reasons.
Is Economics a science or an art? Explain with reasons.
Is economics a positive science or a normative science or both? Explain.
lin
3.
4.
5.
6.
Chapter 2
introductionmeaning and scope
Introduction What is Statistics? Functions of Statistics Importance of Statistics Limitations of Statistics Misuse of Statistics
i introduction
e—, plannW and X ^^^^ ^^ ^
progJsst tcTs   developments and
chemistry, medicine, technology etc) neLTnWr' ^P^^^^^es,
new machines have been devdoS'tLt I f ff ' ^^"^'^es of energy,
because manwhether Indtn T? ^f^^ble. All this is possible
thinking and reasoning which had evo^ '""I' ^^ ^^"^^^eis gifted with
given us civiIisation4he wtef f ^^ has
electricity, machinery etc. We ntw ^^ ^^e irrigation system,
systems, better organisations for the comX bul^^^^^^^
All thic h^c , ""'npiex Dusiness and administration today
wayfoftalSmnrp^^^^^^^ ^PP^'ed itself in'findmg
and scientific man'ner. A meZdo^X b'n ^ things. The empirical methodolog^ consists Jf la^^^^
mformation, analysing the information Th^ ^ observations and collecting
conclusions by fu^er^bserta^TsZs W al ZtZ' ^^^^ ^
knows it or not, he uses this method to f'""'.^'e made Whether a common man
buying vegetabks he looks t dS^lr^u^^^^^^^^^^ decisionmakmg. While
and then mentally calculates, or ^rks out wLT. P."''' ^^ops
observes from his daily ex^elre whT'> ^^^ ^hich shop. A shopkeeper
decides to stock these  demand, a'nd
the pattern of demand and manufactures larle o^T "^^^ufacturer also observes
or manufactures new items acco^SnSr^rthTZL^ ^^^ demand
radio, the television etc., it is possiWe trcXr T f "^^^^"^ediathe newspaper, the
Introduction—Meaning and Scope
\ 11
demand and supply, he collects data (information) systematically, gets it organised in some logical or systematic way, analyses this data according to certain principles and draws conclusions. He has to do it carefully since a wrong judgement can completely ruin him.
Quantitative Data and Qualitative Data : An empirical investigation is an investigation where facts are collected through observation. In Physics, Chemistry and Botany, only those things that can be observed by our senses—seeing, hearing, touching, tasting and smelling—are taken to be reliable and then recorded (noted).
We all agree that the rose is beautiful. How do we reach that conclusion? We all like its colour, shape and above all its smell. In this respect, it is not a subjective or personal conclusion. But I say that I like the rose most of all the flowers, this would be a subjective statement. A scientist however, makes very precise statement—he would say that roses have a sweet smell. Similarly, people might say that theft and robbery have increased these days. This might be a conclusion based on impression people get from the newspaper reports of cases of theft and robbery. This impression may or may not be true. We can find out whether it is true or not only by comparing the number of cases of theft and robbery reported during one year with the number of cases reported in other years. An investigator would collect such information from police records.
When information or observations are recorded in numbers or quantity, we say we have quantified information. For example, the number of people in a state who are strict vegetarians, heights or weights of students, everyday temperature, income of individuals, prices of wheat during this week, number of people in country are really poorrichmiddle class, number of people are illiterate who will not get jobs, number of highly educated and will have best job opportunities, etc. are known as 'Quantitative data'.
However, not all information can be numerically expressed. It is not possible in certain cases to measure or quantify information, e.g., preference of people viewing TV. channels, intelligence of students, appreciation of art, beauty, music etc. Supposing a selection for a post is to be made, candidates are interviewed, some questions are put to them and their qualifications are taken into consideration. The
interview board discusses the comparative merit of the candidates and ranks them for final selection. This judgement is not quantifiable, it is based on impression.
Nonquantifiable/qualitative items can however be measured in percentages. For example, percentage of people watching TV. news in English or Hindi or other regional languages. This information obtained in percentages is called 'Qualitative data'. It may be collected through questionnaire or opinion poll using landline or mobile telephone, internet or newspapers.
Social sciences, such as economics, sociology, management etc., do not always deal with what we call inherently measurable or quantifiable facts.
is smistics ? >
It is necessary to have quantitative measurements even for things which are not basically quantifiable. This is necessary for preciseness of statement. The systematic
h
12
vw 11 ^^'^tistics for EconomicsXl
treatment of quantitative expression is known as 'Statistics'. Not all quantitative expressions are statistics; we will see that certain conditions must be fulfilled for a quantitative statement to be called statistics. We will also consider later the functions and hmitations of statistics. First, let us understand what comes under the name Statistics. Statistics can be defined in two ways :
(a) In plural sense.
(b) In singular sense.
i^nn'W^nir"* 'consider whether figures
1600, 400, 80, 20, 700, 300, 70 and 30 are Statistics? Figures are innocent and do not
speak anythmg. But when they refer to some place, person, time etc., they are called statistics. Let us look at the table given below :
Students in Two Schools (20052006)
Kendria Vidyalaya Govt. Senior Secondary School
Students Number Percentage Number Percentage
Boys Girls 1600 400 80 20 700 30070 30
Total 2000 100 1000 100
The above table gives a numerical description of students in Kendria Vidyalaya and
Govt Senior Secondary School. Students are grouped as boys and girls and percentage is st^
calculated for each group. Now, in this context the figures 1600, 400, 700 etc have a ' of
statistical meaning; we call this statistics of students. Similarly, we find in newspapers ^
statistics of scores in a cricket match, statistics of price, statistics of agricultural production, i sin statistics of export and import etc. j ^ "
martT^^'^Vl Statistics we mean aggregates of facts affected to ^ ^
marked extent by muhtphctty of causes numerically expressed, enumerated or estimated according to reasonable standards of accuracy, collected in a systematic manner for a predetermmed purpose and placed in relation to each other."
The above definition covers the following main points about statistics as numerical presentation of facts (Plural sense).
Statistics are aggregates of facts : A single observation is not statistics, it is a group of observations, e.g., "pocket expenses of Anil during a month is Rs 50" is not statistics. But "pocket expenses of Anil, Prakash, Sunil and Suresh during a month are Rs 50, 55, 80 and 70 respectively" are statistics.
{b) Statistics are affected to a marked extent by multiplicity of causes : Statistics are generally not isolated facts they are dependant on, or influenced by a number of phenomena, e.g., electricity bills are affected by consumption and rate of electricity (c) Statistics are numencally expressed : Qualitative statements are not statistics unless A they are supported by numbers. For example, if we say that the students of a class colle.
Introduction—Meaning and Scope
13
are very good in studies, it is not a statistical statement. But when a statement reads as 40 students got first division, 30 second division, 20 third division and 10 failed out of 100 students, it is a statistical statement expressed numerically.
(d) Statistics are enumerated or estimated according to reasonable standard of accuracy: Enumeration means a precise and accurate numerical statement. But sometimes, where the area of statistical enquiry is large, accurate enumeration may not be possible. In such cases, experts make estimations on the basis of whatever data is available. The degree of accuracy of estimates depends on the nature of enquiry.
(e) Statistics are collected in a systematic manner : Statistics collected without any order and system are unreliable and inaccurate. They must be collected in a systematic manner.
if) Statistics are collected for a predetermined purpose : Unless statistics are collected for a specific purpose they would be more or less useless. For example, if we want to collect statistics of agricultural production, we must decide before hand the regions, commodities and periods for which they are required.
(g) Statistics are placed in relation to each other : Statistical data J»re often required for comparisons. Therefore, they should be comparable periodwii>c, regionwise, commoditywise etc.
When the above characteristics are not present numerical data cannot be called statistics. Thus, "all statistics are numerical statements of facts bui all numerical statements of facts are not statistics."
Statistics defined in singular sense (as a statistical method) : Statistics in its second, singular sense, refers to the methods adopted for scientific empirical studies. Whenever a large amount of numerical data are collected, there arises a need to organise, present, analyse and interpret them. Statistical methods deal with these stages :
PRESENTATION
I Interpretation
Statistics as Methodology
According to Croxton and Cowden, "Statistics may be defined as a science of collection, presentation, analysis and interpretation of numerical data.'"
It
Statistics for EconomicsXI The above definition covers the following statistical tools :
(a) Collection of data : This is the first step in a statistical study and is the foundation of statistical analysis. Therefore, data should be gathered with maximum care by the investigator himself or obtained from reliable pubHshed or unpublished sources
(b) Organisation of data : Figures that are collected by an investigator need to be organised by editing, classifying and tabulating.
(c) Presentation of data : Data collected and organised are presented in some systematic manner to make statistical analysis easien The organised data can be presented : with the help of tables, graphs, diagrams etc.
(d) Analysis of data : The next stage is the analysis of the presented data. There are
large number of methods used for analy sing the data such as averages, dispersion correlation etc.'
(e) Interpretation of data : Interpretation of data implies the drawing of conclusions
on the basis of the data analysed in the earlier stage. On the basis of this conclusion certain decisions can be taken.
Stages of Statistical Study
According to the figure, interpretation of data is the last stage in order to draw some conclusion. One has to go through the four stages to arrive at the final stage; they are — collection, organisation, presentation and analysis. First stage — collection of data refers to gather some statistical facts by different methods. The second stage is to organise the data so that collected information is easily intelligible. This is the arrangement of data in a systematic order after editing. Third stage of statistical study is presentation
of data After collection and organisation the data are to be reproduced by various
niethods of presentation, namely tables, graphs, diagrams, etc. so that different
characteristics of data can easily be understood on the basis of their quality and uniformity.
Fourth stage of statistical study is the analysis of data. Calculation of a value by different
methods and tools for various purposes is made to arrive at the last stage of study viz interpretation of data. ^
In brief statistics is a method of taking decisions on the basis of numerical data properly collected, organised, presented, analysed and interpreted.
in i scie drai relai
( (
functions of statistics
Following are the functions of statistics :
1. Statistics simplifies complex data : With the help of statistical methods a mass of data can be presented in such a manner that they become easy to understand. For example, the complex data may be presented as totals, averages, percentages etc
Stati
I
resea Mars quan;
of stj
Introduction—Meaning and Scope \ 5
2. Statistics presents the facts in a definite form : This definiteness is achieved by stating conclusions in a numerical or quantitative form.
3. Statistics provides a technique of comparison : Comparison is an important function of statistics. For example, comparison of data of different regions; periods, conditions etc., is helpful for drawing economic conclusions. Some of the statistical tools like averages, ratios, percentages etc., are used for comparison.
4. Statistics studies relationship : Correlation analysis is used to discover functional relationship between different phenomena, for example, relationship between supply and demand, relationship between sugarcane prices and sugar, relationship between advertisement and sale. Statistics help in finding the association between two or more attributes, for example, association between literacy and unemployment, association between innoculation and infection etc.
5. Statistics helps in formulating policies : Many policies such as that of import, export, wages, production etc., are formed on the basis of statistics. Some laws such as Malthus' theorj^ of population and Engel's law of family expenditure are based on statistics.
6. Statistics helps in forecasting : Statistics also helps to predict the future behaviour of phenomena such as market situation for the future is predict^).' on the basis of available statistics of past and present. Economist might be interested in predicting the changes in one economic factor due to the changes ir another factor. For example, he might be interested to know the impact of today's investment on the national income in future which is possible with the knowledge of statistics.
7. Statistics helps to test and formulate theories : When some theory is to be tested, statistical data and techniques are useful. For example, whether cigarette smoking causes cancer; whether demand increase affects the price, can be tested by collecting and comparing the relevant data.
importance of statistics
The use of statistical method is so widespread that it has become a very important tool in affairs of the world. It IS indispensable to fields of investigations especially in the sciences, such as Botany, Sociology, Economics, Medicine etc. It helps particularly in drawing research conclusions. Let us examine the importance of statistics in some fields relating to economics and business :
{a) Statistics and Economics (b) Statistics and Economic Planning
(c) Statistics and Business (d) Statistics and Government
Statistics and Economics
A number of economists have given a practical shape to statistical tools for economic research. Famous economists (like Augustin, Cournot, Vilfredo Pareto, Leon Walras, Alfred Marshall, Edgeworth, A.L. Bowley etc.) evolved a number of economic laws by
quantitative and mathematical studies^ In India, Prof. P.C. Mahalanobis, Dr V.K R V Rao R.C. Dcsai, ctc. iiave cgntrmuieu aTOtWtne aeveiopmeiu ui lucuiclk^t'
of statistics.
! H
If
16
Statistics for EconomicsXI
...ril—T tools and the importance of
ZnomV: rr' f" "T™" ^^^^^  ^ relanonLp amo^
Tecorm^rA ? of mathematics andLtistic!
m economics. As a result, a new science has evolved which is called Econometrics
the Z" ^r' have evolved due to statistical analysis in
the f eld of economics, Engel's law of family expenditure, Malthus theory of
population etc. New things are being invented today in all the sciences becauTo7the
econoT^b^""" • T ^^^^ ^ new laws in
T^ Statistical methods have made a contribution to the
development of empirical side of economics; the inductive method of economics is dependent upon statistical methods. economics is
TT'' are the tools and
appliances of his laboratory, m the same way as the doctor uses stethoscope for diagnosis
of a patient. A number of economic problems can easily be understood by the useTf
tatistical tools. It helps m formulation of economic policies.^ Let us unLstand the
importance of statistics keeping in view the various parts of economics
oTrh!" consumption : Every individual needs a certain number
7 K' necessities, then on comforts and luxuries, which
depend on his income; but there is no end to his desires and demands. No sooner does he consume one thing, he desires to obtain the other. We discover how
staS^" T"^' consumption. The
Td/vT r T rt^ ^^e taxaWe liability of
individuals and their standard of living
(b) Statistics and the study of production : The progress of production every year can
easily be measured by statistics. The comparative study of the prod^Lty of
various elements of production (e.g., land, labour, capital and enterprise) is also
done with the help of statistics. The statistics of production are ver^ helpful or
ad^stment of demand and supply. Every developed country executes L census of
production with a view to make a comparative study of various fields of production and economic planning. ^ wuucuou
fn^ttirf = ^^  ^onal and
international demand. A producer needs statistics for deciding the cost of .
P^r ^^^^ ^^ competition afd demand o
^mmodity m a market. The law of price determination and cost price which are :
(d) Statistics and the study of distribution : Statistics are helpful in calculation of national income in the field of distribution. Statistical methods are used in solving the problem of the dismbution of national income. Various problems arise di^I to
o7lttfc7datr " ^^ ^ with"hrhelp
et/eo other econormst, have to make bricks." Marshall
Introduction—Meaning and Scope 1 ^
Thus, statistics is useful in the various fields of economics. It gives statement of facts, direction to solve problems, evolution of economic laws and helps in economic planning. Economic laws in the modern economic world are based on mathematics and statistics which help to form econometric models; these models are helpful in solving economic problems. On this basis we can call economics a Science of Human welfare and statistics as an Arithmetic of Human welfare.
Statistics and Economic Planning
Statistics is the most important tool in economic planning. Economic planning is the best use of national resources, both human and natural. Planning without statistics is a leap in the dark. Every phase in planning—drawing a plan, execution and review is based on statistics. The success of a plan is dependent upon sufficient and accurate statistical data available at all these stages.
The comparison of the stage of development of one country with other is possible only with the availability of statistical data. There are a number of problems of underdeveloped countries, e.g., over population, lack of industries, lack of agricukural development, lack of education etc. These problems can be fully viewed and understood only by getting the actual figures for different areas. Similarly, general review of progress in all fields of economic development needs the help of statistical data and statistical methods. Priorities of expenditure of a national budget can be determined through the comparative study of past performances with the present. Thus, planning without statistics is a ship without radar and compass.
Statistics and Business Planning
Business activities can be classified as under :
f"
L
BUSINESS
1
I
Internal Wholesale Retail
International
Import
Export
i of Trade
^11 t V
Banking Transport Insurance Wterehousing Packmg
Advertisement
Mf
18
vw 11 ^^'^tistics for EconomicsXl
of statmical method have to be followed The it steps
the producer to f„ the pnces of ^o^rd""'
method, of .ta„st,ea.
profttable trade he must know what the ZoZ, '' celling activities. For
would last. This is very .mportant or demand
e„po„ for var.o„s co.mod,t.es and at a^d
he has to forecast when the dLaXodd Ll": °nf "Tf
of reserves he must have. Similarly he ^st t ^We what amounts
h.s deposttors, otherwise his bak wol fT^oT rh
transaction ,s required where statistical toTls tt iXb^^ i
hfe e^rrtt^r that is !
what proportion of the. capS decde' i
payments of matured policies ^ Proportion kept ready for
n>.ght fix for the same. In fact no mod^Xanirr °° they
without analysis of the complex faJ^rthrLr"" ^
business analysis statistical tools are afeolt; Tfential
analys^p tz: p":^lt:tsere 717 ^ ^^ ^
Statistical tools of collection, classification L uncertainties,
data are used in all ma,or functioTstfinterpretation of
material control, budgetary control, fmanc aTclro^f T"'' "
^nd so on. ' control, cost control, personnel management
Statistics and Government
Introduction—Meaning and Scope \ 9
like that of crimes, taxes, wealth, trade etc., so that there is no obstacle for the promotion of economic development. Such policies may develop the economic status of the people and the nation, depending upon the accuracy of the statistical law. Statistics is indispensable for all important functions of the ministries of the state. Above all the ministry of planning takes into account statistics of various fields of economy. Keeping in view the increase of global price rise, the ministry plans and makes policy to import oil in 2010, which depends on expected oil production by domestic sources and likely demand for oil for the year 2010. The policy of family planning can be made effective in controlling the population of country. Thus, statistical techniques are used to analyse economic problems of country, viz., unemployment, poverty disinvestment, price control, etc. Sometimes to make plans and policies, planners require the knowledge of future trend. This trend could be based on the data of past years or recent years. The required data can be obtained by surveys. For example, production poHcy of 2010 depends on the consumption recorded in past years and recent years which decides the expected level of consumption in 2010. This helps the planners to make the production policy for the future. The Ministry of Finance is responsible for preparing the annual budget of the country for which reliable statistical data of revenue and expenditure is necessary. In short, statistical tools are of maximum utility in the governance of state and formulation of various economic policies.
f'i
umitations of statistics
Statistics is very widely used in all sciences but it is not without limitations. It is necessary to know the misuses and limitations of statistics. The following are the limitations of statistics.
1. It does not study the qualitative aspect of a problem : The most important condition of statistical study is that the subject of investigation and inquiry should be capable of being quantitatively measured. QuaHtative phenomena, e.g., honesty, intelligence, poverty, etc., cannot be studied in statistics unless these attributes are expressed in terms of numerals.
2. It does not study individuals : Statistics is the study of mass data and deals with aggregates of facts which are ultimately reduced to a single value for analysis. Individual values of the observation have no specific importance. For example, the income of a family is, say Rs 1,000, does not convey statistical meaning while the average income of 100 families say Rs 400, is a statistical statement.
3. Statistical laws are true only on an average : Laws of statistics are not universally applicable Hke the laws of chemistry, physics and mathematics. They are true on an average because the results are affected by a large number of causes. The ultimate results obtained by statistical analysis are true under certain circumstances only.
4. Statistics can be misused : Statistics is liable to be misused. The results obtained can be manipulated according to one's own interests and such manipulated results can mislead the community.
20
h
(Si.
^ • , . ^t^tistics for EconomicsXI
observations of mass data nXkrT L ^ased on
to rectify them. Therefor^ it^eslnreT" " ...ca, states are a fa/.re m ^
misuse of statistics
statistics by deliberately twisting or man pulat^ne^ ^^ ^e
be mterpreted by a lawyer to prove ^^ZTltTl^^;' " " ^^^ ^^^
tools for grinding their owTa^sf b^
student of commerce and economic , hou^ 1 the
This generally takes place at the time of sefectinZm^^^^^ a
and interpreting analysis of data ^ "^^^ing comparisons
and^^eSpJT~  support knowledge of statistics, the truth with the help of his
exercises
1 D.stm8„,sh between qualitatrve and quantitative data
3 Dto Tr characteristics?
4 M rr and plural sense
 above
obrva'Sn'" cr™"'" d never with a single
llT"" '' counting." Discuss.
■
.............
4.
5.
6.
7.
8.
Introduction—Meaning and Scope
21
16.
17.
18.
19.
20,
Discuss with illustration the importance of Statistics in the solution of social and economic problems.
"Statistical Analysis is of vital importance for successful businessmen, economists, administrators and educationists." Discuss with illustrations.
Write notes on : (a) Importance of statistics in modern economic set up, {b) Statistics in economic analysis.
Define Statistics. Explain its utiHty in the field of economic planning. "Statistical thinking is as necessary for efficient citizenship as the ability to read and write." Explain this statement in about 200 words.
"Statistics in these days is indispensable for dealing with socioeconomic problems". How far is this statement true?
What is the importance of Statistics in modern economic set up? Explain giving
examples. . i u
"Planning without Statistics is a ship without radar and compass." In the light ot this statement explain the importance of Statistics as an effective aid to national planning.
Explain the relationship between Economics and Statistics and discuss how far it is correct to say that the science of economics is becoming statistical in its method.
Explain briefly :
(a) Statistics, (b) Statistical methods,
(c) Statistical data, (d) Statistics in economic analysis.
Statistical methods are no substitute for common sense, comment. "The Government and poHcy maker use statistical data to formulate suitable policies of economic development". Illustrate with two examples. Mark the following statements as true or false. J (i) Statistics is of no use to economics without data. (ii) Statistics can only deal with quantitative data. (Hi) Statistics solves Economic problems.
UNIT 2
}1:
If
K
^lecton and organisation of data 12
« CoUecAion of Primary and S p Organisation of Uata « Presentation of Data
Chapter 3
collection of primary and secondary data
What is a Statistical Enquiry? Sources of Data g Primary and Secondary Data Drafting the Questionnaire Methods of Collecting Primary Data Census and Sample Surveys Sample Surveys Methods of Sampling Random Sampling NonRandom Sampling Advantages of Sampling Reliability of Sample Data How Secondary Data is Collected? Some Important Sources of Secondary Data Census of India
National Sample Survey Organisation (NSSO)
COI
cer are coIJ
Sourct
w
other, statisti
sXI Collection of Primary and Secondary Data
what is a statistical enquiry ?
Enquiry means a search for truth, knowledge or information. Statistical enquiry therefore means a search conducted by statistical methods. There are different subjects on this earth; some are described by the degree of expression (quality) and some by the degree of figures or magnitudes (quantity). The application of a statistical technique is possible when the questions are answerable in figures (quantity),
in other words the first and the foremost condition for the answer to the questions in statistical enquiry should be quantitative, for instance :
Profit of firms measured in rupees;
Income of families measured in rupees;
Weight of students measured in kg;
Age of students measured in years;
Intelligence measured in marks obtained by students in a particular test.
But, there are questions like—How great was Jawaharlal Nehru? How brave was Bhagat Singh? etc., which cannot be answered through statistical methods. Questions that can be answered in quantity lies within the purview of statistics, viz.. What is the average production of rice per acre in India? What is the total population of India? How many students are there in a class?
Thus, statistical enquiry means statistical investigation or statistical survey, one who conducts this type of enquiry is called an investigator. The investigator needs the help of certain persons to collect information, they are known as enumerators, and respondents are those from whom the statistical information is collected. Survey is a method of collecting information from individuals.
Let us observe the following table.
TABLE 1
Production of Finished Steel in India (in Million Tons)
fc^ yearProduction
195051 1.0
198081 6.8
199091 13.5
200001 30.3
200102 31.1
200203 34.5
200304 36.9
Source : Government of India, Economic Survey 200405.
We observe that the production of finished steel in India, is different from one year to other. They are not same. They varies from year to year. They are called Variables in statistics which is represented as X, Y or Z variables. The finished steel production in
ti i.
24 ,
Statistics for EconomicsXI
V V tu f by xvariable and the production of finished
From the followmg text, we will understand :
1. What is the source of data?
2. How do we collect data?
3. By which method of survey is data collected
. sources of data
BefoT^n' Z" ^^ '^^ta. This is the first stage in statistics!
SOURCES OF DATA
dary
^ Governmern departments RaiZr^cf
' ^^ ^reparmg^at •
cantr^ine??""'"'" external data whij
^^ary and secondary '
Collection of Primary and Secondary Data 25
But, you may have the other choice that of visiting the factory accounts department, and record the information from the salary register or, may gather this information from the published report of the factory about the payment of wages. This is secondary source for an investigator but, for the factory it is a primary source.
Thus, primary data is collected originally and secondary data is collected through other sources. Primary data is first hand information for a particular statistical enquiry while the same data is second hand information for an another enquiry. The same data is primary in one hand and secondary in the other, e.g., any Government publication is first hand (Primary) for Government and second hand (Secondary)
for a research worker. Thus, secondary data can be obtained either from published sources or from any other source, for example, a website which saves time and cost.
PIHMARY DATA* PUBLISHED
How Primary Data is Collected
The most popular and common tool is questionnaire/interview schedule to collect the primary data. The questionnaire is managed by the enumerator; researchers or trained, investigators. Sometimes the questionnaire is managed by the respondents also.
MIMTim
Following are the basic principles of drafting questionnaire :
(1) Covering letter : The person conducting the survey must introduce himself and make the aims and objectives of the enquiry clear to the informant. A personal letter can be enclosed indicating the purposes and aims of enquiry. The informant should be taken into confidence. He should be assured that his answers will be kept confidential and he will not be solicited after he fills up the questionnaire. A selfaddressed and stamped envelope should be enclosed for the convenience of the informant to return the questionnaire.
(2) Number of questions : The informant should be made comfortable by asking minimum number of questions based on the objectives and scope of enquiry. More the number of questions, lesser the possibiUty of response. Therefore, normally
\l
(I ■•
IJ
Ni
m
.26
Statistics for EconomicsXI
.nstr„c,.ons about units of measutement shol b^ g.7en questionnaire, 
investigator. numbered for the convenience of the informant and the
''' sraetsr^s r— They
the mformant should ^ abrtr^ve Ae aZ^f "" 'TT' ■"."''ject.ve. For this the blank space, e.g., ®^y "smg a tickmark in
WUch of the folloJng languages you use most for uniting, (Pu, a cross) (.) English p
M Punjabi □ iit,)Vrd>x n
(f) Any other q
Sd fnTu^f: 'ir^'s'ts^;^:' ^^^omd be
■Wrong', e.g.. are answerable m'Yes' or 'No'
or 'Right' or
Are you married? Yes/No
Are you employed? Yes/No
should start from general
These questionsleXle aS In which class do you read? In which subject you are more interested?
'rr ^ooses. such
questions should bet Smlt ^^l^rrt^^ ^^^^^^^^^^ work. Such
Collection of Primary and Secondary Data
A SPECIMEN QUESTIONNAIRE
27
H,.
S.I
Hit:
28 1
Statistics for Economicsxi
Example :
ml" "" " "f P" ^'"dents in Universi^
W How will you solve the wage problem in your mdustry?
(a) Which brand of tea do you take?
(b) Why do you prefer it?
(16) T .ssn'r btk"
(b) Do you love your children?
(c) Do you beat your wife?
t
iiethods of collectiiig priiiiiary data
etc. enq in a not
Mer
7.
8.
We^wmg are the methods of primary data collection which a« in common use
\ Collection of Primary and Secondary Data
COLLECTION OF DAW
of
29
r
lARY
SECONDARY
►Direct Personal Interview ►Indirect Personal Interview ►Telephone Interview •^Information from Correspondents ►Mailed Questionnaires ►Questionnaires Filled by Enumerators
1
Published Sources
1
Unpublished Bourns
»Government Publications ► Publications of Internal Bodies ►Semiofficial Publications Report of Committees and Commissions
—> Private Publications
(a) Journals and Newspapers
(b) Research Institutions
(c) Professional Trade Bodies
(d) Annua! Reports of Joint Stock Companies
(e) Articles, Market Reviews and Reports
etc and collect the desired information. In the same way one can think of personal Imryof collection of information regarding family budget and living conto Ta group area. The investigator must be skilled, tactful, accurate, pleasing and should
not be biased. Merits :
1. Original data are collected by this method.
2. There is uniformity in collection of data.
3 The required information can be properly obtained.
4 There is flexibility in the enquiry as the investigator is personally present.
5" Information can be obtained easily from the informants by a personal interview. 6. Since the enquiry is intensive and m person, the results obtained are normally
reliable and accurate. 7 Informants' reactions to questions can be properly studied. , , . ,
8'. Investigators can use the language of communication according to the educational standard and attitude of the informant.
Limitations : 1 j u „
1. This method can be used if the field of enquiry is small. It cannot be used when
field of enquiry is wide.
J'"
SSf
m
'30
Statistics for EconomicsX
2. It is costly method and consume more time.
3. Personal bias can give wrong results
""erwise
5. This method is lengthy and complex
ifSifMlSi
msmrnrn
Merits :
Kiai^ obtained from the third party, it is more or less free froJ
3 of the investigator and the inforr^ant I
3. It saves labour, time and money i
H iLtJ™"'"" """ ' of P'oHems can properl,'
Limitations :
ar in to oh G( in( wt
Mt
Lin
sXI Collection of Primary and Secondary Data
Thus, we find that both the above methods—direct and indirect personal interviews— have certain plus and minus points. For this reason the choice of the method depends on the nature of enquiry and sometimes we balance the demerits of one method by '.tsing the other method also for the same investigation. This way we can counter chec'. the data collected by one method with the other.
(m) Telephone interview : The investigator asks questions over landhne telephone, mobile telephone and even through website. Various researchers, newspapers, television channels, mobile service providers, banks etc., use telephone service to get information from different people, e.g., exit poll, political or economical opinions, music or dance performance opinion etc. Even sometimes website or internet are used for obtaining statistical data. These days online surveys through Short Message Service, i.e., SMS has become popular.
Merits :
1. Telephone interviews are cheaper than personal interviews.
2. It can be conducted in a shorter period of time.
3. The investigator can assist the respondent by clarifying the questions.
4. Sometimes respondents are reluctant to answer some questions in personal interviews. Telephone interviews are better in such cases.
Limitations :
1. Information cannot be obtained from people who do not have their own telephones.
2. Reactions of respondents on certain issues cannot be judged; but it sometimes becomes helpful in obtaining information from respondents.
(IV) Information firom correspondents : In this method, local agents or correspondents are appointed in different parts of the investigation area. These agents regularly supply the information to the central office or investigator. They collect the information according to their own judgements and own methods. Radio and newspaper agencies generally obtain information about strikes, thefts, accidents etc. by this method. It is adopted by Government departments to get estimates of agricultural crops and the. wholesale price index number. It is suitable when the information is to be obtained from a wide area and where a high degree of accuracy is not required.
Merits :
1. This method is comparatively cheap.
2. It gives results easily and promptly.
3. It can cover a wide area under investigation.
Limitations :
1. In this method original data is not obtained.
2. It gives approximate and rough results.
32
Statistics for EconomicsXI f C
r
fij
3. As the correspondent uses his own judgement, his personal b^as may affect the accuracy of the information sent. ^
nn^'^'TT correspondents and agents may mcrease errors.
the mo " " ^^^ ^^^^ kept con^'entkllt '
Merits :
^^ ^his method in cases where
informants are spread over a wide geographical area.
o^p^eLri 1 ~ ^^ ^^   Jess than the cost
3. We can obtain original data by this method
Limitations :
faift?"" ^^^ ^^^ informants. They may
ques^r misinterpret or may not understand'some
3. There may be delays in getting replies to the questionnaires
4. ms method can be used only when the informants are educated or hterate so that ^ they return the questionnaires duly read, understood and answered 1
' possibility of getting wrong results due to partial responses, and those
IrmrnTre^uir ^^ ^^ ^ ^^ ^^^ ^^e splcm:
6. There may be loss of questionnaires in mail. This method is suitable for the following situations •
cTmpef LrS? '' " questionnaire, Government agencies
compel bank and companies etc., to supply information regularlv to the Government in a prescribed form. ^ ^ regularly to the
(b) This method can be successful when the informants are educated.
inf( The be j org; and the high
Mer
Limit
1
3.
4.
5.
'Ki
33
Collection of Primary and Secondary Data
Following are some suggestions for making this method more effective and successful.
(a) Questions should be simple and easy so that the informants may not find it a
m burden to answer them.  u i
(b) Informants should not be required to spend for posting the questionnaires back
therefore, prepaid postage stamp should be affixed. ic) This method should be used in a large sample or wide universe.
(d) This method is preferred in such enquiries where it is compulsory by law to till the schedule. Thus, there is little risk of nonresponse.
(e) The language of the schedule should be polite and should not hurt the sentiments
of the informants.
(VI) Ouestionnaire filled by enumerators : Mailed questionnaire method poses a tanber oi difficulties in collection of data. Generally, these filled questionnaires received to incomplete, inadequate and unrepresentative.
S The second alternative approach is to send trained investigators^or enumeratoi^m M,rmants with standardised questionnaires wl.ich are to be fiUed^jn
^e im^estigator helps the informants in recording their answers. The invest^a^rs shoidd i honest tactful and painstaking. This is the most common method used by research iSons. They train investigators properly specifically for the purpose of an enqu^ ^d also tram them in dealing with different persons tactfu ly, to get Proper answers to Ac questions put to them. The statistical information collected under this method is
highly reliable.
Merits :
1. It can cover a wide area.
2 The results are not affected by personal bias. ,  , . u
3' True and reliable answer to difficult questions can be obtained through
■ establishment of personal contact between the enumerator and the informant. 4. As the information is collected by trained and experienced enumerators, it is
reasonably accurate and reUable. ,
5 This method can be adopted in those cases also where the informants are illiterate.
6'. Personal presence of investigator assured complete response and respondents can
be persuaded to give the answers to the questionnaire.
Limitations : ^ n ■ ^
1. It is an expensive method as compared to other methods of primary collection of
data, as the enumerators are required to be paid.
2. This method is time consuming since the enumerator is required to visit people
spread out over a wide area. 3 This method needs the supervision of investigators and enumerators. 4" Enumerators need to be trained. Without good interview and proper traming, most
■ of the collected information is vague and may lead to wrong conclusions. 5. It needs a good battery of investigators to cover the wide area of universe and therefore it can be used by bigger organisations.
I,
If.
P
34
Klot Suiwey or PreTe« • p ^f^^stics for EconomicsXi
a Pretest or a guidihg survl; " ^ ^uc.
mam survey. This is done to try out the auetlr before starting th,
the general mformation about L po^Ja"^^^^ thods for obtaL the pilot survey helps in : ^ ^e sampled. The information supplied b, [i) Estimating the eosr of ^ ■
avaihbiii,® of fc 'he ei™ needed for
"" rr. or fan.
0t.ons and a,so .„ .He ..proje.enr „
(w) Training of field staff " rX"  ^ namr,
casus MID SAIWnE SURVEVS
ia) Census method/Census Survey, and (b) Sample method/Sample Survey
.e^VS^td^^.—^ " — Have a Cear n„dersra„d.n, „,
Population and Sample
o all ,he ten,. A par, of .hVlorplZL
ekcon termed as sampling. SupposeTe'^are 500 1 "T'" <>'
School. If we want to know the average wekta Secondary
W'll get the mformation abont all the fLrhnnd f each girl and
obtamed by dividing the total weigte of he tn' " we.ght willbe
CX)VERNMENT SENIOR SECONDARY
I i
f( P
O £ S<
UI
gro' Sun
20
Source
sXI Collection of Primary and Secondary Data
by taking only 50 girls out of 500 and obtain the average of this part of the total population. The average of 50 girls reasonably be representative of average weight of 500 girls. In this case weight of 50 girls is the sample.
Census Surveys
The objective of a census method or complete enumeration is to collect information for each and every unit of the population/universe. In this method every element of population is included in the investigation. Thus, when we make a complete enumeration of all items in population, it is known as 'Census Method" or 'Method of Complete Enumeration'. In above example, collecting weights of all the 500 girls in Senior Secondary School is census method of collection where no student is left over, as each student is a unit.
Following are few examples of census :
1. The population census is carried put once in every ten years in India. Most recently population census in India was carried out in February, 2001 by house to house enquiry to cover all households in India.
2. Demographic data obtained by census method on death rates and birth rates, literacy, work force, life expectancy and composition of population etc. are published by Registrar General of India.
3. The data relating to estimation of the total area under principal crops in India are obtained by using village records maintained regularly by Patwari.
Let us review the following census data in the following Table no. 2 regarding relative growth of Urban and Rural Population in India obtained from Reports and Economic Survey 20022003.
TABLE2
Relative Growth of Urban and Rural Population in India
r................. Year i f r UrhaiP Popuiatinn {tn itorpi) Rural PopttUuioti (m rmrei) Total Ptipuldtion (m Lrine») As Perceraage of Total Popukttidn
Urban popuhtion Rumi PvpuUition
1901 2.58 21.25 23.83 10.9 89.1
1911 2.59 22.62 25.21 10.3 89.7
1921 2.80 22.32 25.12 11.2 88.8
1931 3.35 24.54 27.89 12.1 87.9
1941 4.41 27.44 31.85 13.8 86.2
1951 6.24 29.87 36.11 17.3 82.7
1961 7.89 36.02 43.91 18.0 82.0
1971 10.89 43.93 54.82 19.9 80.1
1981 16.22 52.11 68.33 23.7 76.3
1991 21.76 62.87 84.63 25.7 74:3 .
2001 28.50 74.2 102.7 27.8 72.2
Source : Census Reports and Economic Survey 20022003.
if:.
i t i
I7
I (1
36
Statistics for EconomicsXI
74 " 'rr ^^ Of India's population. In 2001
74 2 crore persons, out of about 102.7 crore total population lived in around 5 5 lakhs'
i: ifo^lVt 2 r   or uian areas
urban a;ea. Th T ' Population of around 24 crores lived in
urban areas The urban population formed about 11 per cent and rural population 89 per
r^om I urban population had gone up to around 28 per cen
n 2001 while still over 72 per cent people lived m rural areas. The above table show the relative growth of rural and urban population m India since 1901.
The net addition to rural population between 19912001 was 1133 crore while urban population increased by 6.74 crore persons. The decadal growth at frru^d
in1he rlr" T  ^ mcrease of 2 1 per cent
in the growth rate of urban population m the decade ending 2001 over the decade^nding
SAMPLE SURVEYS
We may study a sample drawn from the large population and if that sample is adequate representative of the population, we should be able to arrive at val 7corcSn
Method of collecting of data. In above example, collecting the weights of 50 girls out of
500 girls m Semor Secondary School is sample method of collectiol In this method ew students as sample considered for our study. metnod tew
Following are a few common examples of samplin •
{a) We look at a handful of gram to evaluate the quality of wheat, rice or pulses, etc
A ^^^ '^^^ ^P^" ^1tric bulbs out of each lot"
[c) A drop of blood is tested for diseases like malaria or typhoid etc
^ ^ fudtrnfof^'tC ^^^ » P^ion for final
(.) Th^^^elevision network provides election coverage by exit polls and prediction is
nnnT"'' T ""V^^^'^^ical termmology population or universe does not mean the total numbe of people m an area; it means the total number of observations or terns fn
r::att  ^ — — ^^lected from^ a ~
methods of sampling
^^^Broadly speaking, various methods of sampling can be grouped under mam (a) Random Sampling, and (b) NonRandom Sampling.
Collection of Primary and Secondary Data
37
Let us discuss now the various samphng methods which are popularly used in practice.
MiTHODS OF SAMPLING
i
Random Sampling (a)
1
NonRandom Sampling
ib)
Simple or Unrestricted Random Sampling Restricted Random Sampling (f) Stratified Sampling (//) Systematic Sampling
or Quasi Random Sampling (f/f) Cluster Sampling or Multistage Sampling
(a) (to) (c)
Judgement Sampling Quota Sampling Convenience Sampling
random sampling
Random Sampling is one where the individual units (samples) are selected at random.
It is called as probability sampling.
Random sampling does not mean unsystematic selection of units. It means the chances of each item of the universe being included in the sample is equal. The term 'Random Sampling' here is not used to describe the data in the sample but it refers to the process used for selecting the sample. Following are the methods of random sampling.
Simple or Unrestricted Random Sampling
This method is also known as simple random sampling. In this method the selection of item is not determined by the investigator but the process used to select the terms of the sample decides the chances of selection. Each item of the universe has an equal chance of being included in the sample. It is free from discrimination and human judgement. Random sampling is the scientific procedure of obtaining a sample from the given population. It depends on the law of probability which decides the inclusion of items in a sample. To ensure randomness, mechanical devices are used. There are t^vo methods ot obtaining the simple random sample. They are :
(a) Lottery Method, and
(b) Table of Random Numbers.
(a) Lottery Method : A random sample can generally be selected by this simple and popular method. All the items of the universe are numbered and these numbers are written on identical pieces of paper (slip). They are mixed in a bowl and then there starts the selection by draw one by one by shaking the bowl before every draw The numbers are picked out blind folded. All slips must be identical in size, shape and colour to avoid the
biased selection.
IMH'
38
Statistics for EconomicsXI
metal pieces on which nuZ^tT Th! d "" "^en or
device and each time one piece comes rotated by a mechanical
of digits, for instance if the numbe'Ts ZZ u '^^^ ""^ber
This^i^ IS us. m drawi::^^^^:; 
ry large the above procedures if the disks, balls or slips L not XrouThnf' ^^ T^^^^^ " been a marked tendency to usetSroTrindT^^^^ > T T' " ^^^^ ^as
samples. A table of random digkst simply a the purpose of drawing such ,
by a random process. The follLing of Som ^gS tt^^^^^^^^^^^^ ~ (.) Tippet. Random Sampling Numbers. There are 10^00 numbe^t^anged 4 digits
MG. Kendall and Babington Smith's Random Sampling Numbers, having 1 lakh
ic) Rand Corporation's a million random digits (d) Snedecor's 10000 random numbers. ie) Fisher and Yates Table having 15000 digits
Rc
2952 3170 7203 3408 0S60
Tippett Numbers
6641 3992 9792
5624 4167 9524
• 5356 1300 2693
2762 3563 1069
5246 1112 9025
ho
Th
hoi
ave strs ;san [peo
7969 5911
1545 1396
2370 7483
5913 7691
6608 8126
college. We will first nuX aVMOo Tui" f ''"t"'' ^
students, now we will cons J a pag^of ?fp i'" ^^^ ""'"''"ing the
15 successive number either horLLhy or ^^a^ "" ^^
Merits
' w a universe. There are less
selected. ^ bas equal chance of being
Regularity begin to operate ^^^ ^aw of Statistical
(
[Meri 1.
2.
3.
4.
5.
6.
sXI Collection of Primary and Secondary Data
3. This method is economical as it saves time, money and labour in investigating a population.
4. The theory of probability is applicable, if the sample is random.
5. Sampling error can be measured.
Demerits
1. This requires complete list of population but uptodate lists are not available in many enquiries.
2. If the size of the sample is small, then it will not be a representative of a population.
3. When the distribution between items is very large, this method cannot be used.
4. The numbering of units and the preparation of the slips is quite time consuming and not economical particularly if the population is large.
t
Restricted Random Sampling
They are as follows :
(t) Stratified random sampling : In this method the universe is divided into strata or homogeneous groups and an equal sample is drawn from each stratum or layer at random. This method is therefore useful when the population of the universe is not fully homogeneous. For example, suppose we want to know how much pocket money an average university student gets every month will be taken equal sample from various strata, namely : B.A. students, M.A. students and Ph.D. students etc. Stratified random sampling is widely used in market research and opinion polls, it is fairly easy to classify people into occupational, economic, social, religious and other strata. There are different types of stratified sampling
{a) Proportional stratified sampling is one in which the items are taken from each stratum in the proportion of the units of the stratum to the total population.
(b) Disproportionate stratified sampling is one in which units in equal numbers are taken from each stratum irrespective of its size.
(c) Stratified weighted sampling is one where units are taken in equal number from each stratum, but weights are given to different strata on* the basis of their size.
Merits
1. The sample taken under this method is more representative of the universe as it has been taken from different groups of universe.
2. It ensures greater accuracy as each group (stratum) is so formed that it consists of uniform or homogeneous items.
3. It is easy to administer as universe is subdivided.
4. Greater geographical concentration reduces the time and expenses.
5. For nonhomogeneous population, it is more reliable.
6. When original population is not normal (skewed), this method is appropriate.
Statistics for EconomicsXI
V\L
N
m.
40
Demerits
1. Stratified sampHng is not possible unless some mformation concerning ti population and its strata is available. concerning u
2. If proper stratification is not done the sample will have an effect of bias. If differ, strata of population overlap such a sample will not be a representative one
(«) Systematic sampling or quasirandom sampling : Systematic sampling is a simo
by preparing this list m some random order, for example, alphabetical order
SMnlr U the list, « stands for any numl
Suppose we have a universe of 10,000 items and we want a sample of 1000, then ^^^
«  10 The method of selecting the first item from the list is to decide at random f^^t
?hen th "r "'Tu' Suppose we pTck up Z Z t
Then the other items will be 15th, 25th, 35th, and so on unSl we have got oVr fuH sal
fullv rw'' T I u" " that the list of the univers!
fully random and that there are no inherent periodicities in the list.
Merits
1. It ^yystematic, very simple, convenient and checking can also be done quickl]
2. In this method time and work is reduced much.
3. The results are also found to be generally satisfactory.
Demerits
random will not be a determming factor in the selection of a sample.
2. It IS feasible only if the units are systematically managed
3. If the universe is arranged in wrong manner, the results will be misleading 
to divL and sub " ^^
to divide and subdivide a universe according to its characteristics. Thus if a survev ki be conducted in a country it will first be divided into zones or states l region t^^^^^ mailer units cities towns and villages and then into localities and hLseToW; At Jd
nonranoom sampun6
ccessi 3n No. Date.^............
sXI
the
:rent
tipler on is ieved
mber. : take from item, mple. irse is
ickly.
!on as
It m e have y is to ;n into .t each nethoc the hsl
of the s, noning are
Collection of Primary and Secondary Data
(a) Judgement or purposive sampHng
(b) Quota samphng
(c) Convenience sampHng \
'n
Judgement Sampling
This is also called purposive or deliberate sampling. In this method individual items of sampling are selected by the investigator consciously using his judgement. Therefore, it requires that the investigator should have a good knowledge of the universe and some experience in the field of investigation. Obviously, the choice of samples will vary from one investigator to another. For example, from a universe of 10,000 ladies who use a particular brand of hairdye, the investigator will select a sample of say, 1,000. His choice of this sample will be such that it is irrespective of the universe. For this an exercise oi judgement is required.
In order for the judgement sampling to be reliable, it should be free from individual lies or prejudice. Since the choice of sample is not based on probability it does not guarantee accuracy and it makes detecting of sampling errors difficult. However, this methods is useful in solving a number of kinds of problems in universe and economics.
The purposive or judgement sampling is suitable in the following conditions :
(a) The number of items in the universe is small to which some items of important characteristics are likely to be left out.
(b) When small sized sample is to be drawn.
(c) When some known characteristics of the universe are to be intensively studied.
(d) It is also appropriate for pilot survey.
Quota Sampling
It is a method of sampling that saves time and cost and is commonly used m surveys of political, religious and social opinion. Interviewers are allotted definite quotas of the universe and they are required to interview a certain number from their quota. Quotas are decided on the basis of the proportion of persons in various categories. In other words, the investigator is given instructions about
how many interviews should be taken say in a given localitv and what proportion should be from say upper, middle and lower mcome groups, as by some other classification which is predetermined. For example, for a study of truancy (running away) from school in Delhi the investigators are allotted quotas of say 10 schools each out of which two should be public schools (Boys), one public school (Girls), three Boys' Senior Secondary Schools, two Girls' Senior Secondary Schools, two Coeducation Schools and from each school he is asked to interview 50 students, taking 10 students each from Classes VIII, IX, X, XI and XII. The interviewer can select any 10 students according to his own judgement.
It is a kind of judgement samphng and provides satisfactory results only when interviewers are carefully trained and personal prejudice is kept out of theprocess of selection. ' '
hi
!i<
42 ,
Convenience Sampling Statistics for EconomicsXI;
P0.1:;: f'^intfri: l^essr ~ ce ^
example, for the study of truanrvTr ^^e basis of convenience F.
a school or schools in I neiS^ ^^^^^ ^^e invesdgat™selec
schools. This method is used wLn^e " .V ^^ ^^ convenient for hL to g^trthe not clear or complete source hst is t^a^lbl" T ^^e sample unit i
easily available lists, such as teleDhLTW may be obtained W
results obtained by' this m^aTntl^^^^^^
unsatisfactory. ^^^ ^ruly representatives of the universe and are
^^HIAGESOF
(I
• of dae. bee. Jo,
getting quick results. ^ therefore, sampling is very useful in
sxhris fre^ir" —a,
. for fc cV^ "^dr m  «
method. in some ways more reliable than cenLs
2 aliow a samplmg mefcd t
« fc o„,, pos.b,e or E ~ or bote, ma„ufaeLd"« fcTar:^;,,*^ ^dl"'
sampling method. '' P^^^'^le due to the scientific nature of
» appropriate «e,d . neceiary^Srfctr:?"^^^^^^^^
le otl: Thi
the larf
Statistic
The means a
Collection of Primary and Secondary Data
43
^^i^lity of sample data ;
The main purpose of sampling is to collect maximum information with minimum ^nditure of money, time and labour and yet achieve a high degree of ^curacy and Ability. For ensuring reliability certain principles must be followed. In samphng method : is presumed that whatever conclusions are drawn from a sample are also true for the lole population. This presumption is based mainly on the followmg two laws :
(a) The Law of Statistical Regularity, and
(b) The Law of Inertia of Large Numbers. .'r u (a) Law of statistical regularity : The law of statistical regularity is derived from the
mathematical theory of probability. It says that a comparatively small group of items chosen at random from a very large group will, on the
characteristics of the large group. Basically, it applied to rWom se^lection. Thus so in the process of sampling each unit of the universe has an equal chance of being selected. Therefore, the selected items can be said to be representative of the universe. Although the law is not as accurate as a scientific law is, it does insure a reasonable degree of accuracy. Since there is a certam regularity m natural phenomena, we assume a certain uniformity in nature A random samphng is said to follow the law of statistical regularity because of this basic uniformity m a
universe. r , t r
lb) Law of inertia of large numbers : This law is also called the law of stability of mass
data. It is based on the law of statistical regularity. Basica ly, it states that if the
numbers involved are very large, the change in a sample is likely to be very small
in other words, the individual units of a universe very continually but the total
universe changes slowly. That is, large aggregates are most stable than «tnaU
Because of the slow change in the nature of total universe this law is called the law
of inertia (laziness) of large numbers. For example, sugar production of factory will vary significantly from year to year but Ac sugar production of a country as a whole will remain comparatively s able. Or a g eat Inge may take place in the malefemale ratio of family may appreciably bange ove a short period, but the malefemale ratio of a country as a whole will ^^
the period, ^o take another example, if a. coin is tossed 6 times we may get heaj^s f^r ^ Js and tails two times. But if a coin is tossed 5^0 times^ there is a high p^^i^ of getting heads and tails 2,500 times each. This happens due to ^^^
I oplation of this law. That is, when one part of large group is changing m one direction
the other moves in the opposite direction.
Thus, reliability of sampling depends mainly on randomness of selection of data and the large size of universe, expressed by the above two laws.
Statistical Errors
There is a great difference in the meaning of mistake and error in statistics. Mistake imeans a wronfcalculation or use of inappropriate method in the collection or analysis
44
Statistics for EconomicsX.
other words, the difference between the approximated (estimated value) and the actual value (true value) is called statistical error m a technical sense. For examl we make a' estimation that in a particular meeting, 1,000 persons are there. But we clnt persons It may be wrongly counted, as 1,030. There is a difference of 30 between the estimate value and counted va ue. This difference is called '...or' in statistics. But w^en weTak*
aTS''^r VrThey arl knowi as mistake . For example, there is a meeting, we sent a person to count the audience
Sources of Errors
Following errors are likely to occur in collection of data :
ur'l^! origin arise on accoum of inappropriate definitions of statistical
unit scale, or defective questionnaire etc. For example, wrong scale to measun
meLl 'I height to nearest of inch or approximatrTh
differences may also occur due to differences in measuring tapes due tc manufacturing defect. In Physics or Chemistry such errors of mLsurementrwlI occur while taking readings on various instruments.
nZZ incomplete data, madequat.
crsdonna " sample, nonresponse of respondent, incomplete answers i questionnaire, misinterpretation of questions in questionnaire, careless oi unqualified investigators, etc. 'diciesb oi
dr^o I'f f^ "hmetic calculatio
due to clerical errors, arithmetic slips etc. by omitting some figure consideri
wrong value, making wrong totals etc. by respondent L investigator thrjlta^"''''^'"''^''''" ' statisticians for misinterpret!
Types of Errors
(a) Absolute and relative errors : Absolute error is the difference between the actua
true value and estimated approximate value while relative error is the raTo o absoS error to the approximated value. absolut
Absolute error = Actual value  Estimated value Symbolically,
Ue = U' U
wr
the enu mei wh( faul
Relative error = Actual value  Estimated value Symbolically,
Estimated value
e =
U'U U
Sec furthei obtain alreadj are inv b) unf
sXI Collection of Primary and Secondary Data
Here, Ue = Absolute error
e = Relative error U' = Actual value U = Approximate value niustration. Sales of commodity approximated Rs 497 and actual sale Rs 500. Absolute error (Ue) = 500  497 = 3
500497 3
and Relative error (e) 
= .006
500 500
Relative error (e) can also be represented in percentage
X 100 = 0.6%.
500
Relative error is generally used in statistical calculations because absolute error gives wrong or misleading calculations.
(h) Biased and unbiased errors : Biased errors arise due to some prejudice or bias in the mind of investigator or the informant or any measurement instrument. Suppose the Hiumerator used the deliberate sampling method in place of simple random sampling method; then it is called biased error. These errors are cumulative in .ir re and increase when the sample size also increases. Biased errors arise due to fauli^ j^iocess of selection, faulty work during the collection of information and faulty method of analysis.
Unbiased errors are not the result of any prejudice or bias. They are those which arise acccidently just on account of chance in the normal course of investigation. Unbiased errors are generally compensating.
(c) Sampling and nonsampling errors : The errors arising on account of drawing inferences about the population on the basis of few observations (sampling) are called sampling errors. The errors mainly arising at the stages of ascertainment and processing 'of data, are called nonsampling errors. They are common both in census enumeration and sample surveys.
To avoid these errors, the statistician must take proper precaution and care in using itfie correct measuring instrument. He must see that the enumerators are also not biased. Unbiased errors can be removed with proper planning of statistical investigations.
Statisticians should have none of these errors.
i "
how secondary data is collected
Secondary data are those which are collected by some other agency and are used for i^her studies. It is not necessary to conduct special surveys and investigations. We can obtain the required statistical information from other institutions, or reports which are ^eady published by them as a part of their
routine work. It saves cost and time which 'are involved in collection of primary data. Secondary data may be either (a) published or (fc) unpublished.
46
Statistics for EconomicsXI
Ji
hm. Sf.
Published Soiuces
The various sources of pubhshed data are as under :
(/■) Gove^ent pubUcations : Different ministries and departments of Central ar State Governments publish regularly current information along with statistical da on a number of subjects. This information is quite reliable for related studies. ^ examp es of such publications are: Annual Survey of Industries, Labour Gaze Agriculture Statistics of India, Indian Trade Journal, etc.
(«) Publications of international organisations : We can obtain valuable internation s atistics from official publication of different international organisations, like, ti United Nations Organisation (UNO), International Labour Organisation (ILO International Monetary Fund (IMF), World Bank, etc. .
(Hi) Semiofficial publications : Local bodies such as Municipal Corporations, Distri Boards etc: publish periodical reports which give factual information about heal sanitation, births, deaths etc.
(iv) Reports of committees and commissions : Various Committees and Commission are appointed by the Central and State Governments for some special study an recommendations. The reports of .uch committees and commissions contai valuable data^ Some of the reports are : Report of National Agricultu Commission, Report of the Tariff Commission, the Patel Committee Report e
(v) Private publications :
(a) Journal and new^apers. Journals like Eastern Economists, Journal of Industr and Trade, Monthly Statistics of Trade; and newspapers, like Financial Expres Economic Times, collect and regularly puWish the data on different fields ( economics, commerce and trade.
(b) R^earch institutions. There are a number of institutions doing research o allied subjects This is the most importarn source of obtaining secondary dat The National Council of Applied Economic Research and Foundation ( !>cientihc and ^onomic Research are such institutions. Research scholars at ti university level also contribute significandy to the availabihties of secondai
(c) Professional trade bodies. Chambers of Commerce and Trade Associatio, publish statistics relating to trade and commerce. Federation of Indian Chamb of Commerce, Institute of Chartered Accountants, Sugar Mills Associatio Bombay Mill Owners Association, Stock Exchanges, Bank and Cooperath Societies, Trade Unions, etc. pubhsh statistical data.
(d) Annual reports of joint stock companies are also useful for obtaining statistic information. These are pubKshed by companies every year.
^^^ ^md^'' also, provide valuable data for reseat
sXI Collection of Primary and Secondary Data
Unpublished Data
Research institutions, trade associations, universities, labour bureaus, research workers and scholars do collect data but they normally do not pubHsh it. Apart from the above sources we can get the information from records and files of government and private offices. 
Limitations of Secondary Data
One should use the secondary data with care and full precaution and should not accept them at their face value as they may be suffering from the following limitations:
1. They may not have been collected by proper procedure.
2. They may not be suitable for a required purpose. The information which was collected on a particular base may not be suitable and relevant to an enquiry.
3. They may have been influenced by the biased investigation or personal prejudices.
4. They may be out of date and not suitable to the present period.
5. They may not satisfy a reasonable standard of accuracy.
6. They may not cover the full period of investigation.
Precautions in the Use of Secondary Data
The investigator should consider the following points before using th j secondary data : (a) Are the data reliable?
{b) Are the data suitable for the purpose of investigation?
(c) Are the data adequate?
(d) Are the data collected by proper method?
(e) From which source were the data collected? if) Who has collected the data?
(g) Are the data biased?
. Thus, the secondary data should not be used at its face value. It is risky to use such statistics collected by others unless they have been properly scrutinised and found reliable, suitable and adequate. ■
■
ijl^ofrlant sources of secondary dali> of india and national survey organisations)
There are various sources and organisations through which statistical data are being compiled in India. Since India achieved Iiidependence, great and rapid strides have been made in the field of collection of data. In the context of economic planning, importance of statistics (data) in the country has become great. Statistics are necessary for framing and judging the progress of economic planning. The study of Indian statistics is made under following heads :
I. Statistical Organisation of India (CSO)
II. Indian Statistical Material.
48
Statistics for EconomicsXI
This can be studied >finder following sections :
(A) AgricultureStatistics (B) National Income and Social Accounting
(C) Population Statistics (D) National Sample Survey
(E) Price Statistics (F) Industrial Statistics
(G) Trade Statistics (H) Financial Statistics
(I) Labour Statistics
There are some agencies both at the national and state level, which collect, process .^nd tabulate statisticar data. Some important major agencies at the national level are ^ensus of ^dia, Narionai Sample Survey Organisation (NSSO), Labour Bureau, Central Statistical Organisation (CSO), Registrar General of India (RGI), Director General of Commercial Intelligence and Statistics (DGCIS), etc.
census of india
unique experience of undertaking the biggest census in the world in 1981 and has also an unbroken record of more than hundred years of decadal censuses Ihe Indian census is universally acknowledged as most authentic and comprehensive source of information about our land and people. In 1869 Hunter was appointed Director General of Statistical Surveys. He not only elaborated the statistical system but also assisted the statistical surveys of districts and provinces. That later followed into famous Gazetteers. He advised m conducting of census of India which undertook explanatory surveys from 1869 to 1872 and thereafter matured into a decennial census which ever since contmued without interruption. After 1872 the next census was taken in 1881 and ^nce then it has ^become a regular feature of holding census every ten years uninterruptedR The Census of India provides the most complete and continuous demographic record of
T Independecne was held in 1951 and latest one completed
m .001. The study of population is important for several reasons in overall study of economic development. Information of demographic characteristics include birth and death, fertility, sex ratio, agecomposition, migration and literacy etc. The economic Characteristics of ppulation are manifested through workers' participation m economic
classification of workers m various occupations, employment
The data generated by the Census of India 2001 provide benchmark statistics on the
people of India at the beginning of the next millennium. This is a mirror of a fair
^presentation of the socioeconomic and demographic condition of our people which
constitute about onesixth of the human population on this planet. The census statistics
s useful for assessing the^impact of the developmental programmes and identify new
thrust areasTor focussing the efforts on improving the quality of life in our country Basic
population data fmm Primary Census Abstract. Census of India 2001 gives information ot population m India as :
TABLE 3
PersonsMales Females Sex Ratio
1,028,610,328 —^ ^ _" 532,156,772 496,453,536 933
Collection of Primary and Secondary Data
49
national sample survey organisation (ii5s0)
The National Sample Survey (NSS), initiated in the year 1950, is a nationwide, large scale continuous survey operation conducted in the form of successive rounds. It was established on the basis of a proposal from Prof. P.C. Mahalanobis to fill up data gap for socioeconomic planning and policy making through sample surveys. On march 1970, the NSS was recognised and all aspects of its work were brought under a single Government organisation namely the National Sample Survey Organisation (NSSO) under the overall direction of a Governing Council to impart objectivity and autonomy in the matter of collection, processing and publication of the NSS data.
The Governing Council consists of 18 experts from within and outside Government and is headed by an eminent economist/statistician and the membersecretary of the council is Director General and Chief Executive Officer of NSSO. The Governing Council is empowered to take all technical decisions in respect of survey work, from planning of survey to release of survey results. The NSSO headed by a Director General and Chief Executive Officer, has four divisions namely. Survey Design and Research Division
(SDRD), Field Operation Division (FOD), Data Processing Division (DPD) and Coordination Publication Division (CPD). A Deputy Director General heads each division except FOD. An Additional Director General heads FOD.
Functions of NSSO
The functions of National Sample Survey Organisation are :
(i) Collection of data on socioeconomic conditions, production of small scale household enterprises consumption etc. on continuous basis in a comprehensive manner for whole country. A major objective of NSS has been to provide data required to fill up the gaps in information needed for estimation of national income.
[ii) Collection of data relating to the organised industrial sector of the country. {Hi) Supervision of surveys conducted by states in agricultural sector through their own
agencies and also giving guidance to them for analysing and coordinating the results of these surveys. The NSSO took a forward view of the data requirements to planners, research workers and other users and draw up a long term programme. The programme conducts periodical surveys on :
{a) Demography, health and family planning; {b) Assets, debt and investment; (c) Land holdings and livestock enterprises;
{d) Employment and unemployment, rural labour and consumer expenditure; and (e) Self employment in nonagricultural eflterprises.
The data collected by NSSO surveys on different socioeconomic subjects are released tiirough reports and its quarterly journal 'Sarvekshana\ The data comprises different iocioeconomic subjects like employment, unemployment literacy, maternity child care.
■ 1 t r
50
Statistics for EconomicsXI
utUisat^n of public distribution system, utilisation of educational of services etc Th
car Aoar; fTolfll . T 2004)was on morbidity and head
care. Apart from collection of rural and urban retail prices for compilation of consume
pn.e mdex numbers NSSO also undertakes field work of Annual S^ ^dust^ conducts crop estimation surveys. ^ maustries an
exercises
2.
3.
4.
5.
6.
7.
8.
9.
10. 11. 12. 13.
14.
15.
iXT
What do you understand by Statistical Enquiry? Explain
d~ft~f ^ merits anc
Discuss the comparatwe merits of various methods of collecting primary data.
SrsTable^^^^^^^^^^^ " ' itigations,
What are the similarities and dissimilarities between the two methodsl questionnaires to be filled in by informants and schedules to be fild in h enumerators? Explain with examples. *
mat is a questionnaire? Give a specimen of a questionnaire.
Describe the questionnaire method of collecting primary data. What precaution! must be taken while preparing questionnaire? precautionf
Write short notes on :
(a) Census of India
{b) National Sample Survey Organisation (NSSO) mat IS Secondary^data? Discuss the various sources of collecting secondary data, mat precaution should be taken before using secondary data? Explain
iTZn 1 constructing interview schedules and questionnaires
F ame at least four appropriate multiple choice options for following questions (t) How often do you use computers? («) What is the monthly income of your family? (m) Rise in petrol price is justified : (iv) Which of the newspaper do you read regularly ?
Jv) Which of the following most important when you buy a new dress' l<rame two way questions (with 'Yes' or 'No')
Following statementstrue or false.
(/) Data collected by investigator is called secondary data. («) There are many sources of data.
(m) Telephone survey is the most suitable method of collection of data when the population is literate and spread over a large area.
sXI Collection of Primary and Secondary Data
16. Distinguish between population and sample.
17, Distinguish between census and sample surveys. List four important types of sampling : methods. Explain the reasons for preferring sample surveys in the collection of data.
Name the methods of selecting a sample. Describe the method of stratified sampling  with merits and demerits.
19. The Education Ministry is interested in determining the level of education of unmarried girls in the country. How would you organise a survey for this purpose?
20. Does the lottery method always give you random sample? Explain.
21. Do samples provide better resuhs than surveys? Give reasons for your answer.
22. Examine the important types of sampling methods.
23. Distinguish between random sampling and systematic sampling. Give suitable examples.
24. Discuss briefly the following :
(a) Law of inertia and large numbers
(b) Law of statistical regularity
25. What do you understand by 'Census' investigation? Explain its suitability with illustrations.
26. What do you mean by 'Sample' investigation? Explain its suitability with illustrations.
27. Explain briefly the different methods of sampling. Give illustrations.
28. Give a comparative study of stratified sampling and multistage sampling.
29. Write a critical note of random sampling method.
30. How would you distinguish convenience sampling with judgement (deliberate) sampling? Explain.
31. What do you mean by statistical errors?
32. Discuss briefly the following :
(a) Biased and unbiased errors
(b) Absolute errors and relative errors p; :—^ (c) Sampling and nonsampling errors
Give two examples each of sample, population and variable. Which of the following methods gives better result and why? (a) Census (6) Sample
Chapter 4
organisation of data
¥
(b)
Classification
1. Definition
2. Objects of Classification
3. Characteristics of Classification
4. Methods of Classification Statistical Series
1. Definition
2. Types of Series
3. Frequency Distribution
(a) classification
The quantitative information collected in any field of society or science is never uniform. They always differ from one to another, e.g., prices of vegetables, students in different sections, income of families, time in different watches, height or weight of students. A single item out of all the observations of group as numerical may be called variate or variable, e.g..
Price of potato is Rs 10.00 per kg, in a group of vegetable prices.
Students in Section A are 50, in a group of different sections.
Income of family D is Rs 10,000 per month, in a group of families.
Time in HA/IT watch is 10.45 a.m., m a group of different watches.
Height of Rajesh is 60", in a group of students.
Variate can also be called 'variable' or 'magnitude' or 'observation' or 'item' or 'measure' or value'.
The characteristics which are not capable of being measured quantitatively are called attributes. For example, blindness, deafness, literacy, sickness, tall and short, black and blue eyed, intelligence, aptitude for art and music, etc. They cannot be measured numerically in the same way as heights and weights, or, price and incomes. Individuals may be ranked according to quality of attributes. The ranks are sometimes used as their numerical values for purposes of statistical analysis.
The collected data (either by primary or secondary method) are always in an unorganised form in schedules or questionnaires or another written form. The collected data in unorganised form is called RAW DATA. Because of the limitation of human mind
Organisation of Data
53
to understand such a complex, varied and unorganised data, it is necessary to make them available for comparison, analysis and appreciation by proper and suitable grouping and arrangement in condensed form. The process of grouping into different classes or subclasses according to characteristics is called classification. The classified information arranged in a logical and systematic order in a particular sequence is called seriation or statistical series. The classified information presented in precise and systematic tables is called tabulation. In other words, classification is for division of data, seriation is for arrangement of data in a systematic order and tabulation is for presentation of data in a table.
DEFINITION
According to Professor Connor, "Classification is the process of arranging things (either actually or notionally) in the groups according to their resemblances and affinities, and give expression to the unity of attributes that may subsist amongst a diversity of individuals."
According to this definition, the chief features of classifications are :
(1) The facts are classified into homogeneous groups by the process of classification All the units having similar characteristics are placed in one class or group.
(2) The basis of classification is unity in diversity.
(3) The classification may be either actual or notional.
(4) The classification may be according to either attributes or characteristics or measurements.
Classification is grouping of data according to their identity, similarity, or resemblances For example, letters in the post office are sorted out in groups of cities and towns of destination, viz., Delhi, Chennai, Agra, Chandigarh etc. Similarly, students in a school may be grouped as boys and girls, or according to age, in library the books and periodicals are classified and arranged according to subjects, students are classified according to division they secured in certain examination, animals or plants may be grouped according to origin or structure etc.
OBJECTS OF CLASSIFICATION
The chief objects of classification are :
1. To present the facts in a simple form : Classification process eliminates unnecessary details and makes the mass of complex data, simple, brief, logical and understandable. For example, the data collected in a population census is so huge and fragmented that it is not possible to draw any conclusion from them. When these massive figures are classified according to sex, education, marital status, occupation etc., then the structure and nature of the population can easily be understood.
2. To bring out clearly points of similarity and dissimilarity : Classification brings out clearly the points of similarity and dissimilarity of the data so that they can be
Statistics for EconomicsXI
easily grasped. Facts having similarcharacteristics are placed in a class, such as educated, uneducated^ employed, unemployed etc.
3. To facilitate comparison : Classification of data enables one to make comparison, draw mferences and locate facts. This is not possible in an unorganised and unclassified data. If marks obtained by B. Com. students in two colleges are given, no comparison can be made of their intelligence level. But classification of students into first, second, third and failure classes on the basis of marks obtained by them, will make such comparison easy.
4. To bring out relationship : Classification helps in finding out causeeffect relationship, if there is any in the data. For example, data of smallpox patients can help m finding out whether smallpox cases occurred more on vaccinated or unvaccinated population.
5. To present a mental picture : The process of classification enables one to form a mental picture of objects of perception and conception. Summarised data can easily be understood and remembered.
6. To prepare the basis for tabulation : Classification prepared the basis for tabulation and statistical analysis of the data. Unclassified data cannot be presented in tables.
CHARACTERISTICS OF CLASSIFICATION
It is important that the classification should possess following characteristics :
1. Classification should be unambiguous : Classification is meant for removing ambiguity. It is necessary that various classes should be so defined that there is no room for doubt and confiision and must have a class for each item of data in one of the classes.
2. The classes must not overlap : Each item of data must find its place in one class and one class only There must be no item which can find its way into more than one class.
3. Classification should be stable : If classification is not stable and if each time an enquiry is conducted it has to be changed. The data would not be fit for comparison. Therefore, the classification must proceed at every stage in accordance with one principle, and that principle should be maintained throughout.
4. Classification should be flexible : It should be flexible and should have the capacity of adjustment to new situations and circumstances. With change in time, some classes became obsolete and have to be dropped and fresh classes have also to be added.
5. Classification should be suitable to enquiry iClassification should be according to the objects of enquiry. If the investigation is carried on to enquire into the economic conditions of labourers, then it will be useless to classify them on the basis of their religion. »
6. Classification, should have arithmetical accuracy : The total of items included in different classes, should tally with the total of the universe.
55
■ganisation of Data
.O .e. ch—s. . can be .onped
the following bases :
1. According to time, (Chronological Classification)
2 According to area, (Geographical Classification)
3 According to attributes, (Qualitative Classification)
4 According to magnitudes or variables (Quantitative Classification)
For example.
Population of India
1951 35.7
1961 43.8
1971 54.6
1981 68.4
1991 81.8
2001 102.7
OR
2001 102.7
1991 81.8
1981 68.4
1971 54.6
1961 43.8
1951 35.7
Year
Population (in crores)
Year
Population (in crores)
(a) Alphabetical order (names of the countries)
CountryAmerica Brazil China Denmark France India
Yield of .
wheat in . ' 2,25 439 862
kg (Per acre) 1925 127 893
(b) Descending order (figures of the second row above) ^
Country " America China India France Denmark Brazil
Yield of
wheat in ^^^ ^^^ ^^^ 225 127
Note 4«rLes of Ae conntrTes do no. appear in an alphabe.,cal order in example (b).
For example.
56
Twofold classification
POPULATION
Statistics for EconomicsXI
r
Males
c o
to
2 o o E
CO
= iS « «
s u
Employed
Females
Married (1)
I
Unemployed
Employed
Unmarried Married Unmarried
(2) (3) (4)
Married (5)
i
Unemployed
Unmarried Married (6) (7)
1
Unmarrit (8)
Second and AM Iges of cLsffaLT " ""
confusion in classification Th..Q in iu^ u . , . ' cieariy dehned to avoid
we.gh, sales, t'Tts"7rf ^ ^^
formation of searisrical feries "7' classification ,s made in tl,
: "r;"
Tu 1 10 27 58 72
Thus, there are 15 workers in the income group of Rs 100 to 199 77 7 • mcome group of Rs 200299 and so on. ' '1
(b) statistical series
DEHNinON
other, the resu,. ,s s^^r^^ :""" "'
the division of the data I tt^ flTrr' "
.he collected and classified" ir^ rprl^^eJrr;:;': """""
57
Organisation of Data
STATISTICAL SERIES
Jiasis of bharacter
Z Times Series (on the basis of time)
2. Spatial Series (on the basis of space)
3. Condition ^ries (on the basis of condition)
r Series of Individual ^servation 2. Discrete Series ■ 3. Continuous Series
Frequency distribution
Series on die Basis of General Chat«c.« ^ ^^ „
ISa^r tr reference ro sonre .n,e nn..,
,ear,.onrH,weeU, or da. Por example. ^^^ ^ ^^ ^^
Sugar Production of a Factory ^^^ ^^^^ ^^ 2006)
of .he universe under srudy ^^ j,.^coun.ry, sate, cry, town.
village or colony. For example : ^^^^^ ^^ ^^^^^^^
CountryPer Capita
USA France Japan Canada India 5,100 3,900 2,800 2,100 500
City 3 »
Delhi Mumbai Chennai Kolkata Bangalore 792 649 573 532 459
58
Statistics for EconomicsXI
3. Condition series : A series of values of some variable made according to a condition is called condition series. Data are presented with reference to some condition, viz., height, age, weight, income etc. For example :
Weekly Income of 100 Workers
Income (Rs.) No. of Workers
500 999 35
10001499 25
15001999 15
20002499 20
25002999 5
Series on the Basis of Construction
After collection and classification of data it is the most important job now to construct the data in an arranged order that is the formation of series for further study of presentation, analysis and interpretation. This arrangement can be done in three ways :
{a) Series of Individual Observation, {b) Discrete Series, (c) Continuous Series.
{b) and (c) are with reference to frequency distribution.
Series of Individual Observation
Mass data in its original form is called raw data or unorganised data which can be arranged in any of the following ways :
(/■) Serial order of alphabetical order, (ii) Ascending order, {Hi) Descending order.
The mass data when put in ascending or descending order of magnitude is called an array. A series of individual observations is a series where items are listed singly after collection. They are not listed in groups.
Suppose an investigator has obtained the following information from a factory about the payment of daily wages of 30 workers, which is in unorganised form (Raw Data) as shown in Table 1.
TABLE 1
Daily Wages Paid to Workers (in Rupees)
60 102 61 101 92 80
87 72 86 73 96 101
92 56 90 58 85 74
83 63 84 62 92 100
56 84 90 86 67 72
Organisation of Data
TYPES OF SERIES
Statistical series can be clas^i^ed in the following way :
STATISTICAL SERIES
57
^J^sls of "Character
1. Times Series (on the basis of time)
2. Spatial Series (on the basis of space)
3. Condition Series
(on the basis of condition)
1. Series of Individual observation
 2. Discrete Series
 3. Continuous Series
Frequency distribution
Series on the Basis of General Character
1 Time series. A series of values of some variable according to successive points in time is called time series. Data are presented with reference to some time unit, viz., year, month, week, or day. For example :
Sale in Super Bazar
Sugar Production of a Factory
Year Production
(in WO tons)
1999 78
2000 75
2001 94
2002 86
2003 89
2004 92
2005 95
(1st week of Jan. 2006)
Day Sale
(Rs)
Men. 1,892
Tues. 2,757
Wednes. 3,090
Thurs. 2,650
Fri. 2,592 ■
Satur. 3,822
2 Spatial series. A series of values of some variable according to geographical division of the universe under study is called a spatial series or geographical series. Data are presented with reference to some geographical division, viz., country, sate, city, town.
village or colony. For example :
Per Capita Income
Number of Schools
CountryPer Capi^
USA 5,100
France 3,900
Japan 2,800
Canada 2,100
India 500
City No. of Schools
Delhi 792
Mumbai 649
Chennai 573
Kolkata 532
Bangalore 459
58
Statistics for EconomicsXI j
3. Condition series : A series of values of some variable made according to a condition IS called condition series. Data are presented with reference to some condition, viz., height, age, weight, income etc. For example :
Weekly Income of 100 Workers
Income (Rs.) No. of Workers
500999 35
10001499 25
15001999 15
20002499 20
25002999 5
i;
Series on the Basis of Construction
After collection and classification of data it is the most important job now to construct the data in an arranged order that is the formation of series for further study of presentation, analysis and interpretation. This arrangement can be done in three ways :
(a) Series of Individual Observation, (b) Discrete Series, (c) Continuous Series.
{b) and (c) are with reference to frequency distribution.
Series of Individual Observation
Mass data in its original form is called raw data or unorganised data which can be arranged in any of the following ways :
(/•) Serial order of alphabetical order, (ii) Ascending order, (Hi) Descending order.
The mass data when put in ascending or descending order of magnitude is called an array. A series of individual observations is a series where items are listed singly after collection. They are not listed in groups.
Suppose an investigator has obtained the following information from a factory about the payment of daily wages of 30 workers, which is in unorganised form (Raw Data) as shown in Table 1.
TABLE 1
Daily Wages Paid to Workers (in Rupees)
60 102 61 101 92 80
87 72 86 73 96 101
92 56 90 58 85 74
83 63 84 62 92 100
56 84 90 86 67 72
Organisation of Data
59
The above raw data can be arranged either in serial order (Table 2) or ascending order (Table 3) or descending order (Table 4) as given below :
TABLE 2 Arranged in Serial Order
5 No. ^^ y .""  V (Rs) S.No. Wages (Rs, S. No. Wages IRS,
1 60 11 61 21 92
2 87 12 86 22 96
3 92 13 90 23 85
4 83 14 84 24 92 ■
5 56 15 90 25 67
6 102 16 101 26 80
7 72 ■ 17 73 27 101
8 56 18 58 28 74
9 63 19 62 29 100
10 84 20 86 30 72
TABLE 3 Arranged in Ascending Order (Wages in Rupees)
56 62 73 84 90 96
56 63 74 85 90 100
58 67 80 86 92 101
60 72 83 86 92 101
61 72 84 87 92 102
TABLE 4 Arranged in Descending Order (Wages in Rupees)
102 92 87 84 72 61
101 92 86 83 72 60
101 92 86 80 67 58
100 90 85 74 63 56
96  90 84 73 62 56
FREQUENCY DISTRffiUTION
Before discussing anything about frequency distribution it is advisable to know the following important terms of frequency distribution under which the two types of distributions are grouped. The two types are :
(a) Discrete Frequency Distribution.
(b) Continuous Frequency Distribution.
60
Statistics for EconomicsXI
Terminology of Frequency Distribution
Examine the following two sets of illustrations to clearly understand the basic termmology of frequency distribution. ^
Set I. Children in Families t
Children No. of families (f)
0 1 2 3 4 25 45 37 15 8
Total 130
Height in Inches No. of Students (f)
5658 12
5860 16
6062
6264 4
6466 10
Total 57
diffe^r^t' i' arrangement of items mto a particular order or sequence m
sfuZZ '' ' " Height "f
Frequency : The number of times given value in an observation appears is the frequencv
four chiljen ^ 10 students m the group of 64" to 66" and 16 students m group of 58 to 60 etc. So the frequency of famihes having no child is 25, frequency of families havmg 4 children is 8; frequency of students m the group of 6^" to 66•^s 10 and frequency m the group of 58" to 60" is 16.
class^'lrau^nr"'^ ^ Tu^"/^ ^^^ quantitative classes is called the
class frequency^.^., out of the five classes of Set II students in a group of 58" to 60"
to 60 IS 16 and of 62" to 64" is 4. There is no instance of a class in Set I
"^SiT't^TlT ^ J ^^  the total frequency,
e.g., the total 130 and 57 in our set I and set II.
cJJaT^^ distribution : The distribution of observations over the several values is
n ^^ Set n T" ' ^^ children
m tamilies, and Set II is the frequency distribution of heights of students.
Class. It is a decided group of magnitudes, e.g., 56"58", 100200, 1019 48
form fhe b'ndar^T I = ^^^ magnitudes, which
boundaries of a d^ss, are known as the upper and lower limits, respectivdy For
clTn'^f Set^f^^^^^^^^ '' " '^T " upper.limit.'Thu's m thi fir"
andTht hf H / f magnitudes 56, 58, 60, 62, and 64 are the lower limits
respec^v classes ''' ^^ ^^^ ^^^ ^mits of their
Organisation of Data ^^
The above raw data can be arranged either in serial order (Table 2) or ascending order (Table 3) or descending order (Table 4) as given below :
TABLE 2 Arranged in Serial Order
S. No. Wages S. No. Wages S. No. Wages
(Rs) (Rs) (Rs)
1 60 11 61 21 92
2 87 12 86 22 96
3 92 13 90 23 85
4 83 14 84 24 . 92
5 56 15 90 25 67
6 102 16 101 26 80
7 72 • 17 73 27 101
8 56 18 58 28 74
9 63 19 62 29 100
10 84 20 86 30 72
TABLE 3 Arranged in Ascending Order (Wages in Rupees)
56 62 73 84 ' 90 96
56 63 74 85 90 100
58 67 80 86 92 101
60 72 83 86 92 101
61 72 84 87 92 102
TABLE 4
Arranged in Descending Order
(Wages in Rupees)
102 92 87 84 72 61
101 92 86 83 72 60
101 92 86 80 67 58
100 90 85 74 63 56
96  90 84 73 62 56
FREQUENCY DISTRIBUTION
Before discussing anything about frequency distribution it is advisable to know the following important terms of frequency distribution under which the two types of distributions are grouped. The two types are : .
(a) Discrete Frequency Distribution.
(b) Continuous Frequency Distribution.
60
Statistics for EconomicsXI Terminology of Frequency Distribution
Examine the following two sets of illustrations to clearly understand the basic terminology of frequency distribution.
Set I. Children in Families
Children No. of families (f)
0 25
1 45
2 37
3 15
4 8
Total 130
Height in InchesNo. of Students (f)
5658 12
5860 16
6062 15
6264 4
6466 10
Total 57
Series is a systematic arrangement of items into a particular order or sequence in ffe. different classified categories, as Set I for Children in Families and Set II for Height of
Students.
Frequency: The number of times given value in an observation appears is the frequency For example, in the above sets there are 25 families having no child and 8 families having four children; and 10 students in the group of 64" to 66" and 16 students in group of 58" to 60" etc. So the frequency of families having no child is 25, frequency of families having 4 children is 8; frequency of students in the group of 64" to 66" is 10, and frequency in the group of 58" to 60" is 16.
Class frequency : The number of values in each of the quantitative classes is called the class frequency, e.g., out of the five classes of Set II students in a group of 58" to 60" are 16 and students in a group of 62" p 64" are 4, so the class frequency of the class 58" to 60" is 16 and of 62" to 64" is 4. There is no instance of a class in Set I.
Total frequency : The sum (total) of the frequencies is known as the total frequency, e.g., the total 130 and 57 in our set I and set II.
Frequency distribution : The distribution of observations over the several values is called frequency distribution. For example. Set I is the frequency distribution of children m families, and Set II is the frequency distribution of heights of students.
Class. It is a decided group of magnitudes, e.g., 56"58", 100200, 1019, 48, 713 etc.
Upper and lower limits of the classes : The lowest and the highest magnitudes, which form the boundaries of a class, are known as the upper and lower limits, respectively For 1 example, for a class of 6264, 62 is lower limit and 64 is upper limit. Thus in the first column of Set II, left hand side magnitudes 56, 58, 60, 62, and 64 are the lower limits! and right hand side magnitudes 58, 60, 62, 64 and 66 are the upper limits of their] respective classes.
Organisation of Data 6\
Cias mterval : The magnitude spread between the lower and upper class limits is called class interval. It is the span or width of a class which can be obtained by finding the difference between the upper and lower limits of the class. For example, for class 64"66". the class interval is upper limit (l^)  lower limit (/j), i.e., (l^  l^) = 6664 = 2. The class interval in this case is 2, l^ is the lower limit and is the upper limit.
Midpoint : The midvalue which lies half way between the lower and upper class limits is known as midpoint. Thus, in a class of 62 "64" the midpoint is
upper class limit + lower class limit 2
or
/2+/1 64+62
= 63
2 2
Calculated midpoints are the most important values, as being the representatives of the classes, and are taken for use in further statistical calculations.
Variable : A quantity which varies from one individual to another is known as a variable or variate. Quantitative characteristics such as income, height, weight, number of units sold etc., are variables. A variable may be either discrete or continuous.
Discrete and Continuous Variables
Discrete and Discontinuous Variables are those which are exaci or finite and are not normally fractions. They cannot manifest every conceivable fractional value, but appear by limited gradations. For example, children in a family can be either 2 or 3, but cannot be 2.2, 2.8 or 2.7. It is a descrete variable which is not expressed in a fraction. In the same way test scores of a cricket match, rooms in a house, workers in a factory, fans installed in an auditorium, students in a class are all the examples of discrete variables. The occurrence of the observation will be integers, i.e., 1, 2, 3, 4, 5, 6, ... and so on. Thus the variable is said to be of a discrete type when there are gaps between one value and the next. For example, in set I; 0, 1, 2, 3, 4 are discrete variables.
Even fractional values are discrete or discontinuous variables provided there is an uniform difference from one variable to the other variable. For example, if wage rate per unit is 50 paise then workers of a factory may get wages in rupees as : 0.50, 1, 1.50, 2, 2.50, 3, 3.50, and so on.
Continuous variables are those that one in units of measurement which can be broken down into infinite gradations, e.g., weights, heights, incomes, rainfall etc. They are capable of manifesting every conceivable fractional value {i.e., in decimals) within the range of possibilities. They fall in any numerical value within a certain range. For instance covering a distance on a road, by car, say from 0 kilometre to 5 kilometres one never jumps from 0 to 1 km, 1 to 2 km, or 2 to 3 km, but every fraction of distance from 0 km to 5 km is touched. In other words, the car must pass through'all the infinitely small gradations of distance between 0 km to 5 km. All the fractional values are continuous variables. Heights of students from 56" to 58" in our set II for example, cover all the fractional values falling within the limit of 56 and 58.
I I
62
Statistics for EconomicsXI j
(a) Discrete frequency distribution, and
(b) Continuous frequency distribution.
Conslruclion of Discrete Frequency Distribution
1. Prepare a table with three columnsfirst for variable under study, second for 'Tallv
magnitude to another may preferably be tL satne®
column
^^^lustration .. Prepare frequency table of ages of 25 students of XI Class in your Solution.
Frequency Distribution of Ages of 25 Students
Age
15
16
17
18 19
Tally bars Total (/)
mill 7
mi nil . 9
mi 5
III 3
1 1
Total 4 25
Organisation of Data
Class interval : The magnitude spread between the lower and upper class limits is called class interval. It is the span or width of a class which can be obtained by finding the difference between the upper and lower limits of the class. For example, .or c^ass 64"66". the class interval is upper limit (Z^)  lower limit (/,), i.e., (l^/,)  6t,b4  2. The class interval in this case is 2, is the lower limit and is theupper limit.
Midpoint : The midvalue which lies half way between the lower and upper class limits is known as midpoint. Thus, in a class of 62"64" the midpoint is
upper class limit + lower class limit 2
or
l2±k 2
64+62
= 63
Calculated midpoints are the most importam values, as being the representatives of the classes, and are taken for use in further statistical calculations.
Variable : A quantity which varies from one individual to another is known as a variable or variate. Quantitative characteristics such as income, height, weight, number of units sold etc., are variables. A variable may be either discrete or continuous.
Discrete and Continuous Variables
Discrete and Discontinuous Variables are those which are exac. or finite and are not normally fractions. They cannot manifest every conceivable fractional value, but appear by limited gradations. For example, children in a family can be either 2 or 3, but cannot be 2 2 2 8 or 2 7. It is a descrete variable which is not expressed in a traction. In the same way test scores of a cricket match, rooms in a house, workers in a factory, fans mstaUed in an auditorium, students in a class are all the examples of discrete variables. The occurrence of the observation will be integers, i.e., 1, 2, 3, 4, 5, 6, ... and so on. ihus the variable is said to be of a discrete type when there are gaps between one value and
the next. For example, in set I; 0, 1, 2, 3, 4 are discrete variables.
Even fractional values are discrete or discontinuous variables provided there is an uniform difference from one variable to the other variable. For example, if wage rate per unit is 50 paise then workers of a factory may get wages in rupees as : 0.50, 1, 1.5U, Z,
2.50, 3, 3.50, and so on.
Continuous variables are those that one in units of measurement which can be broken down into infinite gradations, e.g., weights, heights, incomes, rainfall etc. They are capable of manifesting every conceivable fractional value {i.e., in decimals) within the range ot possibilities. They fall in any numerical value within a certain range. For mstance covering a distance on a road, by car, say from 0 kilometre to 5 kilometres one never jumps from 0 to 1 km, 1 to 2 km, or 2 to 3 km, but every fraction of distance from 0 km to 5 km is touched. In other words, the car must pass through'all the infinitely small gradations of distance between 0 km to 5 km. All the fractional values are continuous variables Heights of students from 56" to 58" in our set II for example, cover all the fractional values falling within the limit of 56 and 58.
62
Statistics for EconomicsXI j
Discrete series : Any series represented by discrete variahdes is called a discrete series e.g.. Set I of the distribution of children in families is a discrete series.
Continuous series : Any series described by continuous variables is called continuous series, e.g.. Set II of the distribution of heights of students is a continuous series.
It is to be noted that a discrete variable series can be presented in a continuous type of series also, but continuous variables cannot be presented in a discrete series. Whenever the range of values in a discrete series is too wide, one can have the choice of a continuous frequency distribution.
Considering discrete and continuous series, now individual observations can be constructed and condensed in two ways :
(a) Discrete frequency distribution, and
(b) Continuous frequency distribution.
Construction of Discrete Frequency Distribution
1. Prepare a table with three columns—first for variable under study, second for 'Tally bars' and the third for the total, representing corresponding frequency to each value or size of the variable.
2. Place all the values of the variables in the first column in ascending orderbeginning with the lowest and giving to the highest. The gap between one magnitude to another may preferably be the same.
3. Put bars (vertical lines) in front of the values accordingly in the second column keeping in view the number of items a particular value repeats itself. This column IS for facility in counting. Blocks of five bars or mi or W are prepared and some space IS left between each block of bars.
4. Count the number of bars in respect of each value in the variable and place it in the third column made for total or frequency.
Illustration 1. Prepare frequency table of ages of 25 students of XI Class in your school. .
Va 16, 15, 16, 16, 15, 17, 17,
lo, ly, 16, 15
Solution.
Frequency Distribution of Ages of 25 Students
Age Tally hens ■ Total if)
15 mill 7
16 mi nil . 9
17 m^ 5
18 III 3
19 1 1
Total 4 25
63
Organisation of Data
Illustratxon 2. In a aty 45 famUies were surveyed for the number of domestic apphances
they used. Prepare a frequency array ^ 3
2 2 2 2 1 2. 1 2 2  ^ ^ 3
3 3 2 4
Solution.
2 2
3 7
2 4
2 2
based on their replies as recorded below.
2 1 2 2.3 3 3 6 l" 6 2 1 5 1 5 4 3 4 2 0 3 1 4
Frequency Array of Domestic Appliances Used by 45 FamiUes
Number of Appliances
0 1 2
3
4
5
6 7
Tally bars
mill
mi M M miM II M
Total
1 7 15 12 5 2 2 1
45
I »
appliances.
Construction of Continuous Frequency Distribution
"ations are divided mto groups havmg class mtervals. There are two methods of
classifying the data according to class intervals. 
(^i) Inclusive Method, and
is iiiciuuJ <: QQ 1014 9 1519.9 and so on.
uppe. .«s „e e^cludea. "He ''' orrinterva, is the lower o^ f ^ cfass^
second, i.e., 10 to 15.
64 Statistics for EconomicsXI
Sometimes lower limits are excluded from their respective classes. For example, if the students' obtained marks are grouped as 510, 1015, 1520, 2025, 2530 etc., we include in the first group the students whose marks are above 5 and up to 10. If the marks of a student are 10, he is included in the first group. But if a student gets 5 marks, we will have to prepare a group 05 to include.
There are various methods by which class intervals can be designated. They are: {a) By Inclusive method :
1014, 1519, 2024, 2529
1014.99, 1519.99, 2024.99, 2529.99
Marks : 59,
or Prices in (Rs) : 59.99, {b) By Exclusive method : (i) Lower limit excluded :
Marks : 510,
1520,
2025,
2530
These are to be
These are to be
1015,
Lower limits 5, 10, 15, 20, 25, 30 of their respective groups are excluded. («) Upper limit excluded :
Marks : 510, 1015, 1520, 2025, 2530
Upper limits 10, 15, 20, 25, 30 of their respective groups are excluded. However, if the class intervals are given as 510, 1015, 1520, 2025 etc., it is always presumed that upper limits are excluded in absence of any specific instructions.
(c) By mentioning lower limits (followed by a dash) : Marks : 5 10, 15, 20, 25,
read as 510, 1015, 1520, 2025 and 2530.
(d) By mentioning upper limits (preceded by a dash): Marks : 10, 15, 20, 25, 30.
read as 510, 1015, 1520, 2025 2530.
(e) By midpoints of class inter\'al :
Marks: 7.5 12.5 17.5 22.5 27.5
These midpoints are required to be converted into class intervals. Say for first midpoint (12.57.5) and divide the difference by 2, i.e., (5/2). The quotient is added and subtracted to first midpoint we get, (7.52.5 = 5) and (7.5 + 2.5 = 10). We get thus the class interval 510. In the same way intervals of all the midpoints can be obtained, i.e., 1015, 1520, 2025, 2530.
(f) 'Openend' class intervals
In certain frequency distributions 'openend' class intervals are given as we find in the example given below :
Marks Below 10 1015 1520 2025 2530 3035 35 and above Total
Frequency (f) 7 10 13 18 8 5 3 64
In such cases, values are put on the basis of construction of series. In the above series '5' in place of 'below' and '40' in place of 'above' may be put. Thus making the classes as : Marks 010 1015 1520 2025 2530 3035 35^0
Organisation of Data 63
Illustration 2. In a city 45 families were surveyed for the number of domestic appliances they used. Prepare a frequency array based on their replies as recorded below.
13 2 2 2 2 1 2.1 2 2.3 3 ^ 3 33 2 3 2 2 6 1 6 215 1 5 3 242 7 42434 2 03143
Solution.
I
Frequency Array of Domestic Appliances Used by 45 Famihes
Number of Appliances Tally bars Frequency (f)
0 1 1
1 Mil 7
2 MMM 15
3 MMii 12
4 M 5
5 II 2
6 II 2
7 1 1
Total 45
Thus, from the above table it is clear that out of 45 families 1 is not using any domestic appliance, 7 using 1 appliance, 15 using 2 appliances, 12 using 3 appliances, 5 using 4 appliances, 2 using 5, 2 using 6 appliances and only 1 family using 7 domestic appliances.
Construction of Continuous Frequency Distribution
Observations are divided into groups having class intervals. There are two methods of classifying the data according to class intervals.
[a) Inclusive Method, and
[b) Exclusive Method.
(a) Inclusive Method : Under this method upper class limits of classes are included in respective classes. For example, if the students obtained marks are grouped as 59, 1014, 1519, 2024, 2529 etc., in the group 59, we include in first group students whose marks are between 5 and 9. If the marks of a student are 10 he is included in the next class, i.e., 10 to 14. If there are no whole numbers, the classes can be made 59.9., 1014.9, 1519.9 and so on.
(b) Exclusive method : Under this method upper limits are excluded. The upper limit of class interval is the lower limit of the next class. For example, if the marks obtained by the students are grouped as 510, 1015, 1520, 2025, 2530 etc., we include in first group of students whose marks are 5 or more but under 10. If the marks of a students are 10 he is not included in the first group but in the second, i.e., 10 to 15.
78 Statistics for EconomicsXI
Sometimes lower limits are excluded from their respective classes. For example, if the students obtamed marks are grouped as 510, 1015, 1520, 2025, 2530 etc we mclude m the first group the students whose marks are above 5 and up to 10 If the marks of a student are 10, he is included in the first group. But if a student gets 5 marks, we will have to prepare a group 05 to include.
There are various methods by which class intervals can be designated. They are: (a) By Inclusive method :
1014, 1519, 2024, 2529
1014.99, 1519.99, 2024.99, 2529.99
Marks : S9,
or Prices in (Rs) : 59.99, (&) By Exclusive method : (/) Lower limit excluded :
Marks : 5_io,
2530
These are to be
These are to be
1015, 1520, 2025,
Lower limits 5, 10, 15, 20, 25, 30 of their respective groups are excluded. (ii) Upper limit excluded :
510, 1015, 1520, 2025, 2530
Upper limits 10, 15, 20, 25, 30 of their respective groups are excluded. However, if the class intervals are given as 510, 1015, 1520, 2025 etc it is always presumed that upper limits are excluded in absence of any specific instructions, (c) By mentioning lower limits (followed by a dash) : ^'^'ks : 5, 10 15
20 25
readas 510, 1015, 1520, 2025 and 2530.
{d) By mentioning upper limits (preceded by a dash): Marks : 10, 15, 20, 25, 30.
read as 510, 1015, 1520, 2025 2530.
(e) By midpoints of class interval :
Marks : 7.5 12.5 17.5 22.5 27.5
These midpoints are required to be converted into class intervals. Say for first midpoint (12 57.5) and divide the difference by 2, (J/2). The quotiem is added and subtracted to first midpoint we get, (7.52.5 = 5) and (7.5 + 2.5 = 10). We get thus the
class '"t^al 5 10 In jhe same way intervals of all the midpoints can be obtained, lU 15, 15—20, 20—25, 25—30.
(f) 'Openend' class intervals
In certain frequency distributions 'openend' class intervals are given as we find in the example given below :
Marks Below 10 1015 1520 2025 2530 3035 35 and above Total
Frequency (f) 7 10 13 18 8 5 3 64
In such cases values are put on the basis of construction of series. In the above series 5 m place of below and '40' in place of 'above' may be put. Thus making the classes as • Marks 010 1015 1520 2025 2530 3035 3540
ganisation of Data
65
ciples of Grouping
There is no hard and fast rule for grouping the data, but following general principles ay be kept in mind for satisfactory and meaningful classification of data :
[a) It is advisable to have total number of classes between 5 and 15. The preference for the total number of classes depends on the numbers and figures to be grouped, the magnitude of the figure and possibility of simplified calculations of further statistical studies.
[b) Odd figures for example 3, 7, 9, 11, 27, 33 etc. should be avoided for class intervals. The choice for the class intervals should be either 5 or a multiple of 5. It simplifies our further statistical calculations.
[c) Lower limit of the class as far as possible, should be 0 or a multiple of 5.
[d) For maintaining continuity and correct classes exclusive method of preparing classes is adopted.
[e) The class interval should be equal for all classes.
(/) As far as possible openend classes should be avoided. For example,
Marks Below 5 510 1015 1520 Above 20
The first and the last classes are openend classes; the first is open at the lowerend and last at the upper end. For statistical calculations the openends should be closed. Maintaining the regularity of the class intervals we can close these groups as 05 and 2025.
(g) For frequency distribution, we prepare a table having three columns—first for variables, second for 'Tally bars" and the third for the total representing corresponding frequency to each class.
Simple Series and Cumulative Series : We have seen in the above illustrations the erns of simple series of discrete type and continuous type (Using inclusive and exclusive [lods of class intervals). In simple series the frequency is shown against each value or in cumulative series the frequencies are progressively totalled. See the following tration :
Simple Series
r . Disi^^ti'^pe ; V  ■ Continuous Type
I Marks No. of Students Marks No. of Students
10 4 010 4
20 8 1020 8
30 15 2030 15
40 20 3040 20
50 13 4050 13
66
Statistics for Economic
Cumulative Series
Less than
Marks
Less than 10
Less than 20
Less than 30
Less than 40
Less than 50
No. of Students {c.f}
12 (4 + 8) 27 (4 + 8 + 15) 47 (4 + 8 + 15 + 20) 60 (4 + 8 + 15 + 20 + 13)
More than
Marks i No. of Students (i
More than 0 i 60
More than 10 56 (60^)
More than 20 48 (6012)
More than 30 33 (6027)
More than 40 13 (6047)
Now, we can read students getting less than 10 marks are 4, less than 20 marks 12, less than 30 marks are 27 and so on.
In the same way the students getting more than 0 mark are 60, more than 10 mai are 56, more than 20 marks are 48 and so on.
Illustration 3. From the following table given below of monthly household expenditi (m Rs) on food of 50 households;
(a) Obtain the range of monthly household expenditure on food.
(b) Divide the range into appropriate number of class intervals and obtain the frequei distribution of expenditure.
(c) Find the number of households whose monthly expenditure on food is (/■) less than Rs 2000 (ii) more than Rs 3000
Monthly Household Expenditure in (Rupees) on Food of 50 Households
(c
(e (;
1904 1559 3473 1735 2760
2041 1612 1753 1855 4439
5090 1085 1823 2346 1523
1211 1360 1110 2152 1183
1218 1315 1105 2628 2712
4228 1812 1264 1183 1171
1007 1180 1953 1137 2048
2025 1583 1324 2621 3676
1397 1832 1962 2177 2575
1293 1365 1146 3222 1396
(g)
Sit )atten nethoi lass, j (lustra
Solution.
(a) Finding the highest and lowest expenditure on food of 50 households to get range by the following formula.
Range = L  S
■ rj^ . 65
msatton of Uata
.viples of Grouping
There is no hard and fast rule for grouping the data, but following general principles ly be kept in mind for satisfactory and meaningful classification of data : la) It is advisable to have total number of classes between 5 and 15. The preference for the total number of classes depends on the numbers and figures to be grouped, the magnitude of the figure and possibility of simplified calculations of further
statistical studies. _ . , , r ,
(b) Odd figures for example 3, 7, 9, 11, 27, 33 etc. should be avoided for class intervals. The choice for the class intervals should be either 5 or a multiple ot 5. It simplifies our further statistical calculations. ic) Lower limit of the class as far as possible, should be 0 or a multiple of 5.
(d) For maintaining continuity and correct classes exclusive method of preparing classes is adopted.
(e) The class interval should be equal for all classes.
(f) As far as possible openend classes should be avoided. For example,
Marks Below 5 510 1015 1520 Above 20
The first and the last classes are openend classes; the first is open at the lowerend and last at the upper end. For statistical calculations the openends should be closed. Maintaining the regularity of the class intervals we can close these groups
as 05 and 2025. , r f
(P) For frequency distribution, we prepare a table having three columns—first tor variables, second for 'Tally bars" and the third for the total representing
corresponding frequency to each class. Simple Series and Cumulative Series : We have seen in the above illustrations the terns of simple series of discrete type and continuous type (Using inclusive and exclusive lods of class intervals). In simple series the frequency is shown against each value or I, in cumulative series the frequencies are progressively totalled. See the following
ration :
Simple Series
■ Discrete Type Continue
% Marks No. of Students Marks No. of Students
10 20 30 40 50 4 8 15 20 13 010 1020 2030 3040 4050 4 8 15 20 13
66
Statistics for EconomicsXI j
Cumulative Series
Less than
Marks
Less than Less than Less than Less than Less than
10 20 30 40 50
No. of Students {c.f.}
12 (4 + 8) 27 (4 + 8 + 15) 47 (4 + 8 + 15 + 20) 60 (4 + 8 + 15 + 20 + 13)
More than
Marks
More than 0
More than 10
More than 20
More than 30
More than 40
i No. of Students {t \ 60
56 (604)
48 (6012)
33 (6027)
13 (6047)
Now we can read students getting less than 10 marks are 4, less than 20 marks 12, less than 30 marks are 27 and so on.
In the same way the students getting more than 0 mark are 60, more than 10 ma are 56, more than 20 marks are 48 and so on.
Illustration 3. From the following table given below of monthly household expendit (m Rs) on food of 50 households;
(a) Obtain the range of monthly household expenditure on food.
(b) Divide the range into appropriate number of class intervals and obtain the frequeu distribution of expenditure.
(c) Find the number of households whose monthly expenditure on food is (/■) less than Rs 2000 (ii) more than Rs 3000
Monthly Household Expenditure in (Rupees) on Food of 50 Households
1904 1559 3473 1735 2760
2041 1612 1753 1855 4439
5090 1085 1823 2346 1523
1211 1360 1110 2152 1183
1218 1315 1105 2628 2712
4228 1812 1264 1183 1171
1007 1180 1953 1137 2048
2025 1583 1324 2621 3676
1397 1832 1962 2177 2575
1293 1365 1146 3222 1396
Solution.
(a) Finding the highest and lowest expenditure on food of 50 households to get I range by the following formula.
Range = L  S
'^ganisation of Data
where, L = Longest value, and S = Smallest value
Here, L = 5090 and S = 1007
Range = 5090  1007 = Rs 4083 (b) Dividing the class interval of Rs 500, we get
4083
67
500
= 8.166
Now, we decide 9 classes to include all the given values preparing a continuous frequency distribution by exclusive method (excluding upper limit).
Frequency (Excluding upper limit)
Household Expenditure (Rs) Tally bars Frequency (f)
10001500 miMTHiTHl 20
15002000 MM III 13
20002500 Ml 6
25003000 M 5
30003500 II 2
35004000 1 1
40004500 II 2
45005000 0
50005500 1" 1
Total 50
(c) (i) Number of households whose monthly expenditure is less than Rs 2000 (i.e., 1000  2000)
= 20 + 13 = 33 Households (ii) Number of households whose monthly expenditure is more than 3000 (i.e., 3000  5500)
= 2 + 1+ 2 + 0 + 1 = 6 Households 1. Illustration 4. Form a frequency distribution from the following data by inclusive rthod taking 4 as the magnitude of class intervals taking the lowest class as (10  13). obtain class boundries and midvalues.
31 23 19 29 22 20 16 10 13 34
38 33 28 21 15 18 36 24 18 15
12 30 27 23 20 17 14 32 26 25
18 29 24 19 16 11 22 15 17 10
68
Solution.
Statistics for EconomicsM O
Frequency Distribution
Class interval
1013 1417 1821 2225 2629 3033 3437 3841
Tally bars
mini mini mill mi
Frequency (f)
8 7 5 4 2 1
Total
40
Oass Boundaries
In above illustration 1013 1417 io 91 ^c of inclusive method of construction of coin nor f ' ^^^ I™
or discontinuity between upper uZ S Xs ' w I ^e find 'gap
1 between the upper limit of the fct dasfia and rZT ^^
we find a ';gap' of 1. The contmuity of tie vaLtk ^^^^^^^
adjustment m the class interval. classified data is obtained
Steps
""""" class and ehe „pp„
14  13 = 1 2. Divide the difference by 2
1
Midpoint =
2
(c)
m.v. =
li+h
nil
[ methoc Also ol
31 38 12 18
Organisation of Data
where, L = Longest value, and S = Smallest value
Here,' L = 5090 and S = 1007
' Range = 5090  1007 = Rs 4083
(b) Dividing the class interval of Rs 500, we get
4083
67
500
= 8.166
Now, we decide 9 classes to include all the given values preparing a continuous frequency distribution by exclusive method (excluding upper limit).
Frequency (Excluding upper limit)
Household Expenditure (Rs) Tally bars Frequency (f)
10001500 MM MM 20
15002000 MM III 13
20002500 Ml 6
25003000 M 5
30003500 II 2
35004000 1 1
40004500 II 2 i „
45005000 1 0
50005500 1" 1
Total 50
(c) (i) Number of households whose monthly expenditure is less than Rs 2000 [i.e., 1000  2000)
= 20 + 13 = 33 Households («■) Number of households whose monthly expenditure is more than 3000 (i.e., 3000  5500)
= 2 + 1+ 2 + 0 + 1 = 6 Households Illustration 4. Form a frequency distribution from the following data by inclusive bod taking 4 as the magnitude of class intervals taking the lowest class as (10  13). obtain class boundries and midvalues.
31 23 19 29 22 20 16 10 13 34
38 33 28 21 15 18 36 24 18 15
12 30 27 23 20 17 14 32 26 25
18 29 24 19 16 11 22 15 17 10
68
Solution.
Frequency Distribution
Statistics for EconomicsXI j
Class interval 1 Tally bars Frequency (f)
1013 M 5
1417 mini 8
1821 mini 8
2225 m^ii 7
2629 m^ 5
3033 nil i 4
3437 n 1 7
38^1 1 i i 1
Total 40
f^.Iass Boundaries
In above illustration 1013, 1417, 1821, 2225, 2629 and so on are class Um of mclusxve method of construction of contmuous frequencv distribution. We S 'Z or discontinmty between upper limit of a class and lower limit of next class Fo elS
JTnT: trotftr' f ^^^
ifL dat Ii;—^ " ^^
Steps
14  13 = 1
2. Divide the difference by 2
1
 =0.5
3. Subtract the value obtained from lower limits of all the classes ( 0 5)
4. Add the value obtained to upper limits of all classes (+ 0 5)
are itri^Ld'^"'""" " ^^ each cL
Midpoint = Upper class limit+Lower class limit
2
m.v. =
Ink 2
lelativ
It i factual 1
■ganisation of Data Now we get
69
Illustration 5. Prepare a frequency distribution by inclusive method taking class interval of 7 from tbe following data :
imits japs' nple, s 14, d by
lit of
Class ii^erval frequency (f) mid values (m.v.).
9.513.5 5 11.5
13.517.5 8 15.5
17.521.5 8 19.5
21.525.5 7 23.5
25.529.5 5 27.5
29.533.5 4 31.5
33.537.5 2 35.5
37.541.5 1 39.5
Total 40
28 17 15 22 29 21 23 27 18 12 7 2
9 4 6 1 8 3 10 5 20 16 12 8
4 33 27 21 15 9 3 36 27 18 9 2
4 6 32 31 29 18 14 13 15 11 9 7
1 5 37 32 28 26 24 20 19 25 19 20
Solution.
Frequency Distribution (Inclusive Method)
class
Tally bars Frequency (f)
07 miMmi 15
815 miMTHi15
1623 mm nil 14
2431 mimi 1 11
3239 M 5
Total 60
^tive Frequency Distribution
It is sometimes required to show the relative frequency of occurrences rather than ual number of occurrences in each class of frequency distribution. If actual frequencies I expressed as per cent of the total number of observations, relative frequencies are ained.
70
Individum
]
2
3
4
5
Money (Rs)
114 108 100 98 106
Individual
6
7
8 9
10
Money (Rs) Individual Money (Rs)
109 11 131
117 12 136
119 13 143
121 14 156
126 15 169
Individual
■"'igclllisc
frequencies. Solution.
16
17
18
19
20
182 195 207 219 235
Money (Rs)
75100 100125 125150 150175 175200 200225 225250
Frequency Distribution (Excluding upper limit)
OJ
lof 7 :
Tally bars
Frequency (f)
Mil
Total
2 7 4 2 2 2 1
Relative frequency (%)
20
10 35 20 10 10 10 5
100
Soli
(Assuming the class interval of Rs 25) of ,35. We take a ..e,
Frequency Distribution of Money
Money (Rs)
50100 100150 150200 200250
Tally bars
MMii
Total
Frequency (f)
1 12 4
3
Relative frequency (%)
5 60 20 15
dative ]
It is ! (ictual nuj exprei obtained.
anisation of Data
69
Class interval Frequency (f) mid lvalues {m.v.)
9.513.5 5 11.5
13.517.5 8 15.5
17.521.5 8 19.5
21.525.5 7 23.5
25.529.5 5 27.5
29.533.5 4 31.5
33.537.5 2 35.5
37.541.5 1 39.5
Total 40
fflustration 5. Prepare a frequency distribution by inclusive method taking class interval
28 9 4 4 1
Solution.
17 15 22 29 21 23 27 18 12 7 2
4 6 1 8 3 10 5 20 16 12 8
33 27 21 15 9 3 36 27 18 9 2
6 32 31 29 18 14 13 15 11 9 7
5 37 32 28 26 24 20 19 25 19 20
Frequency Distribution (Inclusive Method)
Class Tally bars Frequency (f)
07 mimm 15
815 miMM 15
1623 mm 111! 14
2431 MM 1 11
3239 M 5
Total 60
lative Frequency Distribution
It is sometimes required to show the relative frequency of occurrences rather than Illmber of occuLnces in each class of frequency distribution If actual frequencies ^pressed as per cent of the total number of observations, relative frequencies are
ained.
90 Statistics for EconomicsXI
Illustration 6 In a hypothetical sample of 20 individuals the amounts of money them were found to be :
Individual Money Individual Money Individual Money Individual
(Rs) (Rs) (Rs)
1 114 6 109 11 131 16 182
2 108 7 117 12 136 17 195
3 100 8 119 13 143 18 207
4 98 9 121 14 156 19 219
5 106 10 126 15 169 20 235
frequencies. Solution.
Frequency Disttibution (Excluding upper limit)
Money {Rs) Tally bars Frequency If) Relative frequency (%)
75100 II 2 10
100125 Mil 7 35
125150 nil 4 20
150175 II 2 10
175200 II 2 10
200225 II 2 10
225250 1 1 5
Total 20 100
(Assuming the class interval of Rs 25)
Frequency Distribution of Money
Money (Rs) Tally bars
50100 1 1 c
100150 150200 Mmiii nil 12 4 J 60 20 15
200250 III 3
Total 20 100
XI
vith
ganisation of Data ^^
quency Distribution with Unequal Classes
Data are sometimes given in unequal class intervals. Such series are used when there f great fluctuation in data. For example :
ative
1 Set I Set II Set III
pfass Frequency Class Frequency Class Frequency
05 X 2 X 2A X
510 Y 5 Y 26 X + Y
1020 Z 7 Z 28 X + Y + Z
2030 A 720 A 810 A
3050 B 2040 B 1012 B
5075 C 4060 C 1214 C
x)ss of Information
Raw data is grouped by making equal or unequal class frequency distribution, say 15, 510, 1015 or 05, 57, 712, 1220 and so on. By making such classes there is loss of information of individual observation. Further, the statistical analysis is based on die midpoints of these classes without giving any importance to individual observation. ^ such, the significance of individual observation is lost.
livariate Frequency Distribution
We have so far studied above frequency distributions involving single variable only, uch frequency distributions are called univariate frequency distributions. Often we come aoss data composed of measurements made on two variables for each individual items.
example, we may study the weights and heights of group of individuals, the marks .uiined by a group of students in two different subjects, ages of husbands and wives for group of couples, etc. A frequency table where two variables have been measured in the ue set of items through cross classification is known as 'bivariate frequency distribution" ntervalB 'twoway frequency distribution'. Various values of each variable are grouped into ious classes (not necessarily the same for each variable).
lUustration 7. Following figures give the ages of 20 newly married couples in year, jresent the da ! of husband t of wife {of husband ! of wife
Solution. We are given two variables : (i) age of husbands, and (ii) age of wives. We Id represent the data in the form of a twoway frequency distribution so that we are to show the ages of husbands and wives simultaneously. This is also called bivariate \cy distribution.
24 26 27 25 28 24 27 28 25 26
17 18 19 17 20 18 18 19 18 19
25 26 27 25 27 26 25 26 26 26
17 18 19 19 20 19 17 20 17 18
72
r
. I i '1 i.fV
Age of husband (years)
24
25
26
27
28
Total (/)
17
Bivariate Frequenqr Distribution , Age of wife, (years)
Statistics for EconomicsXll
I (1) III (3) I (1)
(1) (1) (3). (1)
(1) (2) (2) (1)
20
I
(1) (1) (1)
Total (
2 5 7 4 2
20
Illustration 8 Tbe data given below relate to the heights and weights of 20 nersc
66" IT class Tnterval 62^
64 66 and so on and 115 to 125 lbs., 125 to 135 lbs. and so on.
S.N.
1
2
3
4
5
6
7
8
9
10
Solution.
Height S.N. Weight Height
170 70 11 163 70
135 65 12 139 67
136 ■ 65 13 122 63
137 64 14 134 68
148 69 15 140 67
124 63 16 132 69
117 65 17 120 66
128 70 18 148 68
143 71 19 129 67
129 62 20 152 67
Height Inches ,
lbs.) 62—64 64—66 66— 6S—70 70—72 Totali
115125 125135 135145 145155 155165 165175 II (2) i (1) 1 (1) 111 (3) 1 (1) 1 (1) II (2) 1 (1) II (2) 11(2) i (1) 1 (1) 1 (1) 1 (1) 4 5 6 3 1 1 i
Total (/) 3 4 5 4 4 20
71
anisation of Data ^^uency Distribution with Unequal Classes
CData are sometimes given in unequal class intervals. Such series are used when there eat fluctuation in data. For example :
ions. 64",
iO
W Set I Setll Set III
Frequency Class Frequency Frequency
1 05 X 1 X 2A X
510 Y 5 Y 26 X + Y
1020 Z 7 Z 28 X + Y+ Z
t2030 A 720 A 810 A
,3050 B 2040 B 1012 B
J&75 C 4060 C 1214 C
a (f)
KS of Information
Raw data is grouped by making equal or unequal class frequency distribution, say 5, 510, 1015 or 05, 57, 712, 1220 and so on. By making such classes there is loss of information of individual observation. Further, the statistical analysis is based on ' midpoints of these classes without giving any importance to individual observation, such, the significance of individual observation is lost.
,jiate Frequency Distribution
We have so far studied above frequency distributions involving single variable only. 1 frequency distributions are called univariate frequency distributions. Often we come uss data composed of measurements made on two variables for each individual items. • example, we may study the weights and heights of group of individuals, the marks ained by a group of students in two different subjects, ages of husbands and wives for oup of couples, etc. A frequency table where two variables have been measured in the • set of items through cross classification is known as 'bivariate frequency distribution' i'twoway frequency distribution'. Various values of each variable are grouped into
ous classes (not necessarily the same for each variable). Inhistration 7. Following figures give the ages of 20 newly married couples in year.
24 26 27 25 28 24 27 28 25 26
17 18 19 17 20 18 18 19 18 19
25 26 27 25 27 26 25 26 26 26
17 18 19 19 20 19 17 20 17 18
(of husband t of wife ! of husband t of wife
Solution. We are given two variables : [i) age of husbands, and (ii) age of wives. We lid represent the data in the form of a twoway frequency distribution so that we are to show the ages of husbands and wives simultaneously. This is also called bivariate
icy distribution.
72
Bivariate Frequency Distribution
Statistics for Economics
(years)
24
25
26
27
28
17
Total (/)
I (1) III (3) I (1)
18
19
I (1)
I (1)
III (3)
I (1)
6
(1) (2) (2) (1)
20
(1) (1) (1)
Total i
2 5 7 4 2
______——20
You are required ro
nterval 62"
S.N.
1
2
3
4
5
6
7
8
9
10
170
135
136
137 148 124 117 128 143 129
Solution.
Height
70 65 65
64
69 63
65
70
71 62
S.N.
11 12
13
14
15
16
17
18
19
20
Weight
163
139 122 134
140 132 120 148 129 152
Height
70
67 63
68
67 69 66
68 67 67
Bivariate Frequency Distribution Inches
lOrganisation of Data
73
_:
exercises
uestions :
I Distinguish between variable and attribute. Explain with examples, i Define classification. Explain the objects and characteristics of classification. ! What do you understand by classification? Explain the methods of classification of j data giving suitable examples.
; Is there any use in classifying things? Explain with illustrations. ^ Explain discrete and continuous variables with examples. Define series and explain the different types of series. Define Frequency Distribution. State the principles required to be observed in its formation.
8. Explain with illustration the 'inclusive' and 'exclusive' methods used in classification of data.
9. Distinguish between univariate and bivariate frequency distribution.
10. Distinguish between discrete and continuous variable.
11. What is loss of information in classified data?
12. Do you agree that classified data is better than raw data?
13. What is a relative frequency distribution? Illustrate. Write short notes on the following :
I (a) Classification and series.
[b) Geographical and chronological classification.
(c) Exclusive and inclusive classintervals, i (d) Discrete and continuous series. I [e) Simple and cumulative frequency. I (/) Equal and unequal class frequency
blems :
Prepare a statistical table from the following data taking the class width as 7. by ! inclusive method :
28 17 15 22 29 21 23 27 18 12
7 2 9 4 6 1 8 3 10 5
20 16 12 8 4 33 27 21 15 9
3 36 27 18 9 2 4 6 32 31
29 18 14 13 15 11 9 7 1 5
37 32 28 26 24
74
I
i^v
Statistics for EconomicsXI j
50 57 58 51 53 62 64 60 61
51 64 55 55 52 60 65 58 60
52 63 56 56 58 64 63 62 60
54 62 54 54 60 65 60 62 59
56 63 52 53 62 53 61 61 59
the following marks in frequency table taking the lowest classinterval
69 33 91 53 63 69
70 36 80 78 52 51
73 73 92 64 55 49
74 57 95 70 64 57
75 80 42 85 43 29
77 65 73 95 76 53
86 73 40 83 43 76
84 72 75 57 58 59
62 65 67 87 81 84
61 75 85 81 58 81
4.
47 69 78 62 72 43 87 61 84 23
Change the following into continuous series and convert the series into 'less than' and more than cumulative series :
Marks (midvalues) No. of students
5.
5 15 25 35 45 55
8 12 15 9 4 2
Marks obtained by 24 students in English and Statistics in a class are given below
»
S.No. Marks in English Marks in Statistics S.No. Marks in English Marks in ] Statistics j
1 22 16 13 23 16
2 23 16 14 25 17
3 23 18 15 23 17
4 23 16 16 22 17
5 23 16 17 27 15
6 24 17 18 27 16
7 23 16 19 26 18
. 8 25 19 20 28 19
9 22 16 21 25 19
10 23 18 22 24 16
11 24 18 23 23 17
12 24 17 24 25 19
I " ^^
^ganisation of Data
tin a survev it was found that 64 famiUes bought milk in the following quantities a parSar Inth. Quantity of milk (in litres) bought by 64 famthes m a month.
.O 99 9 22 12 39 19 14 23 6 24 16 18 7
i y. p • i I i i iH i i ■■
1 Comrert the above data in a frequency distribution making classes of 59, 1014 and
J so on.  u 1
I: The marks obtained by 20 studends in Statistics and Economics are. given below.
« • • . r _____—. Vviii*i/~vn
Marksin
10 11 10 11 11 14 12 12 13 10
Marks in Economics Marks in Statiaics Marks in Eammcs
20 13 24
21 12 23
22 11 22
21 12 23
23 10 22
23 14 22
22 14 24
21 12 20
24 13 24
25 10 23
8 Prepare 'less than' and 'more than' cumulative frequency distributions of the
^ 140150 15.160 160170 170180 180190 190^200
Ino. of workers : 5 10 20 \ . / ' .
I Find out the frequency distribution and 'more than' cumulative fi^quency^^ble . below : 10 30 40 50 60
Quantity(kg) : 17 22 ^ lociqo
If class midpoints in a frequency distribution of a group of persons are : 125, 132,
139, 146, 153, 160, 167, 174, 181 pounds, find (^i) size of the class intervals, and (b) the class boundaries.
PRESENTATION OF DATA
g^^lfeftwr^ Prcssentaiion
4«nmaiic Presentation
Chapter 5
tabular presentation
' 1.'  Introduction
2. Definition and Objectives of Tabulation
3. Essentials of a Satisfactory Table
4. Parts of a Table Types of Table
J" jjji
w.
b. » ".e da. eUH.
as we,. J^tJZ riLX
There are four methods of presentation. They are : eral people.
(i) Text presentation, (it) Semitabular presentation, (ni) Tabular presentation, and (iv) Pictorial presentation.
(i) Text Presentation
\^Jabular Presentation ^^
increased from an extremely low figure of less than 2 lakhs in 195051 to over 46 lakhs in 199091. There was around tenfold increase in this sphere between 1991 and 200405 as the number of landline connections increased to 4.42 crore besides 4.5 crore mobile phones. Thus the number of telephones
stood 9.7 crore in March ?C05. With Wnifold increase in telephone connections, the teledensity [viz., the number of telephone connections per hundred persons) has increased from 3.6 in 2001 to 6.7 m 2005.
(«) SemiTabular Presentation
■ Semitabular presentation is both through tables and paragraphs, This method is not often used, but is useful when figures are required to be compared along with one or two sentences of explanation.
^ (Hi) Tabular Presentation
Tabular presentation is a systematic presentation of numerical data in columns and rows in accordance with some important features or characteristics.
(iv) Pictorial Presentation
Pictorial presentation is visual form of statistical data in diagrams and graphs.
and objectives of tabuutio» c
Systematic presentation of data is one of the most important consideration in statistical j work and it is done through the use of tables. A statistical table is an arrangement of I systematic presentation of data in columns and rows. Tabulation is the process of fpresenting in tables. Tabulation is a process and the outcome of which are statistical Itables. In brief, tabulation is a scientific process involving the presentation of classified ata in an orderly manner so as to bring out their essential features and chief iracteristics.
According to H. Secrist, "Tables are a means of recording in permanent form the alysis that is made through classification and of placing juxtaposition things that are ,nilar and should be compared". According to Tuttle, "A statistical table is the logical listing of related quantitative ta in vertical columns and horizontal rows of numbers, with sufficient explanatory and alifying words, phrases and statement in the form of titles, headings and notes to make and full meaning of the data and their origin.''
bjectives of Tabulation Statistical data arranged in a tabulated form have following important objectives: I 1. They simplify complex data and the data presented are easily understood.
2. They facilitate comparison due to proper systematic arrangement of statistical data in different columns.
78
I t
Statistics for EconomicsXI j
3. They leave a lasting impression without any confusion.
4. They facilitate computation of different statistical measures namely averat dispersion, correlation etc.
5. They present facts in minimum space and unnecessary, repetition and explanatic are avoided and required figures can be located more quickly.
6. Tabulated data makes easy for summation of various items and errors and omissions can easily be detected.
7. Tabulated data are good for references and they make it easy to present intormation on graphs and diagrams.
^^^m of a satisfactory ti^^
The following are the essentials or characteristics of a satisfactory table :
1. Attractive : A table should be attractive to draw the attention of readers. To ma
It so, care should be taken in determining its size, proportion of columns and rov writing of figures, etc.
2. Manageable size : The size of the table should be neither too big nor too sma loo much of details should not be given in a table. If the table is too large becomes confusing to the eyes and there is great difficulty in following the lir and columiis at a glance. If more details are to be given, then a number of sr tables should be preferred to one big table. So, it should be simple and comp.
3. Comparable : The facts should be arranged in a table as to make comparis. between them easy, because, comparison is one of the chief objectives of tabulatio Whenever it is necessary, average, percentage, proportion, etc., should be given the table to facihtate comparison.
4. According to objective : A table should be according to objective of statistic investigation.
^^ « easily understandable,
should be complete within itself containing all the explanations necessary to mi clear the meanmg to items. Units of measurement must be clearly stated such, price m rupees" or "weight in kilograms". Columns and rows should be numl when It is desired to facilitate reference to specific parts of a table
nSffr '' scientifically prepared. All the requir
ru es of tabulation should be carefully observed. A table should have a suitab
title, proper captions and stubs, source, footnotes etc. Certain figures which are I
be emphasised should be in distinctive type or in a 'circle' or a 'box' or ber^^
thick lines^ A table should have miscellaneous columns for the data which can«
be grouped m the classification made. Large numbers are hard to read and dif
to compare therefore, they should be approximated e.g., up to the nearest
79
Wlabular Presentation
OF ATi
A good table . an art. ^ ^ng a^tference
1 Table number : A table should ^B ^^^ etc.) or numbered (say 1,
in the future. A table must be codihed ^e A, ^ ^ ^
2, 3, 4 etc.) whenever more than or^e table ^s^ prepa ^^^^^ ^^ ^ ^^^^^ ^^^^^^^
either at
the centre on the top anove rnc^^^ Sometimes table number
the table number is given refers to the chapter or section
like 1.2 and 2.4 are also used. In ^ ^ould mean second table m
and second digit to its order. ^^ the fourth table in second first chapter or section and Table 2.4 wo
chapter or section. ^ ^^ or a catch title written 2. Title : There may be a V'^^^^Se^ be W, clekr and self explanatory,
in few words. Title given above of all lettering used m the
The lettering of the title should be ^he most pr ^^ ^^^^^ ^^^ ^^e
table. A complete title explams of classification of data.
field to which the data are ^^ed jf ^a) bas. of ^ ^^^
3. Captions and stubs (Column he^^ ^^^^ by smbs^
headings given to columns a e caUed captio ^ ^^ ^ be numbered
Both stubs and captions ^^ould be « ^^^^^^ ^^^ and stubs
rfof cSt^ons verticaUy and stubs
. ^^rtable :lt con^ms je ^^^^
part of the table. This table ^^ould be mad^^^^^^^ arrangement of items
in view the purpose of b^ excluded,
in columns and rows and : (0 alphabetically W Items in a table may be arranged ^progressively, and (vt)
geographically, (Hi) chronologically, sSnif cance which are to be
million tons. . ^ the table It is a phrase or a statement
6. F«.»o.e, : I. is placed « "om of tte ta^ ^^ ^
which contains systems and keys like puttmg star(s)
table. Footnotes can be identified by vanom sys ^^ ^^ 3 „
or signs (say £ etc.) or ^ Umitattons of data,
a, b, c, d etc.) Footnotes m also necessa.7 to sp y f
« diei is any or to explain »me , ^„rce note should be
7. Source : ta case of the ■'•j'^XalnTst ^taon, name of the pubUsher or
a°t ti:" o^.. is usefu, to the reader to
fi^ and gather additional information.
80
\'L
structure of table
Number Title
(Head note, if any)
Statistics for EconomicsXll
ifcsiEswwaiiisaMJB^ aw
——
■
— _________
Footnote : Source :
 
table 1 Literacy Rates in India
Year
Rural
1951 19.02
1961 34.30
1971 48.60
1981 49.60
1991 57.90
2001 71.40
Source : Economic
45.60 66.00 69.80 76.70 81.10 86.70
Total
27.16
40.40
45.96
56.38
64.13
75.85
Rural ' Females
4.87 urttan 22.33
10.10 40.50
15.50 48.80
21.70 56.30
30.60 64.00
46.70 73.20
Total
(Per cent)
Persons
8.86 15.35 21.97 29.76 39.29 54.16
Rural
12.10
22.50
27.90
36.00
44.70
59.40
The table clearly shows that •
per cent among females. This shows
Urban
34.59 25.40 60.20 67.20 73.10 80.30
Total
18.33
28.30
34.45
43.57
52.21
65.38
per cent in 2001, while it
that there is a general bias
Total
81
globular Presentation
against female education and in our conservative society, girls still get discriminated in the matters like health, nutrition, education, etc. iii) Literacy rate in urban areas was high at 80 per cent in 2001 than rural areas where ^ ^ ^t rs less L 60 per cent. This clearly speaks of inadequate facilities of education av^IbL in the rural areas as well as comparatively lower willingness of the conservative rural folk to go to schools for education. Illustration 1. In a sample study about coffee drinking habits m two towns, the
^tal coffee drinkers were 45% and Males noncoffee
were 55%, Males noncoffee drinkers were 30% and Females coffee drinkers were 15%.
Represent the above data in a tabular form. before
Solution. Let us calculate the missing percentages of the above information before
representing the data in a tabular form.
STOWN A 100
TOWN B 100
1
NonCoffee drinkers drinl(ers
35 40
NonCoffee drinkers drinkers
30 25
TABLE 2
Coffee Drinking Habits in Towns A and B
(in percentages)
Coffee Drinkers
NonCoffee Drinkers
82
Alternative Solution
104 Statistics for EconomicsXI
TABLE 3
Coffee Drinking Habits in Towns A and B
{in percentages]}
SBSSsSiePlSlillBJi Toum A Town B ■ ■ ■■ ^
Coffee NonCoffee Total Coffee Noncoffee Total ]
Drinkers Drinkers Drinkers Drinkers .1
Males 40 20 60 25 30 55
Females 5 35 40 15 30 45
Total 45 55 100 40 60 100
Illustration 2. Of the 1,125 students studying in a school during 20052006, 720 are Hindus, 628 are boys and 440 are science students. The number of Hindu boys is 392, that of boys studying science 205 and that of Hindu students studying science 262; finally, the number of science students among the Hindu boys was 148. Enter these frequencies in a table and complete the table by obtaining the frequencies of the remaining cells.
Solution.
TABLE 4
Faculty Boys Girls Total 1
Hindus NonHindus Total Hindus NonHindus Total Hindus NonHindus Total J
Science Arts 148 24457 179 205 423114 214121 48 235 262262 458178 227440 j 685 1
Total 392 236 628 328 169 497 720 405 1125 j
niustration 3. Census of India 2001 reported that Indian population had risen to 102 crore of which only 49 crore were females against 53 crore males. 74 crore people resid m rural India and only 28 crore lived in towns or cities. While there were 62 crore nc workers Population against 40 crore workers in the
entire country, urban population an even higher share of nonworkers (19 crore) against the workers (9 crore) as comp; to the rural population where there were 31 crore workers out of 74 crore populatic Represent the above information in a tabular form.
I Tabular Presentation Solution.
83
TABLE 5 Growth of Population in India
(figures in crores)
Source : Census of India 2001.
Males : 53 crore Females : 49 crore
fs of tables
Table can broadly by classified as under :
A. From the point of view of purpose :
(i) General purpose tables.
(ii) Special purpose tables.
B. From the point of view of originality :
{/■) Original tables. (ii) Derivative tables.
C. From the point of view of construction
(i) Simple or single tables.
(ii) Complex tables.
library
i «ble provides of » ts^s^SSllt^^^^
it eaiy to make comparisons and clear relationships.
wMcH co„.ai„ —. inro™a.o„ i. .^e «ae fom, in wUch they are origi^lly collected
sSl^ciurX^^^^^^^^ ^a .o. ...era,
purpose tables.
«J,„
SsfT
f
u
84
Simple and Complex Tables
. table 6
of Students in a School
Marks
010 1020 2030 3040
Total
15 12 28 5
60
Double or WWay Tabte (Double Tabulation)
,, , , table 7
Ii" a School
Mirt,
'^"tsZa^^'J (Tble Tabulation)
dlustration : g'^Is mto Sec A. and b in our following j
r^ • , table 8
According .o Mari.. Se. and Section
^ \ : ^ ^ ^^^^^^^^ "■" ...........
ular Presentation
85
The above table can even be called as manifold table, higher order table or ma^ 2lon Z "lh we can increase the number of charactensttcs, more sections,
" rh'aX"is needed when a number of characteristics are to be simultaneously ii But as more characteristics are included, the table becomes more complex, and
, may be confusing to the reader. If the field of investigation is not big, the data have not too many
future use and thirdly when ,he table requirements are varymg. ifabulation wiU be more accurate than the manual process.
EXERCISES
tions :
I State the advantages of Tabular Presentation of data.
i Describe the major functional parts of statistical tables. Draw a structure of a table I ExpLn bnefly the main characteristics of a good statistical
Whai are the points to be taken into accomit while preparing a table? IxpTarand discuss the various types of tables used in a survey after the data have
, ^i:^!rt;ween tabulation and classification. Explain the objects of tabulation.
^ Discuss briefly the importance of tabulation. ^ . , u, '
I What is a statistical table? Discuss briefly the essentials of a good table. ■
1 What are the objects of tabulation?.
i following industries : . . . ■
Fishing, coal mining, ironore mining, cloth and wool industries. ^
hArepare a blank table to show the distribution of population according to sex and ^ four religions in three age groups in Delhi and Mumbai. y ^ five
2006.
Statistics for EconomicsXli we^eTrL"""' op" in  »w„s following d. J
Town A 51% 16% 18%
Town B 54% 28% 20%
p Males in Total Population ' Smokers
Male Smokers ; J Tabulate the above data.
 Preset the following information in a suitable table •
oi ^ ——
Wong ro a trade un,™ 20M T "" ''' """ ""i
of which 1290 were men Sn Z h ^ u •<> 1^80
students according to : mformafon regarding the college]
(a) Faculty
(b) Class
(c) Sex id) Years
8,
Social Sciences, Commercial Sciences. Undergraduate and Postgraduate classes. Male and Female. 2005 and 2006. Tabulate the following •
JXaSSe^—^^^
of the total sales during the yeaT "^P^cfvely. Texnles accounted for 30%
pesttt^
• t
Ls
Town A
Town B
60% people were males 40% were coffee drinkers, and 26% were male coffee drinkers 55% people were males, 30% were coffee drinkers, and 20% were male coffee drinkers
Chapter 6
digrammatic presentation
Introduction
Importance and Uses of Graptis and General Rules for Constructing Diagrams Types of Diagrams
A. Onedimensional Diagrams
B. Pie Diagrams
Limitations of Diagrammatic Presentation
Other methods of presentation.
rORIAL PRESENT^
r
1
Presentation
resentation
—► ONEDIMENSIONA^DIAGRAMS
(/■) Simple Bar Diagram(i7) Subdivided Bar Diagram (i/i) Multiple Bar Diagram (/i^ Percentage Bar Diagram (v) Broken Bar Diagram (vA Deviation Bar Diagram ►TWODIMENSIONAL DIAGRAMS (f) Rectangles (fO Squares
'//A Circles and Piediagrams ►THREEDIMENSIONAL DIAGRAMS (!) Cubes (iO Cylinders (I/O Blocks etc. —► PICTOGRAM —►CARTOGRAMS OR MAPS
♦ GRAPHS OF FREQUENCY DISTRIBUTION (i) Line Frequency Graph
Histogram (Hi) Frequency Polygon (iV) Smoothed Frequency Curve (Frequency Curve)
'Ogive' or Cumulative Frequency Curve GRAPHS OF TIME SERIES (A One Variable Graphs (Ii) Two or more than two Variable Graphs (i/0 Graphs of Different Units
fr
88
r =a
conunon or representing the statisticalT^^^^^ xs the most popular an
are appeahng to the mind through the eyL as the^ of Presentatior
For the purpose of simplifying an^tter^p'Tas ''h ^^^^ W diagrams m this chapter and some iZonZ Znr
used m presenting statistical informal ^ ^ are commonly
and uses of graphs and
1. They are interesting, attractive and impressive • h. . I
fluctuations of the statistical values bv^n? ' ^^^ ^^^nd and interested in going through tiff^ure's^.e^"^ ^^^^^ ho is not ^agrams are used for publicity X^a^aX'"'
• • TW save time and energy of
say without any stram on' mmd a"d knowl^^^^^^^^^ ^^^^ ^^ey warn to
data simple and intelligible knowledge of mathematics as they make the)
quick comparisons. becomes easy Thus, diagrams can be used for ]
4. They have universal utility • Sinr^ c
presentation of graphs anj^ia^^ SLv I T'^ ^^ ^y Ae!
journals, newspapers, board meeting etc D L^T ^^ ^ " exhibitions, fairs, information to the common man TW are wideT Particularly to givj
and other fields. Diagrams play arimorttr'1 " campaigns. ^ ^ ^^ an importarn role m the modern advertising
hke median, quartiles, mode etc. Twf is dSt^^^ ^
Central Tendency of this book. ^^apter on Measures of]
whjes for constructing
I
dCf  Be acu^el through p„„.ee. Uei
general rules are observed : " advantageous if following
conveys main facts depicted by the Sa^m T^n ^^ but short.
given It must be brief, self expLato^ a^d^^^^^^ subheadings can also be
for Identification and for purpose TrefereJe ^e used
Digrammatic Presentation
89
i
. . J The «:i7e of the diagram should be neither too big nor too
paper. It should be attractive, neat and appealing to the eyes, so that peoples attention is automatically drawn towards it. . , ,
'il^ilifp
'poputoton or 'productton' on Yax>s and years' or 'months "n Xax,.
4 Scak ■ A diagram should be drawn with the help of geometric  ^
scale slould be selected to su.t far as possible be in even numbers or multiple ot 5, lU, /u, zo, luo
'months' on Xaxis. . ^u v onH Y
. A i^A^^ ■ The 'sc/j/e' of measurement on both Xaxis ana i
sl'zrrrf ^^ « f U t^aror^rVftX™ ^nJt
through different colours, shades, dotting, crossing, etc., an index must g for identifying and understanding the diagram.
the source from which data have been obtained, more effective than a complex one.
types of diagrams
There are various types of geometric forms of diagrams used in practice as shown on
following two Geometric forms of diagrams :
A. Onedimensional Diagrams
B. Pie Diagram
A ONFDIMENSIONAL DIAGRAMS
™lionaMiagrams are also called ^^^^ JthXiror:!
used in practice. They are called onedimensional because of height of the bar
90
significance and not the width of the bar Foil " u ^^ j
() Simple bar diagram ^^^ ^ypes of bar d^
(b) Subdivided bar diagram
(c) Multiple bar diagram id) Percentage bar diagram (e) Broken bar diagram (/) Deviation bar diagram (a) Simple Bar Diagrams • The
variable can be presented, A Jimple^rdlTlrc^ b'^ T
vertical base. It is used for vistil ? "" horizontal o,
production, population, sa,es,Xt e^dilr
one category either in years, months w«ks et T ^fonnation of
or groups. All the bars L be brau^LZ T u
attractive. or shading to make them more
simpfc bar diagram the scale is detet^ned'^teiTof ^
Illustration 1. Draw a b«r A:, "" " the series,
of computer softw^r " relating to expo,;
Xc^ore. ; T/oo'  2000.. ' .OOf0.
, &o„omfc Survey. 200203 p 144) Solution. ' '
export of computer software (19972002)
Scale : 1 cm = Rs 7,000 crore
Y
42000 
35000
ff 28000S O
r 21000
^ 14000
6,500
36,500
28,350
17,150
10,940
YEARS
Fig. 1
200102
"tr rrr aw———
; vertical base showing horizontal b«s as under :
Alternative solution  Vertical base. ^ „ V ^vU
w T Years on Xaxis; Value (Rupees m ^ores on Yax s W 2 • Years on Yaxis; Value (Rupees m crores) on Xaxis.
Scale : 1 cm = Rs 7,000 crores.
Export of Computer Software (19972002)
Scale : 1 cm = Rs 7,000 crores
200102
200001
199900
199899
199798 0
36,500
■ 28,350
4 17,150
10,940
. 6,500
—r
7000
14000 21000 2^00 3M00 42000 Rupees (in crores)
.. above ™o „e ui^L ^  ^^^^
i„ the export of computer ^^^^^^^ the software export have
,crores) m 199798 to Rs per year for last four years.
TwoXtl^rr;!: o"^^^^ .o^ nomlc survey .00.0. are
, , ....______Dri>>a nhanaes
given below : Poodgrains Production
(in million tons)
Wholesale Price Changes
52WeeteAverageJnflat^^
'199900 01 02 03 04
X (Provisional) Average up to Jan. 14, 2006. e First advance estimates (Khanf only).
.05 06
199900 01 02
.03 04 05 06
Fig. 3
92
ReauirPnt^^t j ' ^^^^t^stics for EconomicsX
.he guZrr/a Sir given
(h) Subdivided Bar Diaeram • TJ, j
In general subdivfdedo;!^^^^^^^ ^Component Ba
values of the given data is to be dividedTo v^no '^e to.
Ft of all a bar representing total s CwrXn Z ^^
proportion to the values given in the dat^ S/ ? , P^^s i, dotting or designs can be'used to d^sttguishosg
remember that the various componentrshouFd be t '
tndex' IS to be given alongwith the Lgram to ? " ^^^^ bar.
Illustration 2. Draw a snir.Kl ^ differences.
Draw a suitable diagram to represent the following mfo^ation :
Year
2001 2002
2003
2004
2005
Trains
Murder
108 131 97 102 75
Solution.
Robbery
82 115 144 70 68
Loot
321 386 352 285 245
Total
511 632 593 457 388
CRIME IN RUNNING PASSENGER TRAINS (20012005)
Scale : 1 cm = 200 crimes
800
600
co m
E  4001 O
a loot e robbery
■ murder
200  _ ' ^^ia
2001 pnno ^
2002 2003 YEARS
2004
jr
2005
Fig. 4
(c)M between t interrelaC of drawin; In this cai spacing isi in a set, d be given. '
93
Qigrammatic Presentation ^ifU^w^ofD^riation
S.E. Asia West Asia Africa
Other Regions Total
d,agram to represent the above data.
Ltion. Suhd.v.aea bar a,a.ta,n .s sn,.ab.e to the ahove data.
Y A
100
Q Other Regions ® Africa pfi West Asia
oE. Asia
200304
200405
YEARS
Fig. 5
94
, Statistics for Economics}
Illustration 4. Draw a suitable diagram of the following data :
Statement of CrimeJnR^g Passenger Trains
Solution.
Year Murder Robbery Loot
2001 108 82 321
2002 131 115 386
2003 97 144 352
2004 102 70 285
2005 75 68 245
CRIME IN RUNNING PASSENGER TRAINS (19982002)
(Scale ; 1 cm = 100)
500
4001
2 300H
tr o
200
loo
ses
321
352
285
B Murder ■ Robbery Q Loot
245
tons during the same fortnight last vear(ronnf TK « ? T'
during the first fortnight of DecemberToo ^ 2 ssloo^f f""
and 41,000 tons for exports as against 1 54 000 ton / . consumption
exports during the sam"! fortnigriast sfasfm^ (t) Present the data in a tabular form
(Hi) Present these data diagrammatically.
95
Digrammatic Presentation Solution.
(/) Presentation of data in a tabular form. • c i Stock
Fortnight Sugar Production, Offtake for Internal Consumption, Export arU Stock
in Sugar Mills in India. (figure in thousand tons)
December, 2000 December, 2001 , (First fortnight)
Production Offtake from Mills Export Stock 378 154 Nil 224 387 283 41 63
Source : muiau du^^i 
export and stock which we have calculated. (Hi) Diagrammatic presentation of above data by
(a) Subdivided bar diagram
(b) Multiple Bar diagram
INDIAN SUGAR MILLS ASSOCIATION REPORT
(Fortnight Sugar production, offtake for internal consumption, export and stools in Sugar Mills in India Scale : 1 cm = 50,000 tons.)
MULTIPLE BAR DIAGRAM
SUBD'VIDED BAR DIAGRAM
400
350
300
250
iction K
8,000 1 200
tories ■150
iption 1
lil for ■ 100
50
grams B0
Dec. 2000 Dec. 2001 (First fortnigh) (First fortnight)
Dec. 2000 (First fortnigh)
Dec. 2001 (First fortnight)
Fig. 13
96
Statistics for EconomicsXI j
Proceeds per Chair Factory A (Rs) Factory B (Rs)
Wages Material Other Expenses 160 120 80 200 300 150
Total Selling Price 360 400650 600
Profit or Loss (±) (+) 40 () 50
1 he percentages are calculated as under :
Ptrcentaj^^ (For Percentage Bar Diagram)
Proceeds per Chair i______ Factory A (%) Factory B (%)
j Wages 1 Material Other Expenses 40 30 20 33.3 50 25
Total Selling Price 90 100 108.3 100
Profit or I oss (+) (+) 10
OT
UJ 4
m ^ a.
3
tr
97
Digrammatic Presentation
% COST Y
chmr 3. pnofit amo loss
100
^ , , . n » PROFIT AMD LOSS
: 1 cm = 20%
factory b
factory a
(t UJ 60
Q
Z
CO 40"
UJ
lU
Q
rj 20'
cc
o:
20
M other Expenses Q Material Wages wm Profit or Loss
Fig. 8
. c .pries in which some values may Broken Bar Diagram : Sometimes we may S™^ reasonable shape
each bar is written on the ™ ° , b„ a suitable dragram.
Year Number of students
2001 2002 2003 2004 ^___ 25 48 375 125
neces^to brsn^ ^ ^
98
Statistics for EconomicsXI 3 NO. OF STUDENTS .N SCIENCE (20012005)
Scale : 1 cm = 25 students
200
175
f2 150
2
LLI Q 125
?
CO 100
u.
o
d 75
2
50
25
0
2002 2003 2004
YEARS
2005
Fig. 9
net 'i^r: ^e^l^ipt tt^ t.e
export, etc., wh,ch have both " "" "
■n plus and minus values to plot th.ron
the base l.ne and negative vales bell'fbatTe"
Year
1998
1999
2000 2001 2002
Export
47 125 20 94 120
Import
30 115 39
no
125
(Rs in Lacs)
Balance of Trade
17 10
/
19 16 5
99
Digrammatic Presentation . Solution.
BALANCE OF TRADE (19982002)
Scale 1 1 cm = 5 lacs
Y
25 20 15 I
CO O
CO lU LU ti. =3 CC
10 5 0
5 10 15
20 25
gg Surplus ■ Deficit
u
t
1998
1999
2000 YEARS
Fig. 10
2001
2002
B. PIE DIAGRAMS _ ^ ^^^^^ ^^ ^^^^^^^ These
Pie diagram or circular diagram is ^^^^ ^ comparatively easier to draw,
diagrams are very useful in emphasising exhibited. Circles can
With circles and sectors, totals as well as comp^em parts ca ^^^^ ^^^ ^^^^^
be drawn by making the^/™ ^IgllmL ^called I p.
angle at the centre is 360 or 2%. I hereio , _ ^ oercentage breakdowns by
Pie diagrams are very POP^^^J/JJ" represent the
portioning a circle into various parts ^ various parts will indicate
Lvernment expenditure and different pornons^^^^^^ ^^^^^^^^ Transport,
the expenditure over different heads l^ke^ heads Namely, food, clothing.
Education, etc. Similar^ expenditure of ^^^ ^^ components or the
rent, education, etc. If the series is diagrams are less effective than
difference among the components is very small, then pie a g
bar diagram.
Steps for Construction of Pie Diagram percentage of respective totals. Pie
helpful for comparison.
100
 «ke„ ,o be equal to 360. ,t ,s „™ te^e^ h" ""
ry to express each part proportionately
™^rees.S.„ceiperce„totthetotaWalne,se,„alto^.3..,.hepercentages
oahe .mponent parrs w„l be now converted to degrees by .n,t,ply.„g each of
Degree of any Component part = Component value
simultaneously for comparison tbrralTrh "
proportional ro the square roots » ^
■ ^^^^^ "a.. . is common
position on the circle. Now, with this K ^^ o'clock
centre with the help ofp^r t] "^a o t ^ ^^^ component, the new line drawn a iTete to f ^
circumference. The sector so obtained wi^ the
component. From this second line a ble nowT""' ^^^^^^^ ^^^
equal to the degree represented by second c^Z ^^ ^^^ ^^re
the portion of the second component S^ih 7 representing
component parts can be coZuTd ' Psenting differen?
Be distinguished
Illustration 9. Constmrr o abreakup of the cost of —
Item Expenditure
Labour Bricks Cement Steel Timber Supervision 25 % 15 % 20 % 15 % 10 % 15 %
Ltgrammauc Fresentation _ ^^^^ percentage into
So.».>o„. Before draw™, » 3.6" —, we
Labour
Bricks
Cement
Steel
Timber
Supervision
Fig. 11
n of three textile items in percentage
Items
Readymade Garments Cotton Textiles Wollens Textiles
Years
200304
Total
52.2 19.1 28.7
20040S
100.0
/
41.7 23.3 35.0
100.0
102
Solution. (Degrees of angle are rounded off)
Statistics for EconomicsXI j
Items
Redymade Garments Cotton Textile WoIIen Textile
Total
200304
%
52.2 19.1 28.7
100.0
Degree of angle
188 69 103
360
20040S
%
41.7 23.3 35.0
100.0
Degree of angle
150 84 126
360
export of textile items
200304
200405
Fig. 12
Illustration 11. Represent the following data by a pte diagram.
basfs of 360 r ,
basis of 360 taken as equal to the total of the values.
Family X Family Y
1. Food 2. Clothing 3. Rent 4. Education 5. Miscellaneous (Including Saving) 400 250 15r 40 160 640 480 320 100 60
Total 1000 1600
103
the
digrammatic Presentation
>ms of Expenditure
Rs
1. Food
2. Clothing
3. Rent
4. Education
5. Miscellaneous (Including Saving)
Total
Square root
400 250 150 40 160
1000
31.6
400
—x360
= 144
Family Y
Rs
_ , ___i
1000
^x360 = 90
1000
^360 = 54 x360 = 144
1000 1000 1000
x360 = 57.6
360
640 480 320 100 60
1600
40
640
1600 480
x360 = 144
1600 1600 1600 1600
x360
= 108
x360 = 72
x360 = 22.50
x360 = 13.50
360
Radii of circle are determined m proportion 3.2 : 4 (31.6 : 40). Wore the radU of arcle accordmg to avaUabUtty of space 3.2
are :
Family X : Radius y = 16 cm
4
Family Y : Radius  = 2 cm
expenorrure of family x and y
Food im] Clothing m Rent B Education B Miscellaneous
FAMILY X
FAMILY Y
Fig. 24
104
limitations of diagrammatic presentation
Statistics for Economicsxl
fo/lowng points „„„ remembered mterpretation of diagrams, tlie
W—HmSOt^^^^^ " a .i.e,r capacity to g,ve
.n the,r basic fnncti^rrsytdTrj *
2. Diagrams can show presentation.
and ™ tje dte^rb^Lrl^^^Tt'''""'"^' in diagrams. facts are not possible to show
by tables etc. they can misrepresent facts
diagram for visual Presentation of" n"^^^^ ^ Picular
and the object of presentation. Therefore, it shLld be m.^ t ^^^ data
A well constructed simple and attractive Ltam sho ^are and caution,
easier to understand at a glance; sucrpresentatiotr^^ ^^^ mformadon is
newspapers, magazines and journals be seen in financial reports in
2.
3.
4.
5.
6.
Questions :
• a:: —
feplam the various rules of drawing a diagram.
TpLin JT "ftheir utility, txplam M bar diagram, and (b) pie diagram
Digrams are less accurate but more effective than tables in presenting the data •
' rc^mtlstc:: " —« "wmg.
W Composition of the population of Delhi by reltgion H Agnculture production of five states of India.
What are the merits and limitations of dtagrammatic representation of stat,st.cal
Explain the following with illustration ■ M Subdivided bar diagrams, and W Multiple bar diagrams.
8
9.
105
iigrammatic Presentation
(Write short notes on the following 1(a) Percentage bar diagram , (c) Deviation bar diagram 1 (e) Multiple bar diagram.
(b) Broken bar diagram (d) Subdivided bar diagram
iLt the following data by simple bar diagram.
PRODUCTION OF COAL (Million Tons)
Production pillion Tons)
bar diagrams
DEMAND AND AVAILABILITY OF STEEL (Thousand Tons)
Exports (Rs in crores)
4. Present the following data by subdivided bar diagram.
total import
Food Fertilizers Mineral oil Others
Total
200102
200203 200304
474 795
125 298
341 1,113
1,789 1,951
2,729 4,167
(Rupees in crores)
200405
988 323 1,042 2,394
4,744 /
Represent the foJlowine • l . for Economicsx\
«e diagram. 'ie He,p Sunp.e Bar diagram. J
Yi^f ■ ■■ ■ .. ■ . ■:—
' 7.
Export (Rs in crores) Import (Rs in crores)
2002 ~~2m _2004
73 80 85
70 72 74
ZOOS
8.
Food Clothing Rent
Education Miscellaneous
Farntly A (Rs)
Family B (Rs)
P'Xpettdiiure
9 TU ——i—^ 1440
' ^'iree year's result of XTT ri T _
• ^  ".e fCowmg rab...
107
Ipigrammatic Presentation
ntmaiic 11 
B (Rs)
3 1
75 100
175 150
30 25
20 25
Price per Unit Quantity Sold Value of Raw Material
Other Expenses
________
Show the following data by percentage bar diagram^^
L^^ of a product . bar
chart :
COST P^OTRPDS AND fROm AND LCTJ^
Cost per table :
(a) Wages
(b) Other Costs
(c) Polishing
Total Cost
Proceeds per table Profit (+) Loss ()
Chapter 7
cmpbic presentation
3.
sw
Construction of Graphs Graphs of Frequency Distribution Line Frequency Graph • Histogram
Frequency Polygon Frequency Curve
:
asiiiiiifa*
Graphic presentation gives i rr 
as a tool of analysis.
" (Line Graphs,.
^ ^ ...  of graphs;
109
Graphic Presentation . „ j of origin 'O' which represent^
axes. (See Fig. D . ^ j equal parts called qaadrants
Quadrants : Gph pa^ys ^mded " ^^^ ^^ ^ Y are posttrve.
Fig. 1
.ea.    
^^^ VTdata. d.«e„„t
.e^lu'rS^^Str;: S^rrXs . ol t„ue, re—. ''Tt^ol senes graph Xaxts ^^^^^^ ^ ^^ ^
110
■ J J Statistics for EconomicsXI
OF FREQOEIICr
scale wid, d,e difference of lO^wS™ T ' " "" wasting too much of space of ^a7h pSr «« ^ S'^Ph
require a lot of space so that X^is is T",T'" "" "" axis
portion of the scL may be t
use of 'kinke, fe' in ^phtc pr^tT^e Figi'r"" " '
frequency graph
Scale : 0.75 cm = 10 Rs on Xaxis
0.75 cm = 10 Workers on Vaxis
S'graphs..
(b) Histogram
(c) Frequency Polygon
(d) Frequency Curve or Smoothed Frequency Curve ie) Cumulative Frequency Curve or 'Ogive'
(a) Line Frequency Graph
fluency array, on graph by which the line is drawn. represents the frequency of that variable on
kaphic Fresentation
111
Heieht in incht » Nc ~df)
60" 90
61" 80
62" 120
63" 140
64" 132
65" 70
66" 40
^ Metlwd
{. Xaxis for variables under study (Heights in inches)
2. Yaxis for frequencies (No. of students)
3. Draw a vertical line on each value equal to the length of each frequency
4. Both the axes must be clearly lebelled and scale of measurement clearly shown. Xaxis can conveniently be determined according to the need of the problem. We can
have three varieties of Xaxis. Taking the above illustration they are :
(a) Use of kinked hne
(b) Starting from 59" , . ^ , . •
(c) Starting from 60" (use thick line to read the data properly). See the graphs given
{d) Both axes must be clearly labelled and the scale of measurement should be clearly shown.
Solution. HEIGHTS OF STUDENTS
Scale
140 
120 
^ 100 
lU
Q
3
is 80
LL
o 60 
d 40 
H
20
(a) using kinked line
1 cm = Frequency 20 Students 1 cm = r on Xaxiis
Fig. 13
il2
Statistics for Econo
>mtcs~}
STARTING FROM 59" (Xaxis)
 iFpu 20 StudL onVaxis ^ I cm _ 1 on Xaxis
Scale 1 starting FROM 60" !
■ ^ = Frequency 20 Students on V^axls^ 1 cm = 1" on Xaxis
yf
(b) Histogram
' which each
and also called a frequency histo^m : '' " ' ^^dimensional diagram
Cases of Constructing Histogram
U) Histogram of Equal Class Intervals {« Histogram when Midpoints are given Histogram of Unequal Class intervals
Method
1. Xaxis for variables under study (Marks) .yaxis for frequencies
are
freq
Thus is pn
(«)K
n
obtai
113
\ Graphic Presentation class with frequency.
3. oe. reW. —
4. Both the axes must oe ucdny clearly shown.
Solution. histogram
Scale : 1 cm = 10 Marks on X^is
1 cm = frequency 4 on Yaxis
1 in for all the classes and the frequencies
In the above illustration 2, class mterval is 10 for all
SSe!rr: Se for each class can be deaded(c^^
frequency)
Class (Marks)
010 1020 2030 3040 4050 5060
Class frequency (f)
4 10 16 22 18 2
10 X 4 = 40 10 xlO = 100
10 xl6 = 160 10 x22 = 220 10 xl8 = 180 10
Total Area = 720
Total frequency^^;^^______^  ^ interval
Histogram = ^en ^ff^^^Tora the following aisnr,bu,ion of total marks
ojrr^"'a'^Sjra Boara H—.
114
Method
Marks (Midpoints) No. of Students (f)
150 160 170 180 190 200 •8 10 25 12 7 3
Statistics for EconomicsXI
Graf
midpoirns different classes from the given
2. Xaxis for variables under study (Marks). 4 iTZ r fr^q^es (No. of Students).
5: with frequency
he clearly shown. ..... ' the measurement should
bet^ sS::^^ first midpomt, get the difference
^vide the difference by 2, /.e., 10/2 = 5 ~
of we get lower and upper lim.
Thus, the class decided is 145 ~to 155 ^ ^ = ^^^  "PP H.^"
U^g the same jet ^h^ midpoints as under :
No. of Students : g IQ ^25
^ ^hri ways : ^
See the Figs, below : ^ ^^^ Starting from 135 marks.
(iii) I
ni
Solution.
histogram kinked line method
Scale : 1 cm = 10 Marks on Xaxis 1 cm = 5 Students on Vaxis
histogram xaxisstarting
from 145 marks
Scale : 1 cm = 10 Marks on Xaxis 1 cm = 5 Students on Vaxis
165 175 185
marks Fig. 7
Sd Nc histogi
Metho 1. 2.
3.
4.
115
^Graphic Presentation
histogram XAXIS^starting from 135 marks
Scale
1 cm = 10 Marks on Xaxis 1 cm = 5 Students on Vaxis
i
UJ
o
3
u. O
d Z
135 145 155 165 175 185 195 MARKS
Fig. 9
205 215
S.
Hit) Histogram of Unequal Class Intervals 'Tst^lo 4. Rep«se„. .he fonowing^^ans of
of Workers
.he Cass — are unequal, frequencies n.us. he adiusred, otherwise .he his.ogram would give a misleading picmie.
'^"I'^Take .he class which has d>e lowes. class in.erval.
2. Do no. adius. .he frequencies of i„,erval.
^rr^^^^^^ each .ecan^e of h.s.ogram hu. ■ widths will be according to class limits.
116
Thus, the adjusted frequencies are :
histogram
Scale : 0.5 cm = Rs 5 on Xaxis
1 cm = 5 Workers on Vaxis
daily wages in rs
Fig. 10
japhic Presentation
Histogram : When Class Intervals are given by " Method
^ IllustrLn 5. Constru^^ '
r Jji^s ^ Students W
117
59 1014 1519 2024 2529 3034
4
17 25 32 13 6
Solution. „ inrliisive method (where lower and upper
Note : Since the class intervals are given ^ .^d upper limits of
Adjustment : Find the difference between lower hmn^ _ J ^^^^^^^^
and so on.
Adjusted Class Limits
Marks Students (f)
4.5 9.5 5
9.514.5 17
14.519.5 25
19.524.5 32
24.529.5 13
29.534.5 6
histogram
Scale : 1 cm = 5 Marks on Xaxis
1 cm = 10 Students on Vaxis
9.5 14.5 19.5 24.5 29,5 marks
Fig. 24
118
(c) Frequency Polygon («) Without histogram.
Statistics for Economic^y,
Method
loISo" 
2030 5
3040 12
4050 15
5060 22
6070 14 4
—"  !>uitaDJe histogram kepnm„
2 the of the '"Ho " ™T""""P'
3. Jotn these mdp„i„„ „f ^.de of each rectangle
ciearly sro^^^"^' ^^^^^ ^^^elied and the scale of the meas Solution. measurement should
histogram and prequencv polygon
JO on Xaxis 1 cm = 5 Students on ya*is
25'
CO 20
&
UJ
s 15
1
co
u.
o 10
d
2
5
0
 Histogram
Frequency Polygon
Fig. 24
Graphic Presentation jj^
While drawing the frequency polygon, we observe that some area which was under the histogram has been excluded and some area which was not under histogram has been included under frequency polygon. This dotted area which was under histogram but is not under the frequency polygon. This dotted are is excluded from the area of frequency polygon. But the shaded area has been included under the polygon. This was not under histogram. Thus there is always some area included under the frequency polygon instead ot the area excluded from histogram. Therefore, the total area excluded from the histogram ts equal to the area mcluded under frequency polygon.
(ii) Frequency Polygon : Without Histogram
und™"^ ^^^ illustration, we can get the frequency polygon without histogram as Method
1. Take the midpoint of each class interval.
2. Scale of Xaxis can either be decided on the basis of class interval or midpoints
3. Join the points plotted for the midpoints corresponding to their frequencies by straight lines. We will get the same figure as obtained by the first method (i.e., with histogram).
■BJ
Hr
Muipomts No. ofstfj^nirw 1
15 5
25 12
35 15
45 22
55 14
65 4
Solution.
frequency polygon
Y Scale : 1 cm = 10 Marks on Xaxis 1 cm = 4 Students on Vaxis
Fig. 13
120
Illustration 7 V St'^t'stics for EconomicsXI
exa JSr ks secured by 25 s,ude„,s i„ an
2.1 in 
  " "     « . « ,, Solution.
Frequency Distribution of Marks Tally Bars
Gi foi
Marks
2029 3039 4049 5059 6069
m(
Total
^xuxc preparing 1 exclusive method, i.e..
No. of Students (f)
2
5 8
6 4
25
~1MS Students (f)
19.529.5 29.539.5 39.5A9.5 49.559.5 59.669.5 2 5 8 6 4
J'WUTtlUN
Scale : 1 cm = lo Marks on *axis 1 cm = 1 Student on Vaxis
» i*
marks Fig. 14
121
Graphic Fresentation
lllusmtion 8. We have the following data on the daily expendttute on food (in rupees) fot 30 households in^alocaU^: ^^^
  s r/o r/s r/o .s
(a) Obtain a frequency distribution using class intervals : 100150, 150200, 200250, 250300 and 300350
(b) Draw a frequency polygon. j u r^nt \c) What per cent of the households spend less than Rs 250 per day, and what per cent
spend more than Rs 200 per month? Solution, (a)
lit
Monthly Expenditure on Food (Rs) Tally Bars No. of households (f) . 
100150 nil 4
150200 mil 6
200250 mim^iii 13
250300 M 5
300350 11 A.
Total 30
(b)
frequency polygon
Scale : 1 cm = Rs. 50 on Xaxis
1 cm = 2 Households on Vaxis
*■ X
100 150 200 250 300 350 EXPENDITURE IN RUPEES
Fig. 15
400
122
fc) Ont of ^sn J, u Statistics for EconomicsX
^^ Hence 76.6% spend less tha.
xtlr '' ^^ spend more than
(d) Frequency Curve or Smoothed Frequency Curve
generalVby^'el^^^^^^^^^ frequency curve. It is drawn
area mcluded ,s ,ust the same as^I Tthf poL^^^^"u ^ ^^^^ ^^^"he required to be done carefully to ge^ co rect "eS Smoothing the frequency polygon
shows neither more nor less area of the rectanLs of .h v drawn with care
frequency polygon to get a smoothed freq^ ett "
histogram for the data given in IllustraZ 8 fo^ U"^^ constructing
frequency curve
Scale : 1 cm = Rs. 50 on Xaxis
1 cm = 2 Households on Vaxis
200 250 300
expenditure in rupees Fig. 16
> X
350 400
We observe that :
123
Graphic Presentation
statistics. It is a unimodal distribution curve.
histogram, frequency polygon and frequency curve
_____1— I _____.ri^ol
124
■ J J Statistics for EconomicsXI
ie) yShaped Curve (Curve E) : In this case, maximum frequency'is at the ends of rh. (e) Cumulative Frequency Curve (Ogive)
I O" the graph paper by rwo
(a) 'Less than' method
(b) 'More than' method
Illustration^^Draw^^ for the following data :
Marks
010 1020 2030 3040
Ma. of StHdents
4 4
7 10
Marks
4050 5060 6070
No. of Students
12
obra,„,„g mar,. S" ^^^^^^^^
125
Graphic Presentation
of each class .g in above illustration, the number of students obtain,ng marks more In 0 .s 50; moi; than 10 is 46; more than 20 is 42; and so on.
Cumulative Frequency Distribution
Marks
Less than 10 Less than 20 Less than 30 Less than 40 Less than 50 Less than 60 Less than 70
No. of Students
(c.f)
4
8
15 25 37 45 50
Marks
More More More More More More More
than 0 than 10 than 20 than 30 than 40 than 50 than 60
Nr. of Students (c.f.)
50 46 42 35 25 13 5
We get a rising curve in than method', if the above
case of 'less than method' and declining curve in case of 'more cuLlative frequencies are plotted on the graph paper.
""Ilet the cumulative frequencies of the given frequencies either by 'less than method'
or 'more than method'. 2 Xaxis — the variables under study
3. Yaxis  calculated cumulative frequencies Cumulative
4. Plot the various points and )om them to get a curve (i.e., ugiv
5. be clearly lebelled and the scale of the measurement should
be clearly shown. „ , 'Ogive on Graph Paper
(Cumulative Frequency Curve)
by 'less than' method
Scale : 1 cm = 10 Marks on Xaxis
1 cm = 10 Students on Vaxis
by
Scale
more than' method
1 cm = 10 Marks on Xaxis 1 cm = 10 Students on Vaxis
>X
10 20 30 40 50 60 70 80 MARKS
10 20 30 40 50 60 70 MARKS
Fig. 18
Fig. 19
126
■ J J Statistics for EconomicsXI
FAv
by less than' method
Scale : i cm = 10 Marks on Xaxis 1 cm = 10 Students on Vaxis
Less than method
Fig. 20
^ c„„e for .he ,oHo„i„, dis„,h.„o„ of
Weekly Wages Workers (f)
100109 7
110119 / 13 15 32 20 8
120129
130139
140149
150159
Method
1. Must the lower and upper of he classes.
>jet cumulative frequencies
Both the axes should be iX l beM T
clearly shown. «ale of the measurement should be
and
127
Graphic Presentation Solution,
Adjustment of da limits and calculation
of cumulative frequencies by less than method.
99.5109.5 109.5119.5 119.5129.5 129:5139.5 139.5149.5 149.5159.5
7
13 15 32 20
8
7 20 35 67 87 95
ogive (less than method)
ocale ■ 1 cm = Rs. 10 on Xaxis
• 1 cm = 20 Workers on Vaxis
be
..........Fig. 21 J
and indicate the value o^Ae^i^ 
128
Solution.
Statistics for EconomicsXI
It I
Marks r\ r Number of Students Cumulative Frequency (Less than) c.f Cumulative Frequency (More than) c.f
05 510 1015 1520 2025 2530 30^35 35^0 1 7 10 20 13 12 10 14 9  1 7 17 37 50 62 72 8695 88 78 58 45 33 23 9
than' ogive
Scale . cm = 5 Marks on Xaxis
1 cm = 20 Students on /axis
or h
Kg. 22
Graphic Presentation
129
^ graphs of time series
, ^ , , .......
Time series can be sbown on the graph paper. The information arranged over a period of time (e.g., years, months, weeks, days etc.) is termed as a time series. Presentation of this type of information by hne or curve on the graph paper is of great use in economic statistics. These graphs are known as hne grapjhs or histograms, or arithmetic hne graph. (a) General Rules to Construct a Line Graph
1. As the time (year, month, week) is never in negative (i.e., in minus figures), there is no need of using Quadrant II and III.
2. Year, month or week according to the problem, is taken on Xaxis. Give titles to Xaxis and Yaxis.
3. Start Yaxis with zero and decide the scales for both the axes. For example, on every 1 cm for Yaxis one may represent an equal gap of 50 students and 1 cm for Xaxis a gap between 2000 arfd 2001. Xaxis can start either from 1999 or 2000 (See Fig. 23).
4. The pair values will give different dots on the graph paper. For example, values corresponding to time factor are :
Years Students
2000 50
2001 150
2002 100
2003 150
2004 200
2005 225
2006 200
These dots obtained of pair values are joined by straight line which is called line graph or histogram (See Fig. 23).
students (200006)
Scale : 1 cm = 50 Students on Vaxis
300
250
CO
H
Z 111 200
Q
Z)
H
OT 150
O
d 100
Z
50
0
/ N s /
/ f S
/
/
2000 2001
2002
2003 2004 YEARS
Fig. 23
2005
2006
130
5. It is not advisable to ■
by a straight iine and not by a curte " J™" Ae dots
is r ^^^^ of un.
m One Variable Graph ^^e given below •
Kendriya Vidyalaya
■
Method
199899
199900
200001 200102
200203
200304
200405
120
400
567
490
760
834
750
Gra Wh
in v (yea timt orig
largi <
year Yas smal line ; two reqa porti line, I
1 Select Xax for the time factor (years).
2. Select Yax. for variables under study (students)
3. G„ble title and scales to XJ Ld C
= ^o Its value and .. the. by
STUDENTSKENDRIYA VIDYALAYA (1998^)5)
Scale : 1 cm = 200 Students on Vaxis
z
UJ Q =3 I
co u.
o d
Sc
200102 200203 200304 200405' YEARS
Fig. 24
J Graphic Presentation
■ What is a False Base Line?
131
't rlvldTuse faUe base U„e according to ne^ of tbe ptoblent. Keeping
doln^—^ .o out tequiretnents by using False ^se Line) mustrafon No. nbet of stude^ ts ^ one t^usaud .n e.b
wmmmmM
P" : . . C T in crranh r nresentation See Fig.
Dortion of the scale may De omiueu wm^n ^ciix  ,
£ that is the use of False Base Line in graphic presentation (See Fig. 25).
illustration 13. Present the following information on the graph paper.
Year__________ Students
2U00 1120
2001 1380
2002 . 1587
2003 1490
2004 1760
2005 1734
2006 1675
STUDENT&<30vt. higher sec. school (20(hm»6)
Scale : 1 cm = 200 Students on raxis
2000 2001 2002 2003 2004 2005 2006 YEARS
Fig. 25
132
Statistics for EconomicsXI
Year (1) Agriculture and allied sectors (2) ■ Industry (3)
199495 199596 199697 199798 199899 199900 5.0  0.9 9.6  1.9 7.2 0.8 9.1 11.8 6.0 5.9 4.0 6.9
Services (4)
7.0 10.3
7.1 9.0 8.3
8.2
Gr (d)
Wt
as
axi
data as * rime series graph. estimated sectoral growth rate ,n gdp at factor cost
t>Cdle : 1 cm = 2 per cent growth rate in years
— Services
Agriculture and allied sectors Industry
YEARS
Fig. 26
133
IGraphic Presentation
1(d) Graphs of Different Units different units, we will have two different scales.
When two values are given into two ^^"erent unn , ^^ ^^^
Year
199798
199899
199900
200001 200102 200203
Quantity (in, '000 tons)
9 10 12 11
14
15
Value (Rs. in crores)
300 596 782 900 762 640
Solution.
Average of Quantity: 12 Approximately Average of Value : 695 Approximately
trade of tea in quantity and value (19972003)
qcale • 1 cm = Quantity 3 thousand Tons
or7axis : 1 cm = value in RS. 150 crore
Quantity
SRupees
199798
134
Two figures of graphic presentation
10000
exports (Provisional)
(US $ IMillicn)
Statistics for EconomicsXl\
are shown below to understand time series graph. ^ IMPORTS (Provisional)
(US $ Million)
5000
Fig. 28
m
Questions :
■ TSL^ Name ehe ,«e.e„.
^   .apb . prepared. 7. berween Bar dra^ar^'irH™
JO Wha, « a 'Cumulat,; ^^ Give iUusrrarion.
! r Tr' '"O'ency curve
fop rXt"iara presented .„ rbe «*en an the class intervals h^trtheTa.t^X''""'"
pS  f the ess than type „„ve and
s.gn,f,cance of Rs 715 f^r the gtven « of da™ " '"e
13.
Sm Vim
Graphic Presentation ^^^
What is a false base hne? Under what conditions would its use be desirable? What is meant by (a) Histogram, and (b) Ogive? Explain their construction with the help of sketches.
Distinguish Histogram and Historigram clearly with illustrations. What is a smoothed frequency curve? Discuss briefly various types of frequency curves.
Explain the importance of graphic presentation of data. 19. Describe the procedure of drawing histogram when class intervals are (i) equal, and (ii) unequal.
i4. 15
16.
17.
18.
Probl^^s :
^ The frequency distribution of marks obtained by students in a class test is given
4050 3
below:
. Marks : 010 1020 2030 30^0
No. of Students : 3 10 14 10
Dtaw a histogram to represent the frequency distribution of marks. Comment on the shape of the histogram.
What is histogram? Present the data given in the table below in the form of a Histogram:
Midpoints : 115 125 135 145 155 165 175 Frequency : 6 25 48 72 116 60 38 3/ Make a frequency Polygon and Histogram using the given data / Marks Obtained : 1020 2030 3040 4050
/Number of Students : 5 12
4. Draw Histogram from the following data : r Marks Obtained : 1020 2030
Number of Students : 6 10
In a certain colony a'sample of 40 households was selected. The data on daily income for this sample are given as follows :
200 120 350 550 400 140 350 85
200
15
3040 15
22
4050 10
185 22
5060 14
5070 6
195
3
7080
4
70100 3
5.
180 170 210 430
110 90 185 140
110 170 250 200
600 800 120 400
350 190 180 200
500 700 350 400
450 630 110, 210
170 250 300
(a) Construct a Histogram and a frequency polygon.
(b) Show that the area under the polygon is equal to the area under the histogram. (Hint. Get a frequency distribution table to obtain a continuous series).
136
■ J J Statistics for EconomicsXI
Frequency  f s '''''
'st: it: S...
1519, 2024 TQ
(b) What percent of th^ hr. u ij '
Size of classes
t'::;   aoz^ . 303.35.„
Students ^ 10 15
"'Z; ^
Workers : 9 12 15
Weekly Wages of B cia .
; 'tr '"^r ".^r
cZ''^""
Frequency : 4 ^ ^^ 1^24 2430 3036
u. I
. 0. 3.30 3..0...0 .0.0 .O.O
Companies : 2 3 j
^ —
aOOO.o^, , 35 3. 3. .0 « ^ ^
iwi
■ 137
Waphic Presentation
I P^ .e foUo™. a —^^^^^^^^^ tr r " 
\ Profit (Rs in ^^ 65 80 95
f i graph from the
Ps "  Imhnrtv t
Year
'T99"o91
199192
199293
199394
199495
199596
199697
,, 12 » 25 31 29 27 35
U. 'p:;:::^ — co. a„a .ot. proauct.o„ of a scooter ma—
; company.
Year ^ Production f? (in units) 1 Total Cost I (Rs in lakh)
2001 8500 24
2001 2003 2004 2005
9990 11700 13300 15600
29 34 45 49
<53t
»TICAL TOOLS AND INTIRPRETATIOlf ii
■ Average p^^
••—ye, of Correla«p„
lrtion to lBde» lV«™be«»

Chapter 8
nusmcs of centum, tendency
Weaning and Importance 
Objects and Functions Of Averages
Characteristics of a RepresentatL Averaoe
Three Orders Of Measurement ®
Arithmetic Average or Mean J'st of Formulae and Abbreviations
MCAMte AND IMPORTANCE
the whole group is ealleH 7 observations. An average is a fi. J generally
briefly average Th^ j / <^entral tendency or ^easur^T, ^^P^^^^^ts
marks of students n a , '' ^^ everyday
income of hctnr , represents the marks of aM . !i' the average
As satisfy
fl
139
Measures of Central Tendency
According to Croxton and Cowdon : "An average value is a single value within the range of the data that is used to represent all of the values in the series. Since the average is somewhere within the range of the data, it is sometimes called a measure of central
value." £ 1
According to KeUoy and Smith : "An average is sometimes called a measure of central
tendency because individual values of the variable usually cluster around it."
and functions of averages
1 To represent the salient features of a mass complex data : It determines a single figure' of the whole series. It is a tool to represent the salient features of a mass of.complex data. It is helpful in reducing the mass information into a single value for drawing.general conclusions. It is difficult to generalise anythmg from the ages of crores of Indian People. But if it is said that the average age of an Indmn is 55 years one can draw conclusions about health conditions of the people. Thus the purpose of an average is to represent a group of individual values in a simple manner, so that the mind can get a quick understanding of the general size of the
individuals in the group.
2 To facilitate comparison : Averages are useful for comparison. The average of one group can be compared with averages of other groups. For example, the average marks of students in section A can be compared with the average marks of students in section B, easily at a glance or the average monthly sales of Department A are compared with average monthly sales of Department B.
3 To know about universe from a sample : Averages also helpto obtain a picture of complete group by means of sample data. In statistical enquiries, very often, sample method is used. The mean of a sample gives a good idea about the mean ot the
population.
4 To help in decision making : Averages are helpful for making decisions in planning ■ in various fields. For example, a sales manager may need to know the average
number of calls made per day by salesman in the field. A railway officer will require information regarding the average number of passengers carried by rails on the various passenger runs. Averages are valuable in setting standards, estimating and planning and other managerial decision areas.
5 To trace mathematical relationship : When it is desired to trace the mathematical relationship between different, groups or classes, an average becomes essential. Definiteness can only come, they are expressed in averages.
i gh^tl^ of a repiuesentjirive average >
As the average represents statistical information and it is used for comparison, it must
satisfy the following conditions : . uiju
1 It should be simple to calculate and easy to understand : An average should be calculable with reasonable ease and rapidity only then it can be wide y used. It should not involve heavy arithmetical calculations. If the calculation of the average
Statistics for EconomicsXI u„de«,„d by pLons of ordln^rSltn 'e. """
are not separated the aver^cotton cloth per mill, if big and small mills cotton mill industry fnTdfa s^pUt'u ^^ ^^^^ ^
female workers, adillt worts ^^^ ^^
OF MEASUREMENT
There are three orders of measurement.
1. Measures of first order.
2. Measures of second order.
3. Measures of third order.
141
Measures of Central Tendency
students).
kfNDS OF STATISTICAI AVERAGE
♦►Moving Average
.....^.........^
Arithmetic Mean] tSeometric Mean or Mean X , (GM)
! Harmonic Mean (HM)
Progressive Average Composite Average ^Quadratic Mean
Specialised Average (Index Numbers)
Of the above mentioned average, mean, median, mode are mo^
'"M^lirt cl of qualitative data which caunot be measured quantitatively for rdatiou to all the values, naturally, median should be the choice.
which has highest demand, most fashionable garment, etc.
142
■ J J Statistics for EconomicsXI
I
1. Meaning
2. Calculation of Arithmetic Mean
3. Mathematical Properties of Arithmetic Mean
4. Miscellaneous Problems
^ Merits and Demerits of Arithmetic Mean
(B) Weighted arithmetic average or Weighted Mean
(b) Short Cut Method (Assumed Mean Method) ic) Step Deviation Method
Let us see the calculations m the following senes.
(A) Series of Individual Observations.
(B) Discrete Series.
(C) Continuous Series.
A. Series of Individual Observations
of v^updLtS^^^^^^^ we .e
1010+1020+1030 3060
= Rs 1020
/.e., average wage taken by the workers is Rs 1020 Direct Method ; Symbolically,
Mea
1 M
/.(
M
Alten
where denote
Worker Wages (Rs) X
A B C 1010 1020 1030 X
N = 3  3 XX = 3060
Special 1. I
these «
ieasures of Central Tendency
1. Obtain IX by adding all the values of variables.
2. Divide the total by number of observations (N). Symbolically,
+ +......x„
X —
143
N
 XX • 3060 A. —
= Rs 1020
N 3
Therefore, average of the workers is Rs 1020
where, X =. Arithmetic mean
SX = sum of all the values of observations
/.e., X, + X, + X3 + ..... X„
N = Number of observations
Alternative equation
 Iv
X = —Sx,
n
where, the symbol X is the 'Greek alphabet called sigma and is used xsi mathematics to denote the sum of values.
n  total number of observations
Lx = the sum of n values
t
= i X (1010 + 1020 + 1030)
3060
= Rs 1020
iSpecial Features of Arithmetic Mean
1. If we replace each item of observation by the calculated mean, then the total of these replaced values will be equal to the sum of the given observations.
Workers Wages (Rs) Mean
V v=' 
A 1010 1020
B 1020 1020
C 1030 1020
N = 3 ZX = 3060 3060
NX = IX 3 X 1020  3060
144
2 The sum  ■  Statistics, for Economicsd
Workerc rv, 7:r.1
A B
C
N = 3
Wages (Rs) X
XX
1010 Xj 1020 X^ 1030 X3
^X = 3060
10 0 +10
2(XX) = 0
Symbolically, ^XX) =0
Alternatively, ^ ~ ^ ^^^^^ ' ^^^O) + (1030  1020) 0
 K + Jf^ + + ...xj nx = (1010 + 1020 + 1030)  3 X 1020 = 3060  3060 = 0
ShortCut Method (Assumed Mean Method)
get the anthmetic. mean the total of tt deZl? ^^^^^ « calculated. To total ,s divided by the number of^^^^"Ited. This
assumed mean. observations and finally the product is added to the
b]
r Worker X ^ — A (d)
A B C lOlu 1020 1030 0 10 20
N=3 4 Id =30
Measures of Central Tendency . 145
 Steps :
1. Decide assumed mean {A), i.e., Rs 1010.
2. Calculate deviations of items from assumed mean (d).
3. Get the sum of the deviation {Jd), i.e., 30.
4. Use of following formula.
X= A + 1010 + — = 1010 + 10 = 1020 N 3
i.e., average wage taken by the workers is Rs 1020.
where, A = assumed mean
d = X  A, deviations of X variables from assumed mean
"Ld = Z (X  A), i.e., sum of the deviations of X variables taken from
assumed mean.
N = number of observations ^
Step Deviation Method
We can further simphfy the short cut or assumed mean method. All deviations taken by assumed mean are divided by common factor.
Symbolically (Assumed mean = Rs 1010)
Worker Wages XA
A 1010 . 0 0
B 1020 10 1
C 1030 20 2
N=3 ■ Id'=3
Steps
1. Decide assumed mean, i.e., Rs 1010.
2. Calculate the deviations of items from assumed mean (d).
3. Divide these deviations by common factor, i.e., C = 10.
4. Get the sum of step deviation (Zd'), i.e., 3.
5. Use of following formula :
Ld'
X = A +
N
= 1010 + J X 10
= 1010 + 10 = Rs 1020 i.e., average taken by the workers is Rs 1020.
146
where v a u for EconomicsXI
wnere, X = Arithmetic Mean
'"lit <iev,a„„„. of X vanabies fron, assumed mean
d' =
brcom'r flz"" ~  
C  Common factor T , ^ = Number of observations
  fo^m. .„us„a,ons, done
(a) Direct method Direct Method
rr^^^^^^ .neome of ten wor.ees
Workers Daily Income (in Rs)
Solution.
in a factory.
A B C D E F G H J T
"3^ J ^6^
l>Jiiy Incuwe [Rst X
ZX = 2400
X = ^ il^
N ~ \o ^ rupees.
The average daily income of workers is Rs 240.
St
1. 2.
3.
4.
'Measures of Central Tendency
147
asures of f^entrai. lenui^nc^y
Dlustration 2. Calculate the anthmetic mean of the marks given m illustration l by he shortcut method (Assumed Mean Method).
Worker
A B C D E F G H I
J
Daih income (Rs.)
N = 10
120 150 180 200 250 300 220 350 370 260
XA
JdJ
50 20 0
+50 +100 +20 +150 +170 +60
Id. = +400
Steps :
1 Decide assumed mean, suppose A = 200. v  A  d
2. Calculate the deviations from assumed mean, /.e., A A  ^
3. Get the total of the deviations calculated from assumed mean (d).
4. Use the following formula :
X = A +
= 100 +
400
N  "" 10 = 200 + 40 = 240 rupees
The average daily wage of workers is Rs 240. „ 1 w
niustration 3. Calculate the anthmetic mean of the marks given m illustration 1 by
step deviation method.
.t^'i
Worker
A B C D E F G H I
J
Marks X
XA <<i) .
120 150 180 200 250 300 220 350 370 260
50 20 0
+50 +100 +20 +150 +170 +60
XA
m
8 52 0 +5 +10 H.+2
+15 +17 +6
Id'  40
148
Steps :
1 Decide assumed mean, suppose 4 = 200.
X ^C
■ J J Statistics for EconomicsXI
N
= 200
X 10
10
= 200 + 40 = 240 rupees The average dady wage of wbrkers is Rs 240
B. Discrete Series
Students .Marks :
Solution.
A 50
B 100
c
50
D
150
E 100
F 50
G
150
H 100
I
50
J 100
tx
200
400 f^x.
HO tTC   each, 4 .00 each and .
ieasures of Central Tendency Steps :
1. Multiply the frequencies with variables (fX).
2. Get the sum of the products (I^fX).
3. Divide the total by number of observations {Lf or N).
/1+/2+/3+••■/«
149
Z/X ^f
900 10
or
S/X N
= 90 marks
where,
E/X = sum of the products of variables and their frequencies f Frequency '
N = L/j i.e.. Number .of observations
fx = Product of variables with their respective frequencies
Direct method :
Illustration 5. Following tables gives the marks obtained by 100 students in a class.
Calculate the arithmetic mean.
Marks : 10
No. of Students : ^
Solution.
20 10
30 40
40 20
50 25
Marks No. of Students ■
10 Xj 5 A 50 f^X^
20 Xj 200 f^X^ ■
30 X3 40/^3 . 1200 /■3X3
40 X, 20/; 800 f^X^
50 X, 15 f^ 1250
E/ = N = 100 IfX = 3500
Steps :
1. Multiply the frequency with the variable X.
2. Get the sum of the product (LfX).
3. Divide Z/X by total number of observations, i.e., S/" or (N).
ISO
t
Statistics for EconomicsXI j
X = or
N
3500 100
If
= 35
••• Average marks of students is 35. Alternative equation :
where,
^ = ^Ifx n '' •
n
' 100 J_
100 100
" + 200 . 1200 . 800 . 1250] " 3500 = 35 Marks
^ 100 x 35 = 3500
= 475 + 475 = 0
Aver Step
taken by calculate
XI
Measures of Central Tendency , ^^^
ShortCut Method (Assumed Mean Method) : We can use this ^method to calculate arithmetic mean in order to simplify arithmetic calculations. The followmg formula is
used :
Ifd
X = A +
N
Here,
A = Assumed Mean N = Number of observations f  frequency
X  A, i.e., deviations of variables taken from assumed mean •Lfd = Sum of the product of frequencies and their respective deviations Dlustration 6. Calculate the average marks of students given in Illustration 5 by short cut method.
SliortCut Method (Assumed Mean Method)
cies
25)]
Marks
10 20 30 40 50
5 10 40 20 25
N  100
20 10 0 +10 +20
100 100 0
+200 +500
= 500
n of
:heir
Steps :
1. Decide assumed mean, suppose A = 30.
2. Calculate the deviations from assumed mean, i.e., X  A = d.
3. Multiple deviations by frequency and get fd.
4. Add the product of deviatioiis and frequency.
5. Use the following formula :
Ifd
X = A +
= 30 +
N
500 100
5)
= 30 + 5 = 35 Average marks of students is 35.
Step Deviation Method : We can further simplify the shortcut method. All deviations taken by assumed mean are divided by common factor. The following formula is used to calculate the arithmetic mean by step deviation method.
152
til; V
Here,
Statistics for EconomicsXI j
X = A.m., C
N ^ ^ A = Assumed Mean N = Number of observations C  Common factor f = frequency
 ' Step deviation
'''' ' fr^f d ehe. .especve
devSrtrd/^^'™'"^ "ents given ,„ I„u.„aeio„ 5 b, seep
Step Deviation Method
Steps ;
1. Decide assumed mean, suppose A = 30.
2. Calculate deviations from assumed mean, /.../x  A = d.
3. Divide deviations by common factors ^Zii. ^ common factor = Q 10 ^
4. Multiply step deviations 'by frequency
5. Add the product of step deviations and frequency
6. Use the following formula.
50
= loo ^ 10 = 30 + 5 = 35
Average marks of students is 35
153
Measures of Central Tendency
C. Continuous Series t
In continuous series, the method of calculations of arithmetic mean  ^e ^^ ^ the case of discrete series. The only difference is that m continuous series midpmnts ot trcSsfLrvals are required to be obtained. The following equation can be used
to get the midpoints.
Midpoint = Here presents lower limit and presents upper limit, ..g., the
midpoint for a class 510 can be obtained as : ^^ = 7.5.
After obtaimng the midpoints, we can use all the ^^e me^ods o^ca^^^^^^^^^^ of arithmetic mean in the same way as we used m discrete series. These methods are.
(i) Direct method, («) Shortcut method, and {Hi) Step deviation method.
Direct Method .. .
Marks
04 4
48 8
812 2
1216 1
lliuil. Marks (X) No. of Students (f) Midpoints (m)
04 48 . 812 1216 4 A 2 A 1/. 2 wij 6 m^ 10 W3 14 m. 8 48 f,m, 20 f,m, 14
X/^ = N = 15 •Lfm = 90
Steps :
1. Obtain midpoints (m) of the classes, i.e.,
Here, /, = lower limit and = upper limit
2. Multiply the frequency with midpoint (fm).
3. Get the sum of products (Lfm).
4. Divide Sfm ky total number of observations (N).
S'ii
ll
154
l'!. V.'
Symbolically,
■ J J Statistics for EconomicsXI
/1+/2+/3+.../■„ X  
Mean marks are 6. Alternative Equation
■ X =
«
where, « = total of frequency (E/)
^ frequeLts ™clpoints of classes and their respective
V = ^ X [(4 X 2) + (8 X 6) + (2 x 10) + (14 x 1)]
J_
15 1
X [8 + 48 + 20 + 14]
x90=90
15 15
= '6 Marks.
Special Features of Arithmetic Mean
1. Thyotal of frequencies multiplied by Arithmetic Mean is always equal to the sum ^of the product of midpoints of various classes and their respLive frequtd"
. NX =Lfm
15 X 6 = 90
IhlT of midpoints from arithmetic mean being multiphed by
their frequencies is always equal to ZERO. ' '""uipuea oy
•EfimX) =0
= 4(2  6) + 8(6  6) + 2(10  6) + 1(14  6) = 16 + 0 + 8 + 8 = 0
Measures of Central Tendency
ShortCut Method (Assumed Mean Method) „ u u ^ ' . ^AK^d
Illustration 9. Calculate arithmetic mean of Illustration 8 by shortcut method.
155
)01U11UU. Marks X No. of Students ■f Midpoint m  m1 d ■ /a —
04 48 812 1216 4 8 2 1 1 6 10 14 0 +4 +8 +12 0 32 16 12
N = 15 60
Steps :
1. Obtain midpoints.
2. Decide assumed mean (A = 2).
3. Calculate the deviation from assumed mean, i.e., m  A = d.
4. Multiply deviation by frequency and get fd.
5. Add the product of deviation and frequency.
6. Use the formula :
IM
N
X = A +
= 2 +
60 15
=2+4=6
Mean Marks are 6.
rcl.«e ..hmetic mean of m„s„a.o„ 8 by rbe deviation
method.
Solution.
I'. ' ■'■
mA
tS6
Steps : for EconomicsXI
1 Obtain midpoints.
2. Decide assumed mean (A = 2)
3. Calculate the deviations from assumed mean M =
4. Divide deviations by conmion faaor
(Common factor = C <■ e 10)
5. Multiply step deviation b; frequency.
7: ui't^^'frrmr fr^ncy.
y 15
= 2 + — X 4
15 ^
A^ w . =2+1x4 = 6 Mean Marks are 6.
worSlfXory"^' ^^ folWin, distHbu^on of daily wages of
Daily Wages (Rs)
Below 120 120140 140160 160180 Above 180
No. of Workers
10 20 30 15 5
80
confidl^;,tfr r^tl^S:; ««   « .„ .ose the ends Calculation of mean.
J?aily Wages (Rs) (X)
100120 120140 140160 160180 180200
Measures of Central Tendency Applying formula, we get
lfm
If
11700
157
X =
or
lfm N
80
= 146.25
Mean wage of workers is Rs 146.25.
Illustration 12. Following information pertains to the daily income of 150 famdies. Calculate the arithmetic mean.
Income (Ks) No. of families
More than 75 150
85 140
95 115
105 95
115 70
125 60
135 40
145 25
Solution. First, get the class frequencies from given more than c inxuiative frequencies.
Jncome jRs) (x)
7585 : 8595 ;:..95105 105115 f15125 125135 135145 145155
(f)
10 25 20 25 10 20 15 25
N = 150
Midpoints m100
(m)
80 20
90 10
100 0
110 +10
120 +20
130 +30
140 +40
150 +50
m~lOO 10 id')
2 1 0 +1 +2 +3 +4 +5
m
20 25 0
+25 +20 +60 +60 +125
Ifd! = +245
Applying formula.
X = A +
Id' N
= 100.1^x10
= 100 + 16.33 = 116.33
ll
Arithmetic Mean is Rs 116.33.
t
iJ'ii 
158 ^
Statistics for EconomicsXI
Charlier's Accuracy Check
CO "P ^y given by Charher while
computing arithmetic mean by the shortcut method and the step deviatiL methodTa
frequency distnbution m discrete and continuous series). The formX i^af le"
^f{d + 1) = Lfd' + Zf
Equal values on both sides of the above formula is a proof of correct calculations We add one more column to a table of calculations prepared in discrete and contiCus eri
?hTcoLn ifrrr' ^^^ calculatrngTmr
^ Illustration 13. Calculate the mean for the following marks obtained in Statistics bv 50 students. Also apply Charlier's accuracy check for verifying calculations '
ff = 010 1020 2030 3040 4050 "5060 Students : 4 6 20 10 7 3
Solution.
to a of tl
Marks Students m mlS d fd' f(d'* 1)
X f 10 ) d'
010 1020 2030 3040 4050 5060 4 6 20 10 7 3 5 15 .25 35 45 55 10 0 +10 +20 +30 +401 0 +1 +2 +3 +4 4 0 +20 +2C +21 +12 0 6 40 30 28 15
N= 50 Lfd = 69 md' +1) = 119
Arithmetic Mean, X = A + ^ x C
N
= 15 +
69 50
X 10
= 15 + 13.8 = 28.8 Hence mean marks are 28.8 or 29 approx. Applying Charher's test :
md' + 1) = Lfd' + Zf 119 = 69 + 50 119 = 119 Hence, the calculation is correct.
H hi 2.
of obi
H« X
ires of Central Tendency  ^^^
ithematical Properties of Arithmetic Mean
The arithmetic mean has the following important mathematical properties : 1. The sum of the deviations of the items from the arithmetic mean is always equal to zero. Mean is a point of balance and sum of the positive deviations is equal ^o the sum bf the negative deviations.
Marks x~x
X
5 10
10 5
15 0
20 + 5
25 + 10
LX = 75 nxx) = o
 _
^ ~ N 5
= 15
Z(X  X) = Le., Ix = 0
Here, Ex or E{X  X ) = Total of the deviations from arithmetic mean. In case of discrete and continuous series Zfx or I,f(X  X) = 0 2. We can calculate the combined arithmetic mean from the means and the number of observations of two or more related groups. The combined mean formula is as under :
(f) Ttvo related groups :
 _ N1X1 + N2X 2
A 1.2  77—~T7
NI+N2
Here,
X J 2 = Combined mean of two groups
X1 = Mean of first group
X2 = Mean of second group
Nj = Number of observations in
the first group N^ = Number of observations in the second group
(«) Three related groups :
 _ N,XI+N2X2+N3X3
X
U,3 
x,=
N, =
Combined mean of three groups
Mean of first group
Mean of second group
Mean of third group No. of observations in first group
No. of observations in second group No. of observations in third group
T
■c
160
T^u £ , Statistics for EconomicsXI
The formula can be extended for more groups
groupraVuIl?''^^ of combined mean of more
Combined Mean
X
1.2.3..
= + N, X2 +N3X3+ ..Nj(„ ■
Ni+N2+N3+..N„
of /o'Sr in^IL™? cllcd^^^^^^^^^ rr ^ 
seaions A and B. lalculate the combined mean of all the students of
Solution.
Section : I. No. of Students
A 40 Xi 60 N,
B ■ 35 X2 40 N,
Here,
N, = 60, N^ = 40, X, = 40 and Xi = 35
^ ^ (60x40)+ (40x35) 60 + 40 ~
_ 2400 + 1400 100
38 the
3800 100
= 38 marks.
" ^ ^ marks.
ttere ar^n"4?s:udrsr^a^dt ^ « 
<n s^ion . are 40, fmd out ^rt s'^lS^r'''^
Section Mean —— —_ No. of Stilts
A 40 Xi 60 Nj
B ? X2 ■ 40.N,
H
Measures of Central Tendency
Combined mean (Xi,2) =38
161
N2X2
where.
V 
 N1 + N2
2 = 38, Nj = 60, N^ = 40 and Xi = 40 (600x40) + (40XX2)
38 = 38 =
60 + 40
2400 + 40X2
100 _ 3800 = 2400 X 40 X2
 40X2 = 3800  2400
40X2 = 1400
X2 = 35 Marks Hence, mean of the students of section B is 35 marks.
Illustration 16. The mean marks of 100 students of combined sections A and B are 38 marks. If the mean marks of section A are 40 and that of section B are 35. Find out the number of students in sections A and B. Solution.
M
Mean No. of Students
100
A 40X1 ? N,
B 35X2 ■ ?N,
Combined Mean (Xi,2) = 38
_ N1X1 + N2X2 N1+N2
Here Xi,2' = 38, Xi = 40, X2 = 35 and N^ + N, = 100
Hint. 1. N^ = (100  Nj)
(Nix40) + (100Ni)x35
=Too
3800 = 40 Nj + 3500  35 N^
2. 40 Nj  35 Nj = 3800  3500 
5 Nj = 300
Nj = 60
Hence, the students in section A are 60 and in Section B are (100  60) = 40.
162
3 The sum of rh for EconomicsXI
1 2
3
4
5
XX = 15
Set I 
(X  3)
(X)
2 4
1 1
0 0
+1 1
+2 4
= 10
X
1 2
3
4
5
Set II ~
X~2 (X2)^
(x') (x'^J
1 1
0 0
+1 1
+2 4
+3 9
= 15
H   N 5 
^ V ~ ^' ' f taken from mean.
W. .. ' 'f"' ''^v'^tions taken from any value
cha^:o, eh.. ^^^^^^^
4. Arithmetic mean i. calculated by a simple formula, ,>., h  E
three values of the formula are known, the third can be calculated"
_ J^X
Tf , X = or IX = NX
if any two of the
X ■ , ^ ' ^^^^^——1 X
10 30
20 30
30 30
40 30
50 30
i:x = 150 150
Measures of Central Tendency
163
150
= 30
N
NX = SX 5 X 30 = 150
150 = 150. ^
This property has great utihty in calculatton of wage bills, e.g., average wage Rs 120.
No. of workers 2000.
... Total wage bill = N. X = 120 x 2000 = Rs 2,40,000.
The relation NX = ZX can be easily used for correcting the value of mean, which is
explained in the following illustration. ^
m„s„ado„ 17. ne arithmetic mean of a series of 40 as Rs 265. Bnt while calculating ii an item Rs 115 was misread as Rs ISO. Fmd the correct
arithmetic mean.
Solution.
Since,
_ EX X =
N
EX = NX Here, X = 265, N = 40
EX = 40 X 265 = 10600
Calculated EX, i.., 10600, is wrong as the us get correct EX by subtracting the incorrect item and adding the correct item
Incorrect EX = 10600
Less : Incorrect item 1^0
10450
Add : Correct item
Correct ^^X = 10565
Hence, corrected arithmetic mean
= Rs 264.12.
40
Illustration 18. Tie arithmetic mean of a given set of data
(i) in terms of rupees, and (it) in terms of paise.
164
f
Solution. Since
Statistics for EconomicsXI j
Here,
N NX
■"''e. of observations,
Calculated ZX i c Tnn values. Let us cor^t M u "" "" observations are Rs 5 less ,1, ^
value = 25). ^ « correct H (5 observati^s x Rs f^st T
Incorrect ^
^dd : Corrected balance
of 5 observations (5x5) 25 Correct —
es .
Corrected ZX 525 ^ ^ .
N "  — = Rs 105
(«) Corrected mean expenditure in terms of n '
Rs 105 X 100 = 10500 ^^
= paise 10500.
™ean of 5 items (1, 2 3 4 5 . • . •
* ' 4, 5,) is 3, I.e.,
1+2+3+4+S EX IS
value, sa. 2. we get tbe
Alisi
(
ll
requ ]
£
Lo to be < examp class M 2.1 Exa
1
2
3
4
__^
IX = IS X = 3
X + 2
3
4
5
6
25 5
X~2
1 0 +1 +2 +3
Xx2
Column 1 : X = 3 (+) Column 2 :X  i: a j j j
(X) Column = 2) = 1
^ = 6 Multiplied 2 = (3 X 2) = 6
5 1
2 4 6 8 10
30 6
We a
JVIaifc Studo
Cunn before ap Thus Marks
Stud^
165
, after aadition. s.b„action and .uUip^on by
b»caon and *a.on by *e same
constant to their means.
Miscellaneous Problems . , ■ u .wUrr^f^uc mean These problems are
re^r rb^^e rS^^e mean.
1. In the case of openend classes Example
Marks
Below 10 1015 1520 2025 Above 25
No. of Students
5 8
3
4
5
Lower limit of the first class and upp. W o^^ j^^r^^L^niS^^ to be defined by marking an —and last classes. Thus, first
example, the same class mterva IS decided M., 5)
class would be 510 and last class 2530.
2. In the case of cumulative frequency distribution
Example
Less than 10 Less than 15 Less than 20 Less than 25 Less than 30
510 5
515 13
520 16
525 20
530 25
We are given Marks :
^tive frequencies ar;:equired to be converted into ascending class frequencies
Thus we get, ^^^^
Marks : 510 1(^5 ^^^^^^ ^^016)
Smdents : 5 =8  ^
= 4
2530 (2510) = 5
■tr
m
166
Example :
Statistics for EconomicsX,
We are given, Marks : Students
Marks No. of Students
More than 5 More than 10 More than 15 More than 20 More than 25 25 20 12 9 5
530 25
1030 20
1530 12
2030 9
2530 5
.ue^citTef: "r " above descending cL„,a.ive
frequencies. They are : Marks : 51 o (2520) Students : =5
1015 (2012) = 8
1520 (129) = 3
2530
2025 (95)
= 4 5
Merits and Demerits of Arithmetic Mean
Merits : Arithmetic Mean ,s the most popularly used because of the following merits ■ . 1. It IS simple to understand and easy to calculate. "
2. It is based on all the observations of the series Therefore it i.' th representative measure.^neretore, it is the most
3. Its values is always definite. It is rigidly defined and not affected by personal bias
4. The calculation of arithmetic mean does not require any specific algement of
^iS m SS^~   ^re mathematical
7 It is fluctuations of sampling and ensues .ability m calculations.
/. tt IS a good base for comparison.
n«h''o7o?:fcuI onTj^^^i^^^^^^^^ '^e average and
be used with caution mean should
ffc^o 000 a " General Manager's sala;, in a firn,
rnfpe^f Sloor"^' P'oveey clerk Rs jjoo, typiS^Rs
167
Measures of Central Tendency
„ ^ Rs on 000+ Rs .S.500 + Rs 4,500+ Rs 2,000 _ g oqO per The average salary will be4
month. Average calculation is not  "presenmive^ I. is affected by an extreme value of Rs 20,000 paid to the General Managet; ,
4 Arithmetic mean can be a value that doe. not extst m the senes at all, ..g., the average of 4, 8 and 9 is = 7, which is not an item of the series.
5 Arithmetic mean gives more impot«,nce to the bigger items and less importance to
titr^cflXaecided inst by observa.on. It needs mathematica. calculations.
(B) Weighted Arithmetic Average or Weighted Mean
1. Meaning
2. Uses of Weighted Mean
3. Calculation of Weighted Mean
(a) Equal Weights
(b) Unequal Weights
arithmetic mean gives equal importance tt/aif the ^^^^^ fact, thL are number
One item may be more impot^mt to different
hi—rlorl^^^^^^^^ ^o^ as jeights. ,n other
wir«=ights are figures to indicate d>e relative mtportauce of ,tems.
1, « » "ve Cual weigh, to different categories of employees in a fa«o^
2. WeStled mean is used for comparison of the results of two or more un,verstt,es
or boards. , . u ^
3. It is used' to calculate standardised birth rate and death rate.
, 4. It is used in the construction of Index Numbers.
168
Statistics for Economics XI
Calculation of Weighted Mean
The formula for calculating weighted arithmetic mean is as under :
IWX
Xw =
where.
Xw = Weighted Arithmetic Mean W= Weights X = The variables Steps. (/) Multiply weights by X and obtain WX
Hi) Divide the total (ZWX) by total weights (XW)
Solution.
of payment of wage per hour by three ways
Simple Arithmetic Mean (x) :
Workers
Man Woman Child 8 6 4
ZX=18
EX
X =
N
8 + 6 + 4
18
: ■: 3 .
= Rs 6 per hour.
(a) Weighted Mean (Equal Weights) : (Xit;)
terSn'   —  ana
•n
Type SWlfittlBil^Bl® Wages (Rs) ■ X Workers W
Man Woman Child 8X, 6 X, 4X3 50 Wj 50 50 W3400 300 X^W^ 200 X3 W3
I.W= 150 IWX = 900
——
'ires of Central Tendency
169
Xw
X,w, + XM+2W  =Rs6
LW 150
Weighted Mean is Rs 6. . n
Thus, weighted arithmetic mean will be equal to the simple arithmetic mean, when all
: items are given equal weights.
Xw = X Rs 6 = Rs 6.
lb) Weighted Mean (Unequal Weights) : (X«^)
Suppose men, women and child workers are 10, 20 and 50 respectively then our
Vorkers
lyfJtr 'x W  V  _U^^^^
Man 8 10 80
Woman6 20 120
Child 4 50 200
l.W = 80 ZWX = 400
 SWX 400 Xw =
= Rs 5
LW 80
Thus, the weighted arithmetic mean will be less than the simple arithmetic mean when items of small vflues are given greater weights and items of big values are given less
weights. _ _
Xw < X
Rs 5 < Rs 6 . u
However, in the absence of given weights, assumed weights can be assigned to the
items on the basis of their relative importance.
But, normally they are not equal. Suppose men, women and child workers are 50, 20 and 10 respectively, then our answer would be different.
Type
Man
Woman
Child
X
Workers W
50 20 10
SW = 80
WX
400 120 40
ZWX = 560
__ ZWX Xw =
560 80
= 7
Weighted Mean is Rs 7.
^ Statistics for EconomicsXl
Thus the weighted arithmetic mean will be greater than the simple arithmetic mean
when items of small values are given less weights and items of big values are given mor^ weights.
Xw > X Rs 7 > Rs 6
niustration 20. Calculate Weighted Mean by weighting each price by the quantity consumed.
Articles of Quantity Consumed Price in Rs
Food (per kg) 3
Flour 11.50 5.8
Ghee 5.60 58.4
Sugar .28 8.2
Potato .16 2.5
Oil .35 20.0
Solution.
fiiBsBslPlSSaHi^B^^^^B Food Articles Price in Rs per kg Qty. Consumed in kg W
Flour 5.8 11.50
Ghee 58.4 5.60
Sugar 8.2 .28
Potato 2.5 .16
Oil 20.0 .35
Total 17.89
WX
iMM
66.700 327.040 •2.296 0.400 7.000
IWX = 403.436
.Xw =
ZWX 403.436
= 22.55
I.W 17.89 Weighted Mean Price is Rs 22.55.
lUustration 21. From the results of the two schools A and B given below, state which or them is better.'
Oass
IX
X
XI
XII
Total
School .A
Appeared
30 50 200 120
400
Passed
25 45 150 75
295
School B
Appeared
100 120 100 80
400
Passed
80 95 70 50
295
171
Measures of Central Tendency
ntf Use Weighted Anthmetk Mean after obtaining homogeneous figures, converting into percentages.
School A
Class Appeared w Passed Pass % X WX
IX X XI XII 30 50 200 120 25 45 150 75 8.33 90 75 62.50 2499 4500 15000 7500
LW  400 LWX = 29499
School B
Class Appeared W Passed Pass % X WX
IX X XI XII 100 120 100 80 80 95 70 50 80 79.2 7062.5 8000 9504 7000 5000
XW = 400 EWX = 29504
School A : LWX 29499 _ 73 75 Xw  400
_ SWX _ School B : Xw 
School B is better.
29504 400
= 73.76
m„s«a»„ 22. An exannnaaon was held to decide the award of a hola*
Subject
Statistics Accountancy Economics Business Studies
Weight
4
3 2
i
Marks of A
63 65 58 70
Marks of B JAarks of C
60 64 56 80
65 70 63 52
DUSIIICSS jiuun^o _________
Of Ihe candidate gening the highest marks .s to be awarded the scholarsWp, who should get it?
172
Statistics for EconomicsXI
Solution.
Subfect Weight Marks of A Marks ofB Marks of C
W X, WX, WX,
Statistics 4 63 252 60 240 65 260
Accountancy 3 65 195 64 192 70 210
Economics 2 58 116 56 112 63 126
Business Studies 1 70 70 80 80 52 52
Total IW = 10 EWXj = 633 SWX^ = 624 EWX3 = 648
EW 10
2:WX2 624
LW 10
EWX3 648
^t Vt ./v O^ I
Weighted Mean of B, Xw^ = ^^^= 62.4 Marks.
Weighted Mean of C, Xw^ = ^^^ = Marks.
The weighted Mean of C is the highest, hence he is entitled for scholarship.
OF FORMULAE AND ABBREVIJmONS
[Arithmetic Mean, Properties and Weighted Mean]
Type of Series 1. Individual Observations (Ungrouped data) Direct Method  ZX N Shortcut MethodSfi^ Deviation Mjethod X = A+'^f'xC N
2. Discrete Series (Grouped data) ^ N XA + ^^f xC N
3. Continuous Series (Grouped data)  Lfm ^ N X = A + N
Mathematical Properties of Arithmetic Mean Here, XX=x Mathematically. (1) 2.(X  X) = 0 ljc = 0 If(X  X ) 0 Ifx = 0 , Properties of ■ Arithmetic Mean (3) E(X  X )Ms the least, i.e., Ix^ is minimum ZWX Weighted. Mean Xw = ^^  N1X1+N2X2 Similarly  N1X1+N2X2 + N3X3 N1 + N2 + N3 (4) NX = IX
ii
Measures of Central Tendency
173
Abbreviations
X = Arithmetic Mean. d = XA, i.e., deviations of X
variable from
x = The variables. an assumed mean.
zx Sum of all the items of the U = Sum of the deviations of X variabic i
variable X. taken from an assumed mean. 
N = Number of observations or (Z/). C = Common factor.
f = Frequency. d' = X — A , i.e., step deviations of C
E/X = Sum of the product of variable (X) Xvariable from assumed mean
and the frequencies (/). and divided by common factor.
m = Midvalues. Yd' = Sum of step deviations.
lfm = Sum of the products of midpoints Lfd = Sum of the product of frequencies
and the frequencies. and their respective deviations.
A = Assumed mean lLfd' = Sum of the product of deviations
and their respec ive step deviations.
XX = X, i.e., deviations of X variable
from the mean.
= Combined mean of two groups. (X Xf = X', i.e., square of the deviations
of X variable from mean.
Xi : Arithmetic Mean of the first group. Xw = : Weighted Arithmetic Mean.
X,  : Arithmetic Mean of second group. W = : Weights.
N. : Number of observations in the first zwx = : Sum of the product of variable
1 group. X and weight.
N, = : Number of observations in the
second group.
EXERCISES
Questions :
1. What is a statistical average? Mention different types of averages.
2. What are the functions of an average? Discuss the characteristics of good average. Which of the average possesses most of these characteristics?
3. What is meant by 'Central Tendency'? Discuss the essentials of a measure of central tendency.
4. Name the commonly used measure of central tendency.
174
StattsUcs for EconomicsXI
5. Define the mean. Also explain properties of mean.
6. Why xs arithmetic mean is the most commonly used measure of central tendency^
Llutionr "" """ ^^ ^  fr^qncy
  ^ sure
' ^of the va^es of the vanable from
in ^ " unweighted mean?
lU. What are the uses of weighted mean?
II.. Write notes on (a) Central Tendency, and (b) Weighted Mean. Problems :
1. Calculate arithmetic averages of the following information :
(a) Marks obtained by 10 students : 30, 62, 47, 25, 52, 39, 56, 66, 12, 24
(b) Income of 7 families (In Rs) : Also show = o 550, 490, 670, 890, 435, 590, 575
ic) Height of 8 students (In cm) :
140, 145, 147, 152, 148, 144, 150, 151
2. .u______________________• " : ^ = 600, c = 147.12 cm.]
Name of batsman Matt ch I Match II Mutch m
1 ■ Inning II InningI Inning 11 Inning I Inning II Inning
A B C D 60 40 100 20 20 50 10 40 26 60 8 46 10 36 18 84 100 70 100 42 40 80 140 52
, , , ••^ = 42.67, B = 56, C= 62.67 and D = 47.33]
Calculate mean of the following series •
5 6 7 8 9
Frequency : 6 12 15 28 20 14
^ fI^J^/"^ expenses of following 10 firms
irmx . . ^ , . ^
65 15
10 5
[X= 7.06]
Firms Sales (Rs in '000) Expenses (Rs in '000)
1 50 11
2 50 13
3 55 14
4
60 16
5 65 16
7 65 15
8 60 14
9 60 13
10 50 13
[X= Sales = Rs 58,000, Expenses = Rs 14,000]
ieasures of Central Tendency
Calculate mean of the following frequency distribution
62 64 67 70 73 82 103 176 212 180
Values Frequency
60 54
77 115
n
Calculate arithmetic mean of the followmg data Profit (in Rs) : 010 1020 2030
No of shops : 12 18 2/
3040 20
175
81 85 89 78 _ 50 21 [X: R> 70.94]
4050 5060
16
[ X = Rs 30.45]
Compare the average age of mal^injhe_ two countries :_____________________
population of U.K.
Age Group
05 510 1015 1520 2025 2530 30^0 4050 5060 6065
(in lakhs)
214 258 222 157 145 161 267 184 120 100
18
19
20 18 16 14 27 25 19 17
8.
[Average Age India = 25.25 years and UK = 29.404 years] Calculate simple and weighted ar^hmetic averages of the folbwing items :
Items Weights Items Weights
68 1
124 9
85 46 128 14
101 31 143 2
102 1
146 4
108 11 151
6
110
7
153 5
112 23 172 2
113 17
[Simple Mean = 121.07 and Weighted Mean = 108.71)
Marks No. of Students
Less than 10 Less than 20 Less than 30 Lesi than 40 Less than 50 : 5 15 55 75 100
_ A = OU iViaiivai
Also get 5:f(XX) = 0
176
11.
12.
13.
14.
15.
Statistics for Economics
There are two branches of an establishment employmg 100 and 80 nerso,
rrS monthly salrfes by thrtwo b^rh
are Rs 27^ and 225 respectively, fmd out the arithmetic mean of the salaries the employees of the establishment as a whole.
[Combined Mean = Xi,2 = Rs 252.81
to be"49 Tth' by a group of 100 students were found
If Too ; H obtained in the same examination by another group
of 200 students were 52.32. Fmd out the mean of marks obtained by both th g^ups of students taken together. ^ ^^ 3,
The mean marks of 1 >0 students were found to be 40. Later on it was discovere
— Ll ' ^^ —^^  correspondingTot
ru • , [Correct X = 39.7 Marksl
The mean weight ot 25 boys in group A of a class is 61 kg and the mean weiS
of .5 boys m group of the same class is 58 kg Find the mean weight of 60 b^!
.. , , . [Xj.2 = 59.25 kg]
Calculate mean ot the following data :
Marks Beloiv : 10 20 30
No. of Students : 5 9 17
40
29
50 45
60 60
16. Calculate Combined Mean
70 80 90 100 j 70_ 78 83 85 [X = 48.41 Marks]
Section Mean Marks No. of Students
A 75 50
B 60 60
C 55 50
17 , ■ [Xi,2,3 = 63.125 Marks]
average ot 31 marks. What were the average marks of the other students.'
ru [X2 = 57.25 Marks]!
R^18r4"T 1000 workers of a factory was found to be
taken af29ranri67" 'T?"' ^^ workers were wron^
taken as 297 and 165 mstead of 197 and 185. Find the correct mean. '
[Corrected Mean : Rs 180.32]!
19. Find the average wage of a worker from the following data •
: Above 300 310 320 330 340 350 360 3701 No. of ivorkers : 650 500 425 375 300 275_ 250 100
[X = Rs 339.23]j
Measures of Central Tendency
177
effip* ........ ^ o. of day
40 to 30 10
30 to 20 28
20 to 10 30
10 to 0 42
0 to 10 65
10 to 20 180
20 to 30 10
[X = 4.29 °C)
21. A candidate obtains tbe followmg percentage of marks : Sanskrit Mathemat^ 84, Economics 56, English 78, Politics 57, History 54, Geography 47. ^ is agreed to give double weights to marks m Enghsh, Mathematics and Sanskrit. What is he weighted and simple arithmetic mean? = 68.8, X = 64.43 Marks]
22. Calculate weighted mean by weighting each price by the quantity consumed:
Food items
Flour
Ghee
Sugar
Potato
Oil
Quantity Consumed
500 kg 200 kg 30 kg 15 kg 40 kg
Price in Rupees {per kg)
1.25 20.00 4.50 0.50 5.50
[Xw = Rs 6.35]
23. Comment on the performance of the students of three universities given below using weighted mean :
No. of Students are in hundreds
Courses of study Mumbai Kolkata Cher tnai
% pass No. of students % pass No. of students % pass Np. of students
M.A. M.Com. B.A. B.Com. B.Sc. M.Sc. Tl 83 73 74 65 66 4 5 2 3 3 S2 76 73 76 65 601 3 6 7 3 7 81 76 74 58 70 73 2 3.5 4.5 2 7 2
[Weighted Mean
Mumbai : 72.55, Kolkata : 7U.6 and Chennai : 72.0; Mumbai is better]
24. A distribution consists of three components with total frequencies of 200, 250 and 300 having means of 25, 10 and 15 respectively Find out the mean o^ combined
: distribution. =
Chapter 9
TOOTioMHTnaet ui. rmmmv«jm
(a) Median,
(b) Partition Values (Quart,les), and
(c) Mode.
median
1. Definition
2. Calculation of Median
3. Mathematical Properties of Median
4. Merits and Demerits of Median
Definition
^ev^r shit t s^r z"   e
sxss
licsXI Positional Average and Partition Values 201
According to AX. Bowley, "If the number of the group are ranked m order according to the measurement under consideration then the measurement of the number most nearly one half ts the median." ^
■ According to Secrist, "Median of a series is the value of the ttem actual or estimated
tvhen a sertes ts arranged in order of magnitude which divides the distribution into the tivo parts.
nL! O™'''''' "" heights of 7 students in a class.
; 'uf "sr
147
151
140
Anurag
Deven
149
M
Suresh
142
At
Mayoor
147
AtuI
144
144
Satish
145
145
Himankar
The first and most important rule for obtaining the median is that the data should be arranged in an ascending (increasing) or descending (decreasing) order. This arrangement facditates locating the central position so that the series may be divided into two parts one less than the central value and the other more than the central value. '
So, we arrange our data in ascending order as follows :
140
142
144
145
147
149
151
mm
Deven
Mayoor
Satish
Himankar
AtuI
Suresti
f
■
Anurag
If we arrange the above data in descending order we get : Name of
smdents : Anurag Suresh Ami Himankar Satish Mayoor Deven Height (cm) : 151 149 147 145 144 142 140
From this ordering also we observe that 145 cm or value of the 4th item is the median.
Calculation of median
(a) Individual observations.
(b) Discrete series.
(c) Continuous series.
Median is the central positional average of given data. That is, median has a position more or less at the centre of the values and it divides the series roughly into equal parts.
180
Statistics for EconomicsXI
{a} Individual Observations
meiarhetht. ^ ^ impute the
Solution.
Name of Students Height (cm.)
Anurag 151
Deven 140
Suresh 149
Mayoor142
Atul 147
Satish 144
Himanka 145
Name of Students Height (cm.)
Deven 140
Mayoor142
Satish 144
Himankar 145
Atul 147
Suresh 149
Anurag 151
Steps :
1. The above data must be arranged either in ascending or descending order to get the value of median. Arrange the data in ascending order.
\th
item
2. Locate the median by finding the size of
3. Applying the formula, we get
fN+^^
Me = Size of
= Size of
fN + V
th
item
Pos
But
fN
h
the h
7 + 1
item
= Size of 4'"' item Median is the Himankar's height, i.e., 145 cm
8th tTm^^ cm, which will be the
»tn item m the list, and calculate the median height.
Solution. When the number of items in an individual series is 3, 5, 7, 9, 11 etc. that
'N+iK ■
th Item will be a whole number.
nil 1
is when it is an odd number, the central item, i.e..
XI the
f ositional Average and Partition Values I g j
But when the number of item in a series is even 2, 4, 6, 8, 10 etc, the central item, /.e.,
N + V
the item will be in fraction.
Arranging the data in ascending order including the height of Rajesh, we get
et
Name of the Students Heii^ht (cm.)
Deven 140
Mayoor142
Satish 144
Himankar 145
AtuI 147
Suresh 149
Anurag 151
Rajesh 152
Me = Size of
= Size of
fN + lX^ .
2 , +
Item
Item
= Size of 4.5'*' item
Medkn is estimated by finding the arithmetic mean of two middle values, i.e., adding the height of Himankar and AtuI and dividing by two. ' &
Size of 4.5"^ item = item + item
2
145 + 147
292
Median height = 146 cm.
Serial No. Marks Serial No. Marks Serial No. Marks
1 17 7 41 13 1 1
2 32 8 32 14 15
3 35 9 11 15 35
4 33 10 18 16 23
5 15 11 20 17 38
6 21 12 22 18 12
JI
182
Statistics for EconomicsXI """sed ,„ an ascending order in the
Serml No. Marks Serial No. Marks Serial No. Marks
1 2 3 4 5 6 11 11 12 ' 15 15 17 7 8 9 10 11 12 18 20 211 22 J 23 32 13 14 15 16 17 1832 33 35 35 38 41
Median = Size of the
item = Size of the
18 + V
th
= 9.5'^ item
The value of 9.5"' item = .Z^lue of the 9"* item + Value of the 10^'' item
= 11^.21.5.
Hence Median = 21.5 (b) Discrete Series
Illustration 4. Calculate median of the followmg distribution :
Solution.
Marks No. of Students
10 1
20 8
30 16
40 26
50 20
60 16
70 7
80 4
Marks
10 20 30 40 50 60 70 80
No. of Students
1 8 16 26 20 16 7 4
N =99
Ctwtulatiue frequencies c.f
1=1 10 = 2 26 = 2 52 = 2 72 = 2 88 = 2 95 = 2 99 = 2
16 16 16 16 16 16
26 26 26 26 26
20 20 20 20
16 16 16
up to (c) Cc
nil
the m(
7
7 + 4
Positional Average and Partition Values
183
Steps :
1. Arrange the data in ascending or descending order.
2. Compute cumulative frequencies.
3. Apply the formula.
Me = Size of
fN + n
th
Item.
4. Median is located at the size of the items in whose cumulative frequency, the value
of
(N + U
th
item falls.
Median = Size of
= Size of
(N + l
th
2
(99 + ^^
Item
= 50th item
Median Marks = 40 Marks.
Illustration 5. Find out the value of median from the following data : Daily wages (in Rs) : 100 50 70 110 80 Number of Workers : 15 20 15 18 12
Solution.
Wages in Number of Cumulative
Ascending Order Workers Frequencies
(Rs) (f) (c.f.)
50 20 20
70 15 35
80 . 12 47
100 15 62
110 18 80
Median is the value of
fN + l
rso+i^i
or
th
or 40.5'*' item. All items from 35 onwards
up to 47 have a value of 80. Thus the median value would be Rs 80. (c) Continuous Series
Illustration 6. The size of land holdings of 380 families in a village is given below. Find the median size of land holdings.
Size of Land Holdings No. of Families
(in acres)
Less than 100 40
100200 89
200300 148
300400 64
400 and above 39
J'
184
IvV
i
Solution.
Statistics for EconomicsXI
Size of Land Holdings (in acres) No. of families (f) Less than cumulative frequencies
0100 40 40
100200 89 129
200300 148 111
300400 64 341
400500 • 39 380
Steps :
1. Compute less than cumulative frequencies.
u
th
item. Do not use
2. Median item is located by finding out size of
the item in continuous series.
3. Locate the median group in cumulative frequency column where the size of
fN^''
the item falls.
4. Apply the following formula to calculate the median from located group :
—— c.f. Median = /j +  x i
where, = Lower limit of median group.
c.f = Cumulative frequency of the class preceding the median class. f = Frequency of the median group.
I = The class interval of the median group. Calculation of Median
Me = size of
= size of
2
380
item
item = 190^'^ item
Median lies in the group 200300. Applying the formula, we get
cf Me = /j + :
X i
Ositional Average and Partition Values
where, /, = 200, f = 190, c.f. = 129 f = 148, /• = 100
185
Me = 200 + 1^1^x100
= 200
= 200 +
148
61x100 148
241.216
148
•. Median size of land holding = 241.22 acres, (ie 50% of the families are having less than or equal to 241.22 acres of land holdingr^d 50% of famihes are having more than or equal to 241.22 acres of land
holdings.)
Illustration 7. Calculate median from the following data :
Age (in years)
5560 5055 4550 4045
Number of Age
Persons(m years)
(f)
'7 3540
13 3035
15 2530
20 2025
Total
Number of Persons (f)
3U 33 28 14
160
«I
Note : If the given question is in deseending otdet of values then Wore giving the question, the dafa is Required to arrange ■„ ascending order to calculate less than
cumulative frequencies. . . ,
Solution. This question has been solved below after arranging the series m ascending
order. "___
Age in years (Ascending order)
2025 2530 3035 3540 4045 4550 5055 5560
No. of persons (f)
14
28, 33 30 20
15 13
7
Cumulative frequency(c.f.)
14 42 75 105 125 140 153 160
186
Statistics for EconomicsXI
In the above example median is the value of lies in 35^0 class interval.
N , Me = + X i
8075
f^l th or ri6o> th
.1) I 2 ; or
or 80''^ item which
= 35
= 35
30
5x5 30
X 5
= 35 + 0.83 = 35.83 .•• Median Age = 35.83 years
Illustration 8. Calculate the median from the following data :
Value Frequency Value Frequency
(f) (f)
Less than 10 4 Less than 50
Less than 20 16 Less than 60 112
Less than 30 40 Less than 70 120
Less than 40 76 Less than 80 125
Solution. If the data are given in the form of cumulative series they have to be converted into simple series in order to find out the frequency of the median class which IS needed m calculation of median. Once it is done that rest of the procedure is the same as in any other continuous series.
Value Frequence (f) Cumulative frequency {c.f.)
010 4 4
1020 12 16
2030 24 40
30^0 36 76
4050 20 96
5060 16 112
6070 8 120
7080 5 125
ha' on
Middle item is
ri25
xth
or 62.5* item, which lies in 3040 group.
licsXI
Positional Average and Partition Values
187
ich
^ c.f. Me = /, +  X i
^^ 62.540 = 30 + —trr X 10
= 30 +
36 22.5x10
36
= 30 + 6.25 Median = 36.25
Illustration 9. Calculate the median from the following data
Size Frequency (f)
More than 50 0
More than 40 40
More than 30 98
More than 20 123
More than 10 165
Solution. e>umulative frequency taoie is oi more man type, in !.u».u eases mc ucua have to be converted into a simple continuous series and median is calculated of ascending order series. ,,
be lich me
Size Frequency (f) Cumulative frequency (c.f.)
1020 42 42
2030 25 67
30^0 58 125
4050 40 165
ri65Y'
Middle item is —^ or 82.5* item which lies in 30^0 group.
th
e I
N
Me = /j +  X i
^ 30 , iM^iZ >, 10
= 30 +
58 15.5x10
58
= 30 + 2.67 Median = 32.67
188
Statistics for EconomicsXI Illustration 10. Compute median from the following data • MMues : 115 125 135 145 155 165 175 185 195 Frequency : 6 25 48 72 116 60 8 22 3
he'^^rdls^^Z^: Th 7fT ^^^ of ^he classintervals of a contmuous
trequency distribution The difference between two midvalues is 10 hence 10/2  5 i.
upper limit ot a class. The classes are thus 110120 170 l^n ^ ^
190200. ' .... and so on up to ——
Uassintervals
110120
120130
130140
140150
150160
160170
170180
180190
190200
Total
Frequency
6 25 48 72 116 60 38 22 3
390
_
6 31 79 151 267 327 365 387 390
The middle item is
(390^
th
or 195"' item, which lies in the 150160 group.
Me = +  ^
195151 = 150 + r^^X 10
116
= 150 +
44
X 10
116
= 150 + 3.79 Median = 153.79
Illustration 11. if the arithmetic mean of the data given is 28 Find rh. I ^ ■ • frequency, and (b) the median of the series. ^ ^^^
Profit per
Retail shop (in Rs) : 010 1020 2030 Number of
Retail shop j2 jg
3040
27
4050 17
5060 6
Positional Average and Partition Values
189
Solution.
{a) Calculation of missing frequency. Let the missing frequency of group 3040 he X.
Profit per Retail shop X Number of retail shops f Midpoint m ' —1 fm
010 12 5 60
1020 18 15 270
2030 27 25 675
3040 X 35 35X
4050 17 45 765
5060 6 55 330
N = 80 + X Y.fm = 2100 + 35X
Applying formula, we get
Ifm
X =
or
28 =
Ifm If " N
2100+ 35X 80 + X
28 X (80 + X) = 2100 + 35 X 2240 + 28 X = 2100 + 35 X 2240  2100= 35 X  28 X 7X = 140
140
X =
7
= 20
Therefore, the missing frequency is 20. (b) Calculation of median :
Profit (Rs) (X) Fret, mcHC\ (f) (cf)
010 12 12
1020 18 30
2030 27 57
3040 20 77
4050 17 . 94
5060 6 100
N = 100
The middle item is
100^
or 50th item, which lies in the 2030 ^group.
190
Statistics for EconomicsXI
Me = /j + ly X
in ^030 20x10
= 20 + —X 10 = 20 +
27 " ■ 27
= 20 + ^ = 20 + 7.407 Median = 27.41.
Illustration 12. In the frequency distribution of 100 famiUes given below, the number of families corresponding to expenditure groups 2040 and 6080 are missing from the table. However, the median is known to be 50. Find the missing frequencies. Expenditure : 020 2040 4060 6080 80100 No. of families : 14 ? TI 15
Solution. Let the missing trequency of the group 2040 be X and the missing frequency of 6080 group be Y.
Now Z/" (total frequency) = 100
i.e., 100 = 14 + X + 27 + y + 15
or X + Y = 100  14  27  15 or X + Y = 44
Expenditure No. of Families (f) Cumulative frequency (c.f.)
020 14 14
2040 X 14 + X
4060 27 41 + X
6080 Y 41 + X + Y
80100 15 100
Median is given in this problem as 50.
^lOOV*'
Middle item of the series is also interval 4060. (Given median = 50)
N
or 50* item, which means it lies in the class
Now,
Me = /. +
c.f.
f
X t
50 = 40 +
50[14 + X] 17
X 20
50  40 =
27^20
f ositional Average and Partition Values 191
10 X 1.35 = 50  [14 + X] 13.5 = 50  14  X
X = 50  14  13.5 = 22.5 Since the frequency in this problem cannot be in fraction so X, i.e., f^ wouId be taken as 23.
X + Y = 44 or /■j + = 44
/; = 44  fj or 44  23 or 21 Thus the missing frequencies in the question are 23 and 21.
Mathematical Properties of Median
1. Median is an average of position and therefore is influenced by the position of items in arrangement and not by the size of items.
2. The sum of the deviations of the items about the median, ignoring ± signs, will be less than any other point. For example :
X : 10 11 12
Deviations from Median : 2 10
Deviations from
any poirit, (say 10) : 0 1 2
The sum of the deviations taken from median (12), less than the sum of the deviations taken from an\
13 1
14
2
= t>
f — »J
ipomt (1®
Merits and Demerits of Median ^^''^i^ilJAN t
Merits
1. It is easy to calculate and understand.
2. It is well defined as an ideal average should be and it indicates the yalue of the middle item in the distribution.
3. It can be determined graphically, mean cannot be graphically determined.
4. It is proper average for qualitative data where items are not converted or measured but are scored.
5. It is not affected by extreme values.
6. In the case of openend distribution it is specially useful since only the position is to be known. It is useful in a distribution of unequal classes.
Demerits
1. For median data need to be arranged in ascending or descending order.
2. It is not based on all the observations of the series.
3. It cannot be given further algebraic treatment.
4. It is affected by fluctuations of sampling.
192
Statistics for EconomicsXI
5. It is not accurate when the data is not large.
6. Interpolation by a formula is required to calculate median in continuous series This reqmres the assumption that all the frequencies of the class interval are uniformly spread which is not always true.
partition values (quartiles)
1. Definition
2. Characteristics of Partition Values
3. Calculation of Partition Values
Definition
When we are required to divide a series into more than two parts, the dividing places are known as partition values. Suppose, we have a piece of cloth 100 metres long an^d we have to cut it into 4 equal pieces, we will have to cut it at three places. Quartiles are those values which divide the series into four equal parts. For getting partition values the most important rule is that the values must be arranged m ascending order only. In the case of finding out the median, we can arrange the data either m ascending or in descending order but here there is no choiceonly ascending order is possible for calculating partition values (Quartiles).
For example, we have the following data of heights of 7 students in a class • Name of students : Anurag Deven Suresh Mayoor Atul Satish Himankar Height (cm) : 151 140 149 142 147 144 145
Therefore, for getting correct results, the data must be arranged in ascending order in all the cases.
Characteristic of Partition Values
The difference between averages and partition values is as follows :
While an average is representative of whole series, quartiles are averages of parts of series For example, the first quartile is the average of first half of the series and third quartile is the average of the second half of the series.
Thus, quartiles are not averages like mean and median. They help us in understanding
how various "ems are spread around the median. Therefore, the special use of partition
values IS to study the dispersion of items in relation to the median, that is in understanding the composition of a series.
Calculation of Partition Values
(a) Individual Series.
(b) Discrete Series.
(c) Continuous Series.
Positional Average and Partition Values
Now we arrange the data (in Illustration 1) in ascending order :
193
Name of Students (cm)
Deven 140
Mayoor142 = Q == First quartile or lower quartile
Satish 144
Himankar 145 = Me = = Second quartile or middle quartile
AtuI 147
Suresh 149 = Qj = Third quartile or upper quartile
Anurag 151
AS we Know tne meuiaii is uic nci^iii. iwuim 
cm. Now, suppose we have to calculate quartiles. By definition quartiles will divide a series into four equal parts and so number or quartiles will be three. They are known as lower quartile, middle quartile and upper quartile. These are also called first, second and third quartiles.
The middle or second quartile (Q^) is the central positional value of the data, i.e., median. The first or lower quartile (Qj) is the central positional value of the lower half, and third or upper quartile (Q3) is the central position value of upper half of the data. In the above data, (Q, = 142, Q, = 145 and Q, = 149.
It must be remembered that Q, is always less than Q^ and Q3 (Q^ < Q^ and Q3) and median falls between Qj and Q3.
(a) Individual Series
Illustration 13. From the following information of wages of 30 workers in a factory calculate median, lower and upper quartile.
S. No. Wages (in Rs) S. No. Wages (in Rs)
1 330 16 240
2 320 17 330
3 550 18 420
4 470 19 380
5 210 20 450
6 500 21 260
7 270 22 330
8 120 23 440
9 680 24 480
10 490 25 520 .
11 400 26 300
12 170 27 580
13 440 28 370
14 480 29 380
15 620 30 350
194
Solution.
1. Arrange the data in ascending order.
2. Locate the item by finding out.
fN+iY^ rN+v""'
and
N + 1
Nth
items
Median
Me = size of
N + l
2
th
Item
r 30 + 1* = size of —J item
= 15.5* item
^ size of 15th item + size of 16th item
380 + 400
Statistics for EconomicsXI
S. No. * Wages (in Rs) S. No. Wages (inRs)
1 120 16 400
2 170 17 420
3 4 5 210 18 440
240 19 440
260 20 450
6 270 21 470
7 o i 300 1 ^ 22 480
0 1 320 23 480
9 330 24 490
10 11 12 13 14 15 330 25 500
330 350 370 380 380 26 27 28 29 30 520 550 580 620 680
F
I
Uj
{b)l I
Calci
= 390 Median is Rs 390.
Positional Average and Partition Values Lower Quartile
195
Qj = size of  size of
rN+n
th
item
(30 + 1
ah
item = 7.75th
= size of item + (size of  size of 7* item)
= 300 + .75 (320  300) = 300 + 15 = 315 .. Lower Quartile is Rs 315. Upper Quartile
Qj = size of
= size of
VN+r"'
item
V3O + 1Y''
item
= size of 23.25* item
= size of 23^" item + ^(size of 24* item  size of 23^" item)
= 480 + ^(490  480)
= 480 + .25(10) = 480 = 480 + 2.50 Upper Quartile is Rs 482.50.
(&) Discrete Series r , u u
Illustration 14. Following are the different sizes and number of shoes m a shoe shop.
Calculate median, first quartile and third quartile.
Size of Shoes
4.5
5
5.5
6
6.5
7
7.5
8
8.5 9
9.5 10 10.5 11
No. of Shoes (f)
4
8 12 15 20 35 50 40 20 15 24 12 5 3
196
Solution.
Statistics for EconomicsXI I
Steps
Sue of shues " 
4.5 4 4
5 8 12
5.5 12 24
6 15 39
6.5 20 59
7 35 94
7.5 50 144
8 40 184
8.5 20 204
9 15 219
9.5 24 243
10 12 255
10.5 5 260
11 3 263
dat:
1. Arrangement of the data in ascending order is necessary.
2. Calculate less than cumulative frequencies.
3. Locate the item by finding out :
fN + U
th
fN + V
th
and
Af + 1
th
Wk " "" " feocy, the tule of
Median
Me  size of = size of
fN + lY'' .
Item
r263+r
th
item = 132th item
First Quartile
= size of 132* pair of shoes = 7.5 size of shoes.
N + l^
th
4
263 + 1^*''
Qj = size of = size of
= size of 66* item = size of 66* pair of shoes
Medial Ap:
item item
licsXI Positional Average and Partition Values
First Quartile = 7 size of shoes. Third Quartile
197
Q = size of
Vn+T''*
Item
= size of
r 263 + 1^
th
item
= size of 198* item Third Quartile =8.5 size of shoes, niustration 15. Calculate Median, First Quartile and Third Quartile from the following data:
Solution.
Income No. of persons
(in Rupees)
800 16
1000 24
1200 26
1400 30
1600 20
1800 5
Income ■ ^H
(in Rs) 1
800 16 16
1000 24 40
1200 26 66
1400 30 96
1600 20 116
1800 5 121
Median :
Applying formula, we get
Me = size of = size of
fN + 1
Nth
item
ri2i+i^
th
item = 61* item
= income 61* person
198
Median = Rs 1200
First Quartile
Statistics for EconomicsXI
Qj = size of
4
Item
= size of
121+n
Item
= 30.5* item = income 35.5* person ••• Qi = Rs 1,000 Third Quartile
Q. = size of
= size of
(N+l^
th
Item
T21 + n
th
4
item
Thus,
= 91.5* item = income 91.5* person Q, = Rs 1,400 Me = Rs 1,200 = Rs 1,000 Q3 = Rs 1,400
(c) Continuous Series
Marks Students
Solution.
Me
3035 14
35^0 16
4045 18
4550 23
5055 18
5560 8
6065 3
Marks
3035 3540 4045 4550 5055 5560 6065
Mo. of students (f)
14 16 18 23 18 8 3
c.f
14 30 48 71 89 97 100
H
licsXI Positional Average and Partition Values
199
Steps :
1. Calculate less than cumulative frequencies.
2. Median, first quartile and third quartile items are located by finding out
th n/»T\th
u.
(N^
v4.
, and
N
Item m continuous series.
3. Locate the median group, first quartile and third quartile group by cumulative
frequency column where the size of respective fall.
4. Apply the suitable formula to get the value :
Me = /j +  X i
N r
— C.f.
2.
th
N 4
th
, and
fN'
4,.
th
Items
fN) 4]
c.f.
X I
Median
Median = size of^ item = = 50* item
Hence, median lies in class 4550
N , Me = /j +  X i
th
100
where, = 45, ^ = 50, c.f = 48, f ^ 23, / = 5
5048
Me = 45 + = 45 +
23
2x5 23
X 5
= 45.43
Hence, median is 45.4% marks.
200
First Quartile
Qj = size of Hence, Q^ lies in class 3540
^ Statistics for EconomicsXl
14 j
item = = 25* item
—c.f.
X I
where, ^ 35, ~= 25, c.f = 14, / = 16, i = 5
= 35 + 1^= 38.43 16
Hence, first quartile is 38.4% marks. Third Quartile
Qj = size of Hence, Q lies in class 5055
(N^ th rioo^i
Item = ^
UJ I 4 J
= 75* item
c.f.
f
X /
v4.
where, = 50, ' — = 75, c.f = 71, f = 18, / = 5
. ^ „ 7571
= 50 + = 51.11
lo
Hence, third quartile is 51.11% marks.
niustration 17. Calculate the Median and Q^ using the following data :
Midpoints marks : 5 15 25
■ No. of
students : 3 10 17
35 7
45 6
55 4
65 2
75 1
Positional Average and Partition Values 201
Solution. Given midpoints are required to be converted into class intervals.
010 1020 2030 3040 4050 5060 6070 7080
3 10 17
7 6
4 .2 1
Calculation of Median and Q3.
Median
Applying formula, we get
Median = size of
th r5o^
U> item = I2 J
= 25* item
Hence, Median lies in class 2030
Applying suitable formula to get the median value
"lcf. Me = /j + ^^— X i
where I, = 20,^ = 25, c.f. = 13, f = 17, i = 10
2S1 3
Me = 20 +  X 10
= 20
17 12x10
Hence, Median is 27.05 marks. Third Quartile
17
= 27.05
n
v4y
item = ^^ = 37.5* item
Qj = size of
Hence, Q^ lies in class 4050 Applying suitable formula, we get
/ X T
/ Nl
Qs = K +
v4.
c.f.
f
X /
3
13 30 37 43
■47
49
50
202
Statistics for EconomicsXI
. where, = 40,
= 37.5, c.f = 37, f= 6, i = 10
03 = 40.^^,10
. 40 . = 40.83
Hence, third quartile is 40.83 marks.
Illustration 18. Calculate the Median and Quartiles for the following : Marks (below) : 10 20 30 40 50 60 70 80 No. of Students : 15 35 60 84 96 127 198 250
Solution. Before calculating Median and Quartiles, first we convert the given cumulative frequencies into class frequencies :
[ W. of^tttdentfi
010 15 15
1020 20 35
2030 25 60
3040 24 84
4050 12 96
50^60 31 127
6070 71 ■ 198
7080 52 250
Total 250
Median. Applying formula, we get
( N^^
Median = size of
v2.
250 ,
Item = — = 125* item
Hence, median hes in class 5060 Applying suitable formula to get median :
Me = /, + 2
f
X t
where / = 50,
N
= 125, c.f = 96, f= 31, i = 10 12596
2
Me = 50
= 50 +
31 29x10
31
 X 10 = 59.35
;. '.Hence, median is 59.35 marks.
• i
Tl
in a
Positional Average and Partition Values First Quartile
rN
2)
Nth
item =
250^
Q^ = size of
Hence, Q, lies in class 3040. Applying suitable formula, we get
E^cf.
= 62.5* item
X t
62.560
= 30 + X 10
. 30 . ^ 31.04
24
Hence, Q, is 31.04 marks. Third Quartile
fN
th
r250
4 J
Q, = size of J item =
Hence, Q, lies in class 6070. Applying suitable formula, we get
4.
= 187.5* item
Q3 = ^
f
X I
203
r"
Hence, Q, is 68.52 marks.
Thns, Q, = 31.04, Q. = Median = 59.35 and Q, = 68.52 marks.
nlustarion 19. The following series relates to the da,ly income of workers employed
in a firm. Compute
(a) highest income of lowest 50% workers,
(b) minimum income earned by the top 25% workers, and
(c) maximum income earned by lowest 25% workers.
204
Statistics for EconomicsXI
2529 3034 3539 20 10 5
Daily Income (in Rs) : 1014 1519 2024 Number of workers : 5 10 15
Before solving it let us understand the question.
1. As the data are of inclusive class intervals, we are required to convert the classes into class boundaries. ^.lasses
2. The data is arranged in ascending order, where
Area of 50% of workers in highest income group
At this point a worker in the centre earning highest daily income of the lowest 50% of workers (;.e., Median value)
3. The data is arranged in ascending order, where
 100% data
1_ .
H4 At this point a worker is eaming minimum daily income of top 25% workers .4. Area of top 25% of workers
4. The data is arranged in ascending order, where
Area of lowest 25% of workers
■ At this point a worker is earning maximum daily income of lowest 25% workers (i.e. lower quartile value = O,)
qutdln"' ^^ ^^^ ^^^to the given
Positional Average and Partition Values SoiutLoa.
205
DaUy Income (m (X)
9.514.5 14.519.5 19.524.5 24.529.5 29.534.5 34.539.5
5 10 15 20 10 5
5
15 30 50 60 65
(a) Computation of highest daily income in lowest 50% of workers. (Median)
Nth
Median is the value of
th (65\
— item or 
Uv I 2 j
or 32.5th item which lies in 24.5
29.5 class interval.
Applying suitable formula to get median value,
N
Me = /j +
c.f.
X t
= 24.5 + = 24.5 +
f
32.530 20 2.5x5
x5
20
= 24.5 + 0.625 = 25.125 .. Highest data income of lowest 50% workers is Rs 25.13. (b) Computation of minimum daily income earned by top 25% workers (Q^)
th
3x65
^ l\
Q, = Value of — item =
Hence, Q3 lies in class 24.5  29.5. Applying suitable formula, we get
.T,
= 48.75* value
Qs = ^
c.f.
f
X t
48.7530 _ = 24.5 + TTx^
= 24.5 +
20 18.75x5 20
= 24.5 + 4.687 = 29.187 Minimum daily income earned by top 25% workers is Rs 29.19.
It,f
Statistics for EconomicsXI (c) Computation of maximum daily income earned by lowest 25% workers (Qj)
Qi = Value of
UJ
item = 4r = 16.25* value
Hence, Q^ lies in class 19.5  24.5 Applying suitable formula, we get
Q. = ^
f
X I
= 19.5 + = 19.5 +
16.2515
15 1.25x5 15
x5
= 19.5 + 0.416 = 19.916 Maximum daily income earned by lowest 25% workers is Rs 19.92.
Graphical Determination of Median and Quartiles
Illustration 20. Determine median and quartiles graphically from the following data : Marks : 05 510 1015 1520 2025 2530 3035 35^0 Students : 7 10 . 20 13 17 10 14 9
Solution.
Secc
Akrrifes . . f ■ Mjr/fes less than Less than cumulative Marks more than More than cumulative
05 7 5 7 0 100
510 10 10 17 5 93
1015 20 15 i7 10 83
1520 13 20 50 15 63
2025 17 25 67 20 50
2530 10 30 77 25 33
3035 14 35 91 30 23
35^0 9 40 100 35 9
N = 100
First Method (only for median). Steps
1. Calculate ascending cumulative frequencies (less than) and descending cumulative frequencies (more than).
2. Draw two ogives—one by 'less than' and other by 'more than' methods.
3
4.
5.
licsXI Positional Average and Partition Values
3. From the intersecting point of two ogives, draw a perpendicular on Xaxis.
4. The point where perpendicular touches Xaxis, median value is determined.
207
Second Method (For Median and Quartiles). Steps
1. Calculate ascending cumulative frequency (less than).
2. Determine the value by the following formulae :
Me = size of
Qj = size of
Q = size of
th
UJ
item, i.e., ^^ = 50* item
u
Item, i.e..
N 4
th
Item, i.e..
100 4
3^100^
= 25* item = 75* item
V ^ /
3. Locate 50, 25, 75 values on Yaxis and from them draw perpendiculars or cumulative frequency curve (ogive).
4. From these points where they meet the ogive draw another perpendicular touching Xaxis.
5. The points where perpendicular touches Xaxis, Qj, Me and Q^ are located.
208
Statistics for EconomicsXI
VehficaUon
Median Group 1520
Me = /j +
N U
cf
f
X t
5037 ^
13x5
Median = 20 Marks Lower Quartile Group 1015
cf
X I
2517 = 10 + —X 5
 10
20 8x5
20
Qi = 12 Marks.
= 12
Upper Quartile Group 25  30
cf
x t
ic 7567 ^ 8x5
= 25.— .29
Q, = 29 Marks
Less than method' cumulative frequency curve is the reminder of the rule that at the
hrst step of calculation of quartiles, the data is arranged in ascending order. Howeven
median can be ocated on graph even by more than 'ogive' or calculated by arranging the data m descending order. ^ h b ^
Positional Average and Partition Values
209
1. Definition
2. Determination of Mode
3. Merits and Demerits of Mode
1. Definition
According to Coxton and Cowden, "the mode of distribution is the value at the point around which the items tend to be most heavily concentrated. It may be regarded as the
most typical of a series of values." x/r j •
The word mode comes from French la mode which means the fashion Mode in statistical language is that value which occurs most often in a senes, that is value which is most typical. If garment manufacturers say that short collars are now in fashion the statement implies that maximum number of people nowadays wear short collar shirts If we say the mode is size No. 7 shoe, it means in a given data maximum number of people wear size No. 7. Thus, mode is that value of observations which occurs the greatest number of times or with the greatest frequency.
For a better understanding of mode let us look at the following information about
frequency of students in relation to marks obtained.
Marks : 5 10 15 20 25 30 35 40 45 50 : 2 3 25 2 1 18 20 24 14 10
According to the explanation of mode given above, the modal marks will be 15 because maximum number of students (25) have obtained 15 marks each. Although 15 have the highest frequency, a more careful examination of the information shows that the highest concentration of the frequency is around 40 marks. That is, m the neighbourhood of 40 marks. There are more frequencies (18, 20, 14, 10) as compared to the neighbourhood of 15 marks (2, 3, 2, 1). Thus 15 marks are not ^yp.c^/ of the series of valLs. For the reasons given above, 40 marks is the mode and not 15. Therefore, to define accurately, mode is that value of observations around which items are most densely
or heavily concentrated.
The mode is defined as the most frequently occurring value. If each observation occurs the same number of times, then there is no mode in that distribution. If two or more observations occur the same number of times (and more frequently than any other observation) then there is more than one mode and the distribution is multimodal, as against unimodal, where there is one mode. If two values occur most frequently then the series is bimodal, in case of three values occurring most frequently then the series is called trimodal. The mode as a measure of central tendency has little sigmficance for a bi or
"Mode is that value of the graded quantity at wh,ch the instances are most numerous. " A.L. Bowley "The value occurring most frequently in a senes (or group) of Hems and around which the other item^ar^ distributed most densely."
210
Statistics for EconomicsXI   .oae .„.
2. Determination of Mode
(a) Series of Individual Observations and Discrete Series
(b) Continuous Series
(c) Graphic Location of Mode
id) Mode from Mean and Median.
(a) Series of Individual Observations and Discrete Series
In a senes of individual observations, the mode can be located in two ways •
"  ^^st
a cl^r""" ^^ ^^^ rks obtamed by 15 students i
Marks : 4 6 5 '
in
9 8
Solution.
10 4 7 6 5 Modal value
8
7
7
7 8 8 9 9 10.
(a) (i) Array : 4 4 5 5 6 6 : Mode = 7 Marks
(«) Discrete Series. Converting the above data into discrete series, we get
Mode = 7 Marks
licsXI Positional Average and Partition Values 233
(b) Discrete Series. In discrete series the mode can be located by two ways :
(i) By Inspection.
(ii) By Grouping.
(i) By Inspection. The mode can be determined just by inspection in discrete series, the size around which the items are most heavily concentrated will be decided as mode. Illustration 22. Find out mode from the following data :
Wages (in Rs)___ ' No. of Persons ■
125 3
175 8
225 21
275 6
325 4
375 2
Solution. By inspection, we can determine that the modal wage is Rs 225 because this value occurred the maximum number of times, i.e., 21 times.
{ii) By Grouping. In discrete and continuous series, if the items are concentrated at more than one value, attempt is made to find out the item of concentration with the help of grouping method. In such situations it is desirable to prepare a grouping table and an analysis table for ascertaining the modal class.
In grouping method, values are first arranged in ascending order and the frequencies against each item are properly written. A grouping table normally consists of six columns Frequencies are added in twos and threes and total are written between the values. It necessary, they can be added in fours and fives also.
Column 1. The maximum frequency is observed by putting a mark or a circle.
Column 2. Frequencies are grouped in twos.
Column 3. Leaving the first frequency, other frequencies are grouped in twos.
Column 4. Frequencies are grouped in threes.
Column 5. Leaving the first frequency, other frequencies are grouped in threes.
Column 6. Leaving the first two frequencies, other frequencies are grouped in threes.
After observing maximum total in each of these cases, put a mark or circle on every total. An analysis table is prepared after completing grouping table in order to find out the item which is repeated the highest number of times. If the same procedure is adopted in continuous series, we shall be in a position to determine the modal class.
We shall now see how mode is determined by grouping method in a discrete series.
212
Statistics for EconomicsXI Illustration 23. Find out mode of a data given in Illustration 20 by grouping.
Grouping Table
Wages (m Rs) No. of persons
• <l) (3) m ■ ' (6)
125 3
11
175 8 32
225 21 29
■ 27 i i 35
275 6 31
325 4 10 12
6
375 2
Analy.sis Table
Column No. 125 "■ 225  275, 32S
1
2 1 1
3 1 1 1
4 1 1 1
5 1 1 1
6 1 1 1
Total 1 3 6 3 1
^^ Smce the value 225 has come largest times, 6 times, hence the modal
visage IS
lUustration 24. Compute the mode from the following :
Size of the item . 2 3 4 5 6 7 8 9 10 11 12 13
Frequency : 3 8 10 12 16 14 10 8 17 5 4 i
beW ^^^ be done as shown
licsXI
Positional Average and Partition Values
213
Grouping Table
2 3
3 8
4 10
5 12
6 16
7 14
8 10
9 8
10 17
11 5
12 4
13 1
11
22
30
18
22
J:
18
28
21
24
25
_42
35
10
30
40
30
38
26
The analysis can be done separately also as shown below :
!
1 2
3
4
5
6
Total
to
■ 1
m
32 I
It
j The value of 6 has come the largest times (5), hence mode is 6.
12
5 3 1
13
^ Statistics for EconomicsXl
214
(b) Continuous Series
applying the following formula? " determined by
Mo = I +
or
Mo = / +
X t
ifl~fo) + {fl~f2)
X /■
where, Mo = Mode
/j = lower hmit of modal class /", = frequency of the modal class /o = frequency of the class preceding the modal class = frequency of the class succeeding the modal class i = class interval of the modal class
The above formula can also be expressed in the following way :
or
Mo = Mo = /j +
X /
A1+A2
"/"iZol + l/iZil
X t
where. Mo = Mode
/j = lower hmit of the modal class
modal class and the frequency of the class before the modal class . precedmg class (ignoring signs)' "
A, = (Read delta 2), .. \f _ f^l Jhe difference between the frequency of the
niustration 25. Fmd out the mode from the following frequency distribution • Central snes : 1 , 3 , ^ ^ ^ ^^^
Frequency ; g ^
10
12 20
12
215
Positional Average and Partition Values
Solution. Since the central sizes are given, we must convert them into class intervals.
Grouping Table
Qass Imervai
0.51.5
1.52.5
2.5.3.5
3.5^.5
4.55.5
5.56.5 6.57.5
7.58.5 8.59.5
9.510.5
(V
6
10
12
20
12 5
3 2
14
22
32
16
32
17
24
44
10
28
37
Analysis Table
42
20
1 2 3 4 5 6 1 1 1 1 1 1 1 1 1 1 1 1 1
Total 1 3 6 3 1
By Inspection Mode lies in the group 4.55.5. To determine the value of Mode, we should apply the following formula.
fifo
X t
where, /j = lower limit of the modal class (4.5) /j = frequency of the modal class (20)
216
Statistics for EconomicsXI
fo = frequency of the class preceding the modal class (12) fi = frequency of the class succeeding the modal class (12) i = class interval of modal grdkp (1)
2012
Mo = 4.5 + = 4.5 +
= 4.5 +
2x201212 8
X 1
401212 8
X 1
16
Mode = 4.5 + 0.5 = 5. Illustration 26. Find the mode of the distribution from the following data :
Below 15 . ........
20 10
" 25 26
" 30 38
" 35 47
40 52
" 45 55
Solution. For calculation, mode of the given distribution first convert the given data into class intervals.
Grouping Table
1015 1520 2025 2530 3035 3540 4045
3 
10
7 
23
16  
28
12 
9 21
14 
5 
8
3 
Positional Average and Partition Values
217
Analysis Table
Column No. lO^IS 1520 202S 2530 303S 3540
1 2 3 4 5 6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Total 1 3 6 4 2 1
The mode lies in the class 2025. Applying the formula, we get
fifo
where.
Mo = /, + ^ X I
/j = 20,/j = 16,/; = 7,= 12, I = 5 167
Mo = 20 +
2x16712
X 5
= 20 + ^ X 5
= 20 + 3.46 Mode = 23.46
Mode when Class Intervals are Unequal
The formula to calculate the mode from the modal class discussed above, is apphcablt in a series where there are equal class intervals. When the class intervals are not equal, before calculating the value of the mode, we must take them equal and the given frequencies should be adjusted presuming that they are equally distributed throughout the class.
Illustration 27. Compute the mode from the following data :
Class Frequency
Class Frequency
03 36 610 1012 1215 1518
4 8 10 14 16 20
1820 2024 2425 2528 2830 3036
24 14 16 11 10 6
218
Statistics for EconomicsXI
Solution. The class intervals are not equal. They are made equal by combining two or more classes.
Grouping Table
Class Frequency
(1) ; a; ; (V (4) (V (6)
06 4 + 8 = 12 ]
36
612 10 + 14 = 24  1 72
60
• 1218 16 + 20 . = 36   
74 98
1824 24 + 14 = 38  _ 
75 Ill
2430 16 + 11 + 10 = 37  _ 81
43
3036 = 6  
Analysis Table
Column No. 612 1824 2430 3036
1 2 3 4 5. 6 1 1 1 1 1 1 1 1 1 1 1 1 1
Total 1 3 6 3 1
The mode lies in the class 1824 Applying the formula, we get
X /
where
/j = 18, f^ = 38, /•„ = 36, = 37, i = 6 3836
Mo = 18 +
2x383637
X 6
= 18 + J X 6 = 18 + 4 = 22. Mode is 22.
219
Positional Average and Partition Values
(c) Graphic Location of Mode
The value of mode can be determined graphically in a frequency distribution. Followmg
are the steps of locating mode on graph. .
1. Prepare a histogram of the given data.
2. The highest rectangle will be the modal class.
3. Draw two lines diagonally inside the modal class rectangle to the upper corner of the adjacent bar.
4. From the point of intersection of these lines, draw a perpendicular of Xaxis which gives the modal value.
Illustration 28. Determine the value of mode of the following distribution graphically and verify the results.
1020 2030 3040 4050 5060 12 14 10 8 6
Marks
No. of Students Solution.
.010 5
GRAPHIC LOCATION OF MODE
Scale: 2 cm = 10 Marks on Xaxis 1 cm = 2 Students on Vaxis
30 40 MARKS
Verification :
Mode lies in the class 2030
Mo = /, + ,
' Ififofi
X i, where = 20, f^ = 14, f, = 12, f^ = 10, / = 10
Mo = 20 +
1412
2x141210
X 10
= 20 + — X 10 = 20 + 3.33 6
Mode = 23.33 Marks.
220
Statistics for EconomicsXI
is ^ ^"'Tt't^ ^ perpendicular,
luchZt? u " ^^^ the per^ndicular
touches the Xaxis, gives the modal value. Mode cannot be determined graphicSSy if two
id) Mode from Mean and Median
(vJ'uvrj'^l't T'r of distribution curves in Chapter 7'
intl W^ a symmetrical distribution mean, median and mode are
orZZ Ll ^fl ^^ ' distributicm of frequencies on either side of the maximum;
vakr.L h I 1 ^^ber of cases above the mean
value and below the mean value are equal. This relationship does not exist in moderately
—' will pull^a^lt
Asymmetrical Distribution, (Negatively Skewed Curve)
Negative
X < Me < Mo
Symmetrical Distribution, (BellShaped Curve) Peak
Asymmetrical Distribution, (Positively Skewed Curve)
Positive
X = Me = Mo
Mo < Me < X
of tie (^'^^^"caly distribution, if the distribution tails off towards higher value
lit r conint r" ""'T ^ ^^'s, positively skewed) and ha
greater concentration m lower values mean and median will be more than the Lde (X
and Me > Mo). In other words, mode is lowest, i.e., X > Me > Mo.
valufbf Ae^dl'Vyi"^^^^^ distribution, if the distribution tails off towards lower value of the data and has greater concentration in higher values, (i.e., negatively skewed),
mean and n^edian are less, then mode (X and Me < Mo). In other words, mode is
highest,X < Me < Mo.
The relationship between mean, median and mode reveals that in a moderatelv assymetrical (skewed) distribution the median lies between the mode and the arS^Sc mean, approximately 2/3rd distance from the tpode and l/3rd from the lafTS relationship is expressed as follows which is given by Karl Pearson
Positional Average and Partition Values 221
Mode = Mean  3(Mean  Median) = Mean  3 Mean + 3 Median = 3 Median  2 Mean
Mo = 3 Med  2X
In most of the cases if the distribution is moderately asymmetrical, the value of mode calculated from mean and median would not differ significantly from the value calculated by other methods. Inhere may be two values in a series which occur with equal frequency, this IS called b,modal series. In case of bimodal distribution or mode is illdefined, its value may be determined by the above formula which is based upon the relationship of mean median and mode. If we know any of the two values out of the three, we can calculate the third value from the above relationship.
Dlustration 29. {a) In an asymmetrical distribution mean is 58 and the median is 61 Calculate mode.
{b) If mode in a tolerably asymmetrical distribution is 12 and median is 16, what would be the most probable mean?
Solution.
Mode = 3 Median  2 Mean = (3 X 61)  (2 X 58) = 67
Mode = 67.
Mode = 3 Median  2 Mean 
12 = (3 X 16)  2 Mean 12 = 48  2 Mean 2 Mean = 4812 = 36
Mean = ^ 2
= 18
Mean =18.
musttation 30. The following table gives production yield in kg per hectare of wheat ot 150 farms m a village. Calculate the mean, median and mode production yield.
Production (in kg) : 5053 5356 5659 5962 6265 6568 6871 7174 7477
No. of farms : 3 8 14 30 36 28 16 10 5
222
Solution.
Statistics for EconomicsXI
Production No. of terms f Midpoints m m  63.5 d fm63.5] fd' 
yield X I 3 ) d' c.f
5053 5356 5659 5962 6265 6568 6871 7174 7477 3 8 14 30 36 28 16 10 5 51.5 54.5 57.5 60.5 63.5 66.5 69.5 72.5 75.5 . 12 9 6 3 0 +3 +6 +9 +12 4 3 2 1 0 +1 +2 +3 +4 12 24 28 30 0 +28 +32 +30 +20 3 11 25 55 91 119 135 145 150
N = 150  i  lfd' = 16
Median :
N
= 63.5 + 0.32 = 63.82 Mean = 63.82 kg per hectare
' \7\th
Median = the size of f —
Item
150
th
Item
= the size of
V 2
= the size of 75* item Median hes in group 6265.
To interpolate median, we use the following formula :
N
T C.f.
Me = /, + X
f
where.
N
/, = 62, — = 75, c.f =55, f= 36. i =
Positional Average and Partition Values
7555
Mode
= 62
36
■x3
= 62 + 1.666 = 63.67
Median = Rs 63.67 kg per hectare
Grouping table
Analysis Table
223
Rupees No . of receiver
X (V (2) (■V (4) iS)
5053 3 
5356 8 11 25
5659 14  22 . if 
5962 30 44 *« 80
6265 36 66  
6568 28 64 94 i!0 1
6871 16 44 54
7174 10 26  31
7477 5  15 
Column No. 5962 6265 6568 6871
1 2 3 4 5 6 1 1 1  1 1 1 1 1 1 1 1 1 1 1
Total 3 6 4 1
224
Statistics for EconomicsXI
^Jy .nspection the o,ode hes i„ the group applying the Mowing formula, we
Mo = L + _fi~fo_
1 _/■_
X i
Here,
2fifof2 I, = 62, f^ = 36, /■„ = 30, f^ = 28, / = 3
3630
Mo = 62 +
= 62 +
2x363028
x 3
14
= 62 + 1.285
x 3
Mode = Rs 63.29 kg per hectare.
3. Merits and Demerits of Mode Merits
compared to mean LSian Inf^o f A?""' " most typical and cogent ues orr ^rm^^
2. Mode is not A i. ' ' of shoes etc. vduts L nottt™."'' " if the extreme
3. Mode can be determined in openend distribution.
"" hmetic mean can not be ascertained
5. M^e is helphU in describing the quaUtative character of the product
Demerits
1. Mode cannot be decided in bimodal and multimodal distributions
2. Mode . not sm^table when relative importance of itemst fctaT
tnt —• " "Ot c:;a'ble of algebraic
4. Mode is not based on every item of the series.
  may differ .om one
"Lg"' " Ae size of the class interval decided
licsXI Positional Average and Partition Values 225
Comparison of Mode with Mean and Median
We find that as compared to mean and median, mode is less suitable. Mean is simple to calculate, its value is definite, it can be given algebraic treatment and is not affected by fluctuations of samphng. Median is even more simple to calculate and is almost as stable as mean, although it is influenced by fluctuations and cannot be given algebraic treatment. Mode is the most popular item of a series and is also easy to calculate and simple to understand. But it is not suitable for most elementary studies because it is not based on all the observations of the series and is unrepresentative. Mode has its own uses and advantages as we have seen, but as compared to mean and median, it is not so precise and accurate.
OF FORMI
Individual Series and Discrete S
1. Median Me = Size of
fN+l^
V 2
th
Item
2. Lower Qj = Size of Quartile
3. Upper Qj = Size of Quartile
fN + 1}
th
I 4 ;
rN+n
Item
4 J
th
Item
Me = Size of
th
Item
X, I N/2c.f. Me = /j + j.—— X I
Qj = Size of
l4j
Item
Q3 = Size of
f
fN
.4;
Item
Q3 = + f '
Mode :
1. Grouping method for discrete series.
2. After grouping, decide the Modal Group and use the formula to find modal value in continuous series
Mos:
X f
226
exercises
Statistics for EconomicsXI
Questions :
Define median. Discuss ks merits and dements.
3 wTT """" 1s.
•5. Write short notes on :
4 De&rr"'?' of a dismbution.
■ "  can be read W ,be
5. What is meant by the following ?
(a) Mode (b) Median, and (c) Anthmetic mean
7. Deftne mode. Explain how mode can be read on graph paper> • nr^a—ft .an and
tendency
■ "raTr^ndelf ^ d,a„ as measures of
and mode of a frequency
" rtrsLr ^^   
<b) Average inrelligence of srudents in a 'class, and (c) Average production per shift in a factory .
Median W Arithmetic Mean]
Problems :
Calculate median of the following data :
145
257
130 260
200 300
210 345
198 360
234 390
159
160
178
Fmd out median of the following information : Marks : 10, 70, 50, 20, 95, 55,
42,
[Me = 210]
60, 48, .80 [Me = 52.5 marks]
Positional Average and Partition ^^^ues 227
'e have the following frequency distribution of the size of 51 households. Calculate the arithmetic mean and the median.
; Size Number of households :
4./Find out median (a) Serial No.
4
9
21
11
7 5
Total
51
1 2 3 4 5
2 4 10 8 15
5 10  15 20 25
2 • 4 6 8 10
[X = 5, Me = 5]
6 20
7 12
25
9 30
..........[Me (a) = 12, (b) = 20]
out median, furst quartile and third quartile of the following series :
Height (in inches) : 58; 59 60 61 ^62 63,. 64 ' 6^ 66
No. of Persons 2 3 6 15 10 .5 4 3 1
/? [Me = 61, Q, = 61 and Q3 = 63]
6. /The percentage of marks obtained by 68 students in an examination are given below  ' Compute the median.
= Below 20 20^0 4060 6080 Above 80 No. of Students : 0 5 22 25 16
_ ^ , , , [Me = 65.6]
7. Calculate the mean of the following distribution of daily wages of workers in a factory:
Daily Wages (in Rs) : 100120 140160 160180 180200 Total
No. of Workers : 10 30 15 5 80
41so, calculate the median for the distribution of wages given above.
. / [X = 146.75, Me = 146.67]
8./The following table gives the marks obtained by 65 students in statistics in a certain examination. Calculate the median.
Marks No. of Students
More than 70% 8
60% 18
50% 40
40% 45
"  30% 50l
20% 63,)
10% 65
[Me = 53.4 Marks]
228
j„ . Statistics for EconomicsXI
■ 26 8 ^ 2 50
j 1619 2029 3039 4049 5059 6064
1 ^^ 46 49 32 28 14
■ — —iiicuian or tne above data,
(ft) Draw a histogram and indicate mean and mode therein.
Get Mode on Histogram. Mean cannot be obtain!? o^ HiLl^^rT:'
Marks No. of Students
3035 14
3540 16.
4045 18
Compute mode from the following series :
4550 5055 5560 6065 23 18 8 3
[Me = 45.43, Q, = 38.4, Q^ = 51.1]
Size of items Frequency Size of items Frequency
1 3 8 10
3 8 9 8
4 10 10 17
5 12 11 5
6 16 12 4
13.
Calculate mode for the following data
7 14 13 1
[Mo = 6]
1 26
2 113
3 120
4 95
5 60
6 42
7 21
8 14
9 5
10 4
[Mo = 3]
Positional Average and Partition ^^^ues
229
^^ind out the Mode from any of the following two distributions :
X : 3040 4050 5060 6070 7080 8090 90100
f ■■ 6 10 16 14 10 5 . 2
And
Marks : 09 1019 2029 3039 4049
No. of Candidates : 6 29 87 181 247
Marks : 5059 6069 7079 8089 9099
No. of Candidates : 263 113 49 9 2
is/uk of electric lamps is given in the following table. Calculate the median and the mode.
Below 400 4
400800 12
8001200 40
12001600 41
16002000 27
20002400 13
24002800 9
Above 2800 4
[Mo
:e Mean, Median and Mode from the following data :
59 1
61 2
63 9
65 48
67 131
69 102
71 40
73 17
Total 350
[X= 67.9, Me = 67.75, Mo = 67.48]
i Followiiig is the distribution of marks of 50 students in a class : [Marks (Kore than) : o\ 10 2030 40 50
50 \
Calculate theNMedian Marks. If^60% of students pass this examination, find out the I minimum marl^btained by a pass tandidate. . [Me = 27.5, 25.5%]
46
40
20
10
230
j„ . Statistics for EconomicsXI
32 20 43 11
61 31 47 .15
52 56 64 20 35 21 50
22 10 43 42
49 62
75 77
persons is given below :
' 97 35 30 30 95
60 27 53 31 9
45 22 36 13 46
73 81 40 40 55
67 54 23
42 25 51
modal age.
19. Determine the value of mode for the foil ^ = ^^
Mode = 3, Median = 2 M^n :
21.
[Modal age = 42 years]
Ma No. of Students
Less than 10 Less than 20 Less than 30 Less than 40 Less than 50 Less than 60 Less than 70 Less than 80 Less than 90 5 15 98 242 367 405 425 438 439
20.
c 1 , LA = jy.j:). Me = SX 44
For the data given below find graphically the folWing • '
[a) The two quartiles. '
(b) The central 50% limit of the age
W The nnntber of workers falhttg ,n .he age gronp of 2S to 57 .ears
Mo = 36.22]
Age in years 2024
iVo. of workers : 5
Age in years : 5054
No. of workers : 23
2529 10 5559 10
3034
15 6064 5
3539
25 6569 2
4044 65
4549 40
Draw a less than' ogtve front the following data and hence find out the value of
Class
2025
2530
3035
3540
4045
4550
5055
5560
Frequency
6 9 13 23 19 15 9 6
231
Positional Average and Partition ^^^ues 253
22. The following table gives the distribution of the wages of 65 employees in a factory.
Wages (in Rs) : 50 60 70 80 90 100 110 120
(Equal to ormore than) " ^^0
Number of employees 65 57 47 31 17 7 ' 2 q
23. Draw the histogram and estimate the value of mode from the following data :
Marks No. of students
010 0
1020 2
2030 3
3040 7
4050 13
5060 11
6070 9
7080 2
8090 1
24. Represent the following data by means of a histogram and find out mode. Weekly wages : No. of workers :
1015
7
1520 19
2025 27
4045
2530 3035 35^0 , 15 12 12 8
W ind^idual incomes, the 'less than type' Ogive and the
;:. 7i?'r ti!^™ ^sir ^^ 
Chapter 10
measures of dispersion
Introduction
Objectives of Measuring Disperelon Metliods of Measuring Dispereion
(A) Dispersion from Spread of Values
(B) Dispersion from Average
(C) Graphic Metiiod—Lorenz Curee Absolute and Relative Measures of
(A) Absolute Measures
(B) Relative Measures Graphic Method
Comparison of Measures of Dispersion List of Formulae
.eafr^r:?:^^^^^ a single fig^e. These
the form of an average. These av^Les Sf uf frequency distribution in
magnitude of the distriLion but ZrS tfir ^^put the general level of Measures of central tendency a^e som^i^r^^^^^
happens when the extent of variationf ^iZl f ^his
relation to the other values is lajr^n any ^^^^^^^^ ? ^^ 
not only to know the average
^ow about the measures of
He knew that the average depth of the r^ter w^ Too 1 uTl' ^
family was 130 cm. Hef therLre decidl7thTh Tu""^" ^^ his
mer on foot. But it so Wpen^af hf ma^^^^^ ^^ ^^
Ae height of youngest child^of the ^s^v 12? m W ""
had happened to the statistician famity ^ ^^ ^^^t must
r^ 7 ^^ there may be great may be below poverty line. TTiere ifnTed to nt ^^^^of a majority of the people
Measures of Dispersion 233
Definitions
According to D.C. Brooks and W.F.L. Dick. "Dispersion or spread is the degree of the scatter or variation of variables about a central value."
According to Prof. L.R. Connor. "Dispersion is a measure of the extent to which the individual items vary."
According to A.L. Bowley. "Dispersion is a measure of the variation of the items."
Now, look at the following data about salaries paid to employees of three different departments of an organisation.
Dept. A
■
Deviation
t. B
mmmmi
Dept. C
Deviation
fWf
MS
5000 5000 5000 5000 5000
0 0 0 0 0
4500 5500 6000 5000 4000
 500 + 500 + 1000 0
 1000
10000 2000 4000 4500 4500
5000
 3000
 1000  500 500
Total : Mean X :
25000 Rs 5000
25000 Rs 5000
25000 Rs 5000
We find from the above table that the average salary paid to employees in each department is the same, i.e., Rs 5000. In department 'A' the salary paid to each employee is the same, i.e., Rs 5000, hence mean is fully representative of the values of the items in the series. In department 'B' though the mean is Rs 5000, but the constitution of series is quite different. In this case lowest value is Rs 4000 and the highest value is Rs 6000 and the difference between the highest and the lowest value is Rs 2000, and the highest deviations from the mean are 1000 and +1000. The mean in this case, does not adequately represent the values of the items in the series of department 'B'. In department 'C though the mean is the same,
but there is wide gap between the values of items. The lowest value is Rs 2000 and the highest value is Rs 10,000, which deviate from mean by 3000 and +5000 respectively. The difference between the highest and the lowest value is Rs 8000. Not a single item in the series is represented by its mean.
From the above illustration we observe that some deviations are positive and some are negative. Similarly, some deviations are large and others are small. Therefore, we are required to make an overall summary of these differences (scatteredness) in all values about the central value. This summary is called the measures of dispersion or measures of variation. It is clear that we must not only know the composition of a series but also observe how the composition of a series differs from another. For such a study we have, a statistical tool called measures of dispersion or measures of variation. ,
234
j„ . Statistics for EconomicsXI
bjectives of measuring uispmm
Before we go on to describe the specific methods of studying variabihty, we must clearly define the objectives.
ia) To Test the Reliability of an Average : Measures of dispersion enable us to know whether an average is really representative of the series. If the dispersion of variabdity m the values of various items in a series is large the average may be unrepresentative of the series. If on the other the variability is small, the average would be a representative value. This point has already been made clear in the above dlustration, wherein different series of three departments the mean was a common value and the variations differed.
(b) To Serve as Basis for Control of Variability : The study of variation is done also for the purpose of analysing why large variations happen or occur and this may help to control the variation itself.
For example, in some major human health problems the blood pressure, the heart and pulse beat are recorded and an attempt is made by the doctors to control these through provision of medicines. Similarly, in industrial production to control the quality of the product and the causes of variations in product are obtained by inspection and quality control programmes. In social sciences where we have to study problems relating to inequality in income and wealth, measures of dispersion are of great help.
(c) To Make a Comparative Study of Two or More Series : Measures of variability are also useful in comparing two or more series with regard to disparities or differences. A greater degree of dispersion or variability would mean lack of uniformity or consistency or homogeneity of the data. While a low degree of variability would indicate high uniformity or consistency or stability. Comparative studies of varmbihty are very useful in many fields like profit of companies, share values, performance of individuals and studies relating to demand, supply and prices, etc.
id) To Serve as a Basis for Further Statistical Analysis : Measure of variability which IS measure of second order is very useful in the use of higher measures such as skewness, kurtosis correlation, regression etc.
Note: Characteristics of a representative average are explained on Page 139 and 140 ot this Book. Same points are for characteristics of a good Measure of dispersion.
Following are the important methods of measuring dispersion :
M^ODS OF MEASURING OlSPERSiON
i
from Spread ilues
Range
Interquartile Range and Quartile Deviation
from
IMean Deviation or Average Deviation Standard Deviation
MethodCurve
Measures of Dispersion
235
First two measures, viz.. Range and Quartile Deviations are from spread of values, termed as positional measures. They are calculated from the values of the variable at a particular position of the distribution. They are not based on deviations from any particular value. While the mean deviation and standard deviation are from an average defined in terms of deviations from a central value. Lorenz curve is graphic method of studying dispersion/variability.
Measures of dispersion in terms of spread and position are as under :
(A) Dispersion from Spread of Values
(a) Range
(b) Interquartile Range and Quartile Deviation
(a) Range
1. Meaning
2. Calculation of Range
3. Merits and Demerits of Range
4. Uses of Range
1. Meaning
Range is the simplest measure of dispersion. Range is the difference betu/een the largest and the smallest value in the distribution. It is determined by two extreme values of observations. In case of the grouped frequency distribution range is defined as difference between the upper Hmit of the highest class and the lower limit of the smallest class. In case of a frequency distribution, the frequencies of the various classes are immaterial since range depends only on the two extreme observations. Range as defined is an absolute measure of dispersion and expressed in the units of measurement of the given data. Thus if we want to compare the variabihty of two or more distributions with the same units of measurement, we may use absolute measure. Symbolically, range is located by the following formula :
Range = L  S
where, L = Largest item
S = Smallest item
Relative Measure
To compare the variability of two or more distributions given in different units of measurement, we cannot use absolute measure but we need a relative measure which is independent of the units of measurement. This relative measure is called coefficient of Range. It is common practice to use coefficient of range even for the comparison of variability of the distributions given in the same units of measurement. It is obtained by applying the following forniula :
Coefficient of Range =
LS L + S
236
j„ . Statistics for EconomicsXI
Range = LS Here, L = 30, S = 5
Range = 30  5 = 25 Range is 25 Marks. Coefficient of Range
= kzS
L + S _ 305 30 + 5 = 0.714
.11 ^35
Range = L S Here, L = 60, S = 0
Range = 60  0 = 60 Range is 60 Marks. Coefficient of Range
= IlzI
L + S _ 600 ^ 60 + 0 = 1
Age (in year) : l^^o 2125 2630 3135
No. of Persons : 10 15 17
Calculate range and the coefficient of range. ^
last class will become 30 5  35.? " ^^^ ^ '
Absolute Measure of Range au ■ , r
Range = LS Alternatively (from midvalues)
T , Midvalue of highest class =33
..Range = 35.515.5 = 20 years .. Range = 33  18 = 15 years.
ai
ci
Measures of Dispersion
Relative Measure of Range
237
Coefficient of range =
L5 L + S
35.515.5
20
= 0.39
Alternatively Coefficient of range ^ 3318 _ 15 _ 0 29 33 + 18 ~ 51
~ 35.5 + 15.5 51
Illustration 3. The following are the marks obtained by 50 students in Statistics. Calculate the range of marks obtained by middle 50% of the students.
. Marks No. of Students
Less than 10 4
Less than 20 10
Less than 30 30
Less than 40 40
Less than 50 47
Less than 60 50
Solution. We arrange the data in continuous series.
Np. of Students
010 4 A
10  20 6 10
20  30 20 30
3040 10 40
40  50 7 47
50  60 3 50
lb get marks ot miame ou /o stuucms, wc ^it ..v,  
and 37.5* student (i.e., 1 and 3«» Quartiles) Q, and Q,. Marks of 12.5* student hes m class 20  30 and marks of 37.5* student in class 30  40.
U.S'^c.f.
Marks of 12.5* student = l^ + ^
X t
20
= 20 + ^ X 10 = 21.25 Marks
Marks of 37.5* students = /j +
= 30 +
20
37.5'*' Smdentc./". f
37.530
X I
10
X 10
= 30 + X 10 = 37.5 Marks
10
Largest Value Smallest Value Range Marks
238
j„ . Statistics for EconomicsXI
= 37.5^arks = 21.25 Marks = LS
= 37.5  21.25 = 16.25 Marks Thus, range of marks obtained by middle 50% students is 16.25 marks
Solution. We arrange the data in ascending order 61, 64, 65, 66, 67, 67, 68, 68, 69, 70, 72 Range height = L  S 72  61 = 11 inches.
When shortest man (61 inches is omitted) the range will be = L  S = 72  64 = 8 Change in the range = 11  8 = 3
Percentage change in the range ^ x 100 = 27.27%
11
3. Merits and Demerits of Range Merits
1. Range is simple to calculate and easy to understand th^ rr, r j •
ban a very accurate picture of variability one may compute'rangl ' "
2. It gives broad picture of the data quickly.
3. It is rigidly defined.
4. It depends on unit of measurement of the variable It has the Demerits
le es
:st he
ies )m
239
Measures of Dispersion
160 to 180 centimetres, if a dwarf (shortest) student whose height is 100 cemimetres is admitted in our data, the range would shoot up from 20 to 80 centimetres. Thus, a single variation in the value of an extreme item affects the value of the range. 3 It is influenced very much by fluctuations of sample. Range is subject to P uctuations ■ of values from sample to sample. However in small samples, it is uscxul in certain
circumstances.
4. It cannot be calculated in case of openend distributions because extreme values ot
the distribution are not known.
5. It does not tell anything about distribution of items in the series relative to a
measure of central tendency. Thus, the range is very unsatisfactory measure of dispersion and should be used with
great care and caution. 4. Uses of Range
Despite various limitations, the range is useful in the following areas: (a) Quality control : Range is used to study the variation in the quality of the items produced of a manufacturing concern. Range has a great significance in quality
control measures.
lb) Measure of fluctuations : It is a very useful measure to study fluctuation.: of series Variations in the prices of share, other commodities arJ money rates pnd rate ot exchange can easily be studied with the help of ran^e. " (c) Use in daytoday life : Range is by far the most widely .'sed measure of variabihty in our daytoday life. For example, the answer to the problems hke daily sales in a departmental store', 'monthly wages of workers in a factory'^ or the expected return of fruits from an orchard', is usually provided by the probable limits in the
form of range.
Id) Use in meteorological department: Range is also used in a very convenient measure by meteorological department for weather forecast since the general public is interested to know the limits within which the temperature is likely to vary on a particular day.
(b) Interquartile Range and Quartile Deviation
1. Meaning
2. Calculation of Quartile Deviation
3. Merits and Demerits of Quartile Deviation
1. Meaning i i ■£ u
Just as in case of range the difference of extreme items is obtained,^similarly if the
diffLnce in the two values of quartiles is calculated, it would give us what is called the 'Interquartile Range'. It is also a measure of dispersion. It is an advantage over range m as much as, it is not affected by the values of the extreme items. In fact 50% of the values of a variable are between the quartile (i.e., Q, and Q,) and as such the interquartile range gives a fair measure of variability.
240
Statistics for EconomicsXI
Semiinterquartile Range or Quartile Deviation ■ A. tU
I.Q.R = Interquartile Range = Q _ q
, Q3 = Third Quartile, Q. = First Quartile SemiInterquartde Range or Quartile Deviation
or
Q.D. =
Q3Q1 2
Symbolically,
QazQi
Coefficient of Quartile Deviation = _2__
Q3 + Q1 2
Coeff. of Q.D. = fc^
1. It IS simple to calculate and easy to understand. f
2. It IS rigidly defined.
3. It does not depend on all the values of the data v^rLblf ^he quartile deviation are the same as those of the
2. Calculation of Quartile Deviation
(a) Series of Individual observations
(b) Discrete Series
(c) Continuous Series !
(a) Series of Individual Observations
the ijtr ^^^^ 
145 130 200 210 , 198
234 159 160 178 257
260 300 345 360 390
fromi
Measures of Dispersion
SJiTiti.
241
' Y S NoC ' ■ Income (Rs) Income (Rs)
1 130 9 234
2 145 10 257
3 159 11 260
4 160 12 300
5 178 13 345
6 198 14 360
7 200 15 390
8 210
Steps
1. Arrange the data in ascending order to get the value of lower and upper quartiles.
2. Locate the value by finding out Qj = size of
item and Qj = size of
fN + 1
th
item.
3. Apply the formulae to get interquartile range, quartile deviation and coefficient of quartile deviation. Thus, we get
Qj = Size of = Rs 160
Q, = Size of
item = 4* item
ri5+n
th
item = 12* item
= Rs 300 Interquartile Range = Qj  Qj Here, Q, = 300 and Q, = 160
= 300  160 = Rs 140
Quartile Deviation =
Q3Q1
Q.D. = Hzl^ = Rs 70
Coefficient of Q.D. =
2
Q3Q1 Q3 + Q1
300160 140 300 + 160 460
= 0.304
242
j„ . Statistics for EconomicsXI
Solution. We are given Quartile deviation (Q.D.) = SlZ^
= 15 Marks
r ■ . •■•  = 30 Coefficient of Quartile deviation
...(1)
QizQL Qs+Qt
= 0.6.
30
Q3+Q1
= 0.6
...(2)
= Q3  Q = — 0.6
••• Qs + Qa = 50 Now equation (1) and (2) are solved, Q3  Qi = 30
80
Q3 = Y = Marks
Putting the value of Q^ in equation (i)
QsQ, = 30 40  = 30
Tu tt = 40  30 = 10 Marks
Thus, Upper Quartile = 40 Marks and Lower Quartile = 10 Marks
(b) Discrete Series
series : coethcient of quartile deviation from the following
Heights
'I '' 63 64 65 66
^ 6 15 10 5 4 3 1
f
Coe
(in inches)
No. of Persons :
Solution.
58 2
Range = L  S = 6658 = 8 inches
ai
(c) G HI
coeffii Ai Nc
Sol
I'l
Measures of Dispersion Quartile Deviation Steps :
1. Arrangement of items in ascending order is necessary.
2. Calculate cumulative frequencies.
3. Locate First Quartile and Third Quartile by
(n+IY"
and — .
V ^ y V ^ y
4. Values are located at the size of item in whose cumulative frequency the value of item falls.
5. Apply the formula to find quartile deviation.
Qj = size of
fN+l^
Item =
(49 + 1^
th
Item
= 12.5* item = 61 inches
Q, = size of
= 63 inches
Q.D. =
Q3Q1 _ 6361
3x50 . Item = = 37.5* item
= 1 inch
243
Height No. of
in inches persons (f) c.f.
58 2 2
59 3 5
60 6 11
61 15 26
62 10 36
63 5 41
64 4 45
65 3 48
66 1 49
Coefficient of Quartile Deviation
Coefficient of Q.D. = = = 0.OI6.
Q3+Q1 63 + 61 124
Thus,
and
Range = 8 inches Q.D. = 1 inch Coeff. Q.D. = 0.016
(c) Continuous Series
Illustration 8. Calculate range and quartile deviation and compare them. Also calculate coefficient of quartile deviation of the following data.
Age (years) : 2030 3040 4050 5060 6070 7080 8090 No. of members : 3 61 132 154 140 513
Solution. Range
Range = L  S = 9020 = 70 years
244
Quartile Deviation Steps :
1. Calculate cumulative frequencies.
2. First quartile and third quartile items are
(NY' 3/».T\th
j„ . Statistics for EconomicsXI
located by finding out (] and
V ^ y
.4
item
in continuous series.
3. Locate, the first quartile and third quartile group m cumulative frequency column where
size of respective — and
UJ
N
UJ
item falls.
4. Apply the following formula :
 + J X I
Thus, we get First quartile
+ JX /
= Size of
= size of
U
(544^
4
item
th item = 136* item
Hence, Q^ lies in the group 4050
'i + J X /
where.
h  40, ^ = 136, c.f. = 64, f= 132, / = lo
n  An 13664
an 72x10 " " ~13r = years
Hence first quartile is 45.45 years.
Age No. of t.f
(years) members (f)
2030 3 3
3040 61 64
4050 132 196
5060 154 350
6070 140 490
7080 51 541
8090 3 544
KI
Measures of Dispersion Third Quartile
245
Qj = Size of
= Size of
th
Vn
.4;
3x544
Item
= 408* item Hence Q3 lies in the group 6070
where.
h = 60,
N
c.f.
f
.4.
X I
= 408, c.f = 350, f = 140, i = 10
xlO
= 60 +
140 58x10
140
= 64.14 years
Quartile Deviation (Q.D.) =
^ 64.1445.45 2
= 9.345 years Coefficient of Quartile Deviation
Coefficient of Q.D. = ^^^
Q3 + Q1
_ 64.1445.45 64.14 + 45.45 _ 18.69 ~ 109.59 = 0.17
Thus Range = 70 years
Q.D. = 9.345 years Coeff. of Q.D. = 0.17
246
j„ . Statistics for EconomicsXI
Absolute value of dispersion
Quartile Deviation (OD ) Q3Q1 45,00018,000
vv ■]  ~ _  ^ j^g 13,500
100% Persons
O3 = Rs 45,000
Relative value of dispersion Coefficient of O.D. =
Q3Q1 Q3+Q1 "
45,00018,000 45,000 + 18,000
27,000
63:000 = 0428
and^rlti^etatr^tn?:^^^^^ ^ ^^^ ^ ^ ^^e data (Rupees)
of dispersion (0.428). percentage or coefficient of the absolute measure
3. Merits and Demerits of Quartile Deviation Merits :
2 'r'""" " "derstand.
3. It ,s also useful where extreme values are likely to affeet the results
4. T^S measure ,s useful when it is desired to know variah.lity in L Itral part o,
Demerits :
1. It ignores half the timeslst 25% and the last 25%
2. h IS a so not possible to give it further algebraic treatment.
4 2Tntb iT" " by fluctuations of sampling.
4^ As an ah olute measure it is not sufficient for comparison.
variX.'"^"'^ " ^^ for a rough study of
247
Measures of Dispersion
(B) Dispersion from Average
The range, the interquartile range and the quartile deviation suffer from common defect. They are calculated by only two values of a serieswither extreme values m case
of range or the two values of the quartiles as in case of quartile deviation. This method of studying dispersion by location of limits is also called the 'Method of Limits .
It is, therefore always better to have such a measure of dispersion which is based on all the observations of a series and is calculated in relation to a central value. Range and Quartile deviations are not calculated in relation to any average. If the variations ot items are calculated from an average, such measure of dispersion throws light on the formation of the series and the scatteredness of items around a central value. This method ot calculating dispersion is called the 'method of averaging deviations'.
Let .us examine from the following illustration about the salaries paid to employees of a departmental store :
Monthly Salaries of Employees
Employee A B C D E Total (Rs)
Salary (in Rs) m 10,000 2,000 4,000 6,500 4,500 EX = 27,000
Deviations from Mean 4,600 3,400 1,400 +1,100 900 £(XX) = 0
 IX 27000 Mean (X) = —  —
= Rs 5,400
We observe that the salary of A (Rs 10,000) is more from arithmetic mean (Rs 5,400) and the salary of B (Rs 2,000) is quite less than the arithmetic mean. In gerieral, some deviations are positive and some are negative. Similarly, some are large and some are
small.
If we consider an average of these deviations calculated from arithmetic mean, we can get an idea of a measure of dispersion. As we know the sum of the deviations calculated Lm arithmetic mean is always zero. Here, positive deviations and negative deviations cancel out each other. Therefore, adding these deviations directly does not help us. Alternatively, we may consider either the 'absolute deviations' or 'squanng deviations . Thus, the measures of dispersion in terms of deviations from central value (average)
are as under :
(A) Mean deviation or Average deviation
Where absolute deviations are obtained from average (ignoring plus and minus signs).
(B) Standard deviation
Where deviations obtained from arithmetic mean are squared.
248
Statistics for EconomicsXI
2. Calculation of Mean Deviation
3. Ments, Demerits and Uses of Mean Deviation 1 Meaning
^^'"■'^""ring plus (.) and nUnJsAsS^^^l^^^ rith^^ean or
Measures of Dispersion where.
249
ElDl = (Read sigma D modulus), sum of the deviations taken from mean or median ignoring ± signs N or M = Number of observations f = frequency X = Mean Me = Median Relative Measure of Mean DeviaHon
Coefficient of M.D. = ^
M.D.
XorMe
2. Calculation o£ Mean Deviation
{a) Series of Individual observations
[b] Discrete series
(c) Continuous series
(a) Series of Individual Observations . j
BtasMrio. 10. Calculate mean devation and its coefficient from median and mean fro^TL following yeld of rice per acre for 10 districts of a state as under:
Districts
Rice Yields (in tons) Solution.
1
22
2
29
3 12
4
23
5 18
6
15
7 12
8
34
9 18
10 12
Calculation of Mean Deviation
12 12 12 15 18 18 22 23 29 34
N = 10
6 6 6
3 0 0
4
5 11 16
LIDI = 57
22 2.5
29 9.5
12 7.5
23 3.5
18 1.5
15 4.5
12 7.5
34 14.5
18 1.5
12 7.5
EX = 195 EIDI =
250
1. Arrange the data in ascending order.
2. Calculate the median of the series
fN + lY'
j„ . Statistics for EconomicsXI
Me = size of
Item.
3. Take deviations of item from median Jgxonng ± smgs and denote the column as
4. Cdculate the sum of these deviations,
1. Calculate the total item of finding arithmetic mean. '"umg
2. Take deviations of items from mean
Jg^ormg ± signs and denote the column by
3. Cdcufate the sum of these deviations,
4. Divide the total obtained by number of items. Formula :
EIDI
5. Divide the total obtained by number of items. Formula :
ZIDI
M.D. =
N
M.D. =
N
Si"""'' »'
Coefficient of M.D. = ^
XT Mean
Now, we get
Mean =
Median
fN + V th item
4 J
rio+1^ th
< 2
Now, we get Median = Size of
= Size of
= 5.5* item = Value of 5* item + 0.5 (Value of
item  Value of 5* item) = 18 + 0.5 (18  18)
= 18 + 0.5 (0) = 18 tons
Absolute Measure :
JV ~ 10  'on®
Relative Measure : Coefficient of M.D.
M.D. 57
N 195
Median ~ =
~ 10
= 19.5 tons Absolute Measure :
N = —
~ 10 = 6 tons
Relative Measure : Coefficient of M.D.
= m.D.
Mean
~ 19.5 = 0.307
251
Measures of Dispersion
Note It is better to calculate M.D. from median than that from mean because the sum of the deviations taken from median ignoring ± signs is less than sum of deviations taken
from mean.
Illustration 11. the yield of wheat per acre for 10 districts of a state is as under:
District : 1 2 3 4 5 6 7 8 9 10
Yield of wheat : 12 10 15 19 21 16 18 9 25 10 (in tons) ,
Calculate : .
(i) Range and coefficient of range. («) Quartile Deviation and its coefficient. {Hi) Mean Deviation about Mean and coefficient. (iv) Meati Deviation about Median and coefficient.
Solution. In order to calculate the quartile and median we arrange the yield of wheat in the ascending order of magnitude.
(i) Calculation of Range (Absolute Measure)
Range = L  S
L = 2S,S = 9 Range = 259 = 16 tons
(ii) Calculation of Quartile Deviation
Calculation of Coefficient of Range (Relative Measure)
LS
Coefficient Range =
259_16^0.47
25 + 9 34
Qj = size of
N + l
item = size of
rio+1
Mh
item
9 10 10 12
15
16 18 19 21 25
= size of 2.75th item
= Value of 2nd item +  (values of 3rd item  value of 2nd item) = 10 + 0.75 (10  10) = 10 + 0 = 10 tons.
Qj = size of
rN+n
th
item = size of
srio+i"!
th
item
= size of 8.25th item
= Value of 8th item + ^ (Value of 9th item  Value of 8th item)
= 19 + 0.25 (21  19) = 19 + 0.25 (2) = 19 + 0.5 = 19.5 tons
252
Absolute Measure : ■•■ Quartile deviation Q.D. = Sl^
_ 19.510
 =4.75 tons
of Mean deviation
Statistics for EconomicsXI Relative Measure :
Coeff. of Quartile deviation
Coeff. Q.D = QiJ:^ Q3+Q1 ^ 19.510 _ 9.5 l^XTlO ~ 2^ = 0.322
Absolute Measure :
Mean Deviation from Mean
Arithmetic Mean, =
N ~ IF tons
Applying formula, we get
M.D. =
Mean Deviation = 4.2 tons ^^
= 4.2
Mean Deviation from Median
Me = Size of
= Size of
4
10^ y''
item
item
Relative Measure :
Coeff. of M.D. =
Mean 15.6 ~ 0.269
= Size of 5.5th item = Value of 5th item
2 (Value of 6th item
Value of 5th item) = 15 + 0.5 (16  15) = 15 + 0.5 = 15.5 tons
M.D.=
^ 10" ••• Mean Deviation = 4.3 tons
Relative Measure :
M.D. 4.3
Coeff. of M.D. =
Median = 0.277
15.5
Measures of Dispersion
, 253.
Range = 16 tons . Q.D. = 4.75 tons M.D. = 4.3 tons
(from median)
and " 
that ,s why we ate g tfj™ h^ XeTah T "'data,
deviation,wh,chasaLaRe of Ae d^er„rh l Q""™''
all the observations of se™s ttn«25 " f"
the value 4.75 tons. "f series, giving us
^^ ^^ ~g of absolute
distribution and thus cVcu^ ^^^ regularities in the
true measure of dispersion accurate and
Relative Calcularioas of Illustration 10 and 11 Refer to Illustration 10 and 11
p R^eY^U ■ Wl^at Veld
M.D. = 5.7 tons Coeff. of M.D. = 0.316 M.D. = 4.3 rons Coeff. of M.D. = 0.277
y.eld has greater variati" 6)1" callations we decide rife
has lesser variation is more rehable Therefore the vSh 7" ^^^ ^^^P ^hich
the yield of rice. ifteretore, the yield ot wheat is more reliable than
{b) Discrete Series
Also"tS:^^^^ lismbution.
254
Solution.
Statistics for Economics~XI
0 1 2
3
4
5
6
7
8 9
10 11 12
15
16 21 10 16
8 4 2 1 2 2 0 2
IDt fm
15 2 30
31 1 16
52 0 0
62 1 10
78 2 32
86 3 24
90 4 16
92 5 10
93 6 6
95 7 14
97 8 16
97 9 0
99 10 20
Z/'IDI  194
Total
N = 99
Steps :
1. Calculate cumulative frequencies
2. Locate the item by finding out
th
item.
3. Value is located at the size of the item in whose cumulative frequency the value of « Item falls.
4. Find out the median,
5. ^ke deviations of items from median ignoring ± signs and denote the column as
6. Multiply frequencies with deviation and get f\D\.
7. After getting the total of f\D\ column apply the following formula :
I.f\D\
N
M.D. =
Median
Me = size of
= Size of
(N + lf
2 J
item
item = 50^^ item
Median = 2 Accidents
Measures of Dispersion Mean Deviation :
255
M.D. =
Z/IDI 194
N
Coefficient of Mean deviation :
M.D.
Coefficient of M.D. =
99
1.96
Median
= 1.96
= Approx 2 Accidents = 0.98
(c) Continuous Series '
Illustration 13. Calculate Mean Deviation from mean and its coefficient of the following data :
Marks : 010 1020 2030 3040 4050
No. of Students : 5 8 15 16 6
Solution.
Calculation of Mean Deviation from Mean
Marks X No. of Students Midpoints m. m25 10 d' fd' m 17 ^IDI •
010 5 5 2 10 22 110
1020 8 15 1 8 12 96
2030 15 25 0 0 2 • 30
3040 16 35 +1 +16 8 128
4050 6 45 +2 +12 18 108
N = 50 Lfd'^ 10 •Lf\p\ = 472
Steps :.
1. Calculate Arithmetic Mean by step deviation method.
2. Take the deviations of midpoints from mean ignoring ± sings and denote them by ID!.
3. Multiply these deviations by respective frequencies and find out /IDI.
4. After getting the total of f\D\ column apply the following formula :
If\D\
M.D. =
Now, we get Arithmetic Mean :
N
Zfd'
X = A + ^x C N
256
where. Mean Deviation :
A = 25,Lfd' =10, N = 50, C = 10
V ir 10
^ ^ "" 50 = 25 + 2 = 27 Marks
Statistics for EconomicsXI
M.D. =
If\D\ N
where, I.f DI = 472, N = 50, M D = ~
50
Mean Deviation = 9.44 Marks Coefficient of Mean Deviation =
Here,
M.D.
X
M.D. = 9.44 and X = 27 9.44
>27
= 0.349
010 5
1020 10
2030 20
3040 5
4050 10
Marks :
No. of Students : Solution. (/) Calculation of Range Absohite Measwe :
Range = L S ^here, L = 50, S = 0
= 50  0 = 50 Marks
Relative Measure :
Coefficient of Range =
L5 L + S
500 50 + 0
= 1
(ii) Calculation of Quartile Deviation
First, we will calculate and Q^
Qj = size of = size of
V4y (50
item
th
item = 12.5th item
Measures of Dispersion 279
Qj lies in the class 10  20. Applying the following, formula :
257
Q,I,*
N ■ 4
cf
■XI
Here,
N
/j = 10, — = 12.5, c.f = 5, f= 10 and i = 10
= 10 + ^^xio
= 10 + ^^^^ = 17.5 Marks
Q = size of
N 14
Item
= size of
'50
.4,
th
item : 37.5* item
Q lies in the class 30  40
Here, 1, = 30,
Vn)
v4y
= 37.5, c.f = 35, /■ = 5 and i = 10
Q, = 30+37.535^^^
= 30 +
5
2.5x10 10
= 32.5 Marks
Absolute Measure :
Quartile Deviation, Q.D. = ^^^^^
32.517.5
= 7.5 Marks
Relative Measure :
Coefficient of Q.D. = Q^Qi
Q3+Q1
32.517.5 15 32.5 + 17.5 50 ■
258
(iii) Calculation of Mean Deviation from Median
Statistics for EconomicsrXI
Marks Stud^ts f c.f. Midpoitos. m f«25 IDI f\m
010 5 5 5 20 100
1020 10 15 15 10 100
2030 20 35 25 0 0
3040 5 40 35 10 50
4050 10 50 45 20 200
N^ 50 Zf IDI = 450
Steps :
1. Calculate median of the given data.
2. Take the deviations of midpoints from median ignoring ± signs and denote IDI.
3. Multiply deviations by respective frequencies and find out Z/'IDI.
4. After obtaining the total of f\D\ column, apply the following formula :
n\D\
M.D. =
N
Now we get. Median :
Median = Size of = Size of
Median lies in the class 2030 Apply the following formula :
N
l2j 12J
item
item = 25th item
Here,
Me = + ^ .
/, = 20, J = 25, c.f = 15, /■ = 20 and / = 10 Me =.20 + ^^x 10
20 +
20
10x10 20
= 25 Marks.
Measures of Dispersion
Absolute Measure : Mean Deviation
259
M,D. =
If\D\ N
Here, Zf \D\ = 450 and N = 50 450
M.D. = Relative Measure
50
= 9 Marks
Coefficient of M.D. = ^ = ^ = 0.36
Illustration 15. Calculate the mean deviation from mean for the following marks obtained by 10 students.
Marks : 24 46 68 810
Student : 3 4 2 1
24 3 3 2 ■ '6 2.2 6.6 6
46 4 5 0 0 0.2 0.8 0
68 2 7 +2 4 1.8 3.6 4
810 1 9 +4 4 3.8 3.8 4
N = 10 lfd = 2  I/IDI = 14.8 I/^WI = 14
Steps
1. Calculate Arithmetic Mean by assumed mean method.
2. Take deviations of midpoints from mean ignoring ± signs and denote them by IDI.
3. Multiply these deviations by respective frequencies and find out f IDI.
4. After getting the total of /IDI column apply the following formula:
S^IDl N
M.D. =
Now, we get Arithmetic Mean :
X = A +
M
N
where, A = 5, = 2 and N = 10 2
= 5 +
10
= 5.2
260
Mean Deviation:
j„ . Statistics for EconomicsXI
Here, where.
M.D. =
If\D\ N
X/IDI = 14.8, N = 10
14.8
= =1.48 Marks
Alternatively: (Shortcut Method) apply the follo„i°; ZnZ, ' ob«,ni„g Xf W,,
where.
Now, we get
M.D. = l/l^^llP^z^KI/B^ N
^f\d\=14,A = 5,N=10 •
class (the class m which mean hes), i.e., 4 + 3=7 If A = Sum of all class frequencies after the mean' class, 2 + 1=3
MD =
10
= 14 + (0.2) (4) 14 + 0.8 14.8
10
10
M.D. = 1.48 Marks
Note : Take care that assumed mean is close to the true mean.
3. Merits and Demerits and Uses of Mean Deviation Ments :
d i.. value .s P.ec,se and
u
spite statist deviat cycles, studiei
(B) SI 1. 2.^
3.<
4.J 5.1 6.1
261
Measures of Dispersion
3 Based on all items. It is based on all the items of the series, hence it is affected by ■ every value of the distribution. Thus mean deviation is a better measure of dispersion than range and quartile deviation. 4. Less affected by extreme values. It is not affected very much by value ot extreme
items.
5 Absolute measure. The averaging of absolute deviations for an average takes out the irregularities in the distribution and thus mean deviation provides an accurate and true measure of dispersion.
6 Calculated value. Mean deviation is not based on limits like range and quartile deviation. It is a calculated value based on the deviations about an average. It provides a better measure for comparison about the formation of different
distributions.
Demerits : u u i
1 Ignoring the signs. The strongest objection against mean direction is that while
■ calculating its value we take the absolute value of the deviations about an averap and ignore the ± signs of the deviation. The step of ignormg the signs of the deviation is mathematically unsound and illogical. Therefore this method is nonalgebraic, for this reason it is not in further statistical calculations.
2 Not well defined. Mean deviation is not a welldefined measure since it is calculated
■ from different averages (mean, median and mode). Mean deviation calculated from various averages will not be the same.
3 Harder calculations. Mean deviation involves harder calculation than the range ' and quartile deviation. Its calculation by an arbitrary origin makes the calculation
tedious. 1 1 ■ • u
4. Cannot be calculated. Mean deviation cannot be computed for distribution with
openend classes.
Uses. Despite so many demerits, mean deviation is not a totally useless measure. In spite of its mathematical drawbacks, it has found favour with economists and business statisticians because of its simplicity, accuracy and also on account of the fact that standard deviation gives greater importance to deviations of extreme values. For fj^^^sting ^u^s cycles, this measure has been found useful than others. It is also good for small sample studies where elaborate statistical analysis is not required.
(B) Standard Deviation
1. Meaning
2. Calculation of Standard Deviation
3. Other Measures from Standard Deviation
4. Mathematical Properties of Standard Deviation
5. Relation between Measures of Dispersion
6. Merits and Demerits of Standard Deviation
262 ■ '
Statistics for EconomicsXI
1. Meaning
It is^e mnf^' Standard deviation was introduced by Karl Pearsons in the year 1893 It i^he most commonly used measure of dispersion. It satisfies most of the properties iLd down for an ideal measure of dispersion. properties laid
rr " mathematically illogical as in its
calculation signs are ignored and absolute deviations are taken. This drawback is
removed m die calculation of standard deviation. One of the easiest ways of dXg a way
devil"'me^r^^^^ ^^ ^^^^^ ^^^ ^^^
Standard deviation is also known as root mean square deviation because it is the square root of the means of squared deviations from le arithmetic La„
.nnT!?"'; deviation, first the arithmetic average is calculated
and the deviations of various items from the arithmetic average are square! ^e ql^d
deviations are totalled and the sum is divided by the number of iteL tL sqnaTrrot Symbolically,
where
<y
XX = *

2. Calculation of Standard Deviation
(A) Series of Individual Observations
(B) Discrete Series
(C) Continuous Series
(A) Series of Individual Observations
Standard deviation may be calculated by any of following methods •
(«) Actual mean method (b) Direct method (c) Assumed mean method
(a) Actual Mean Method
Illustration 16. Calculate Standard Deviation of the following data • 25 50, 45, 30, 70, 42, 36, 48, 34, 60
Measures of Dispersion Solution.
Calculation of Standard Deviation Steps :
1. Calculate the actual mean of the observations.
2. Obtain deviations of the values from the mean, i.e., calculate (X  X). Denote these deviations by
X.
3. Square the deviations and obtaiti the total Ix^.
4. Divide Ix^ by number of observations and find out the square root.
5. Apply the following formula :
263
Here,
Now we get.
Here,
= (X X)
X =
i:X 440
= 44
N 10 Ix^ = 1710, N = 10
jrm
= iir ^ \ 10
Values X XX *
25 19 361
50 +6 36
45 +1 1
30 14 196
70 26 676
42 2 4
36 8 64
48 +4 16
34 10 100
60 +16 256
i:X=r440 1x^4=1710
= jl7i = 13.076
(b) Direct Method
Illustration 17 Calculate the standard deviation of data given in Illustration 16 by direct method. Solution.
Calculation of Standard Deviation
Steps
1. Calculate the actual mean of observations.
2. Obtain the sum of square of values.
3. Apply the following formula :
= iw ■i
{Xf
. N .
Vaiues
X
25 625
50 2500
45 2025
30 900
70 4900
42 1764
36 1296
. 48 2304
34 1156
60 3600
2:X = 440 IX^ = 21070
264
Now we get,
Statistics for EopnomicsXl
X =
EX 440
= 44
N 10
Here, ZX' = 21070, N = 10 and X = 44
a =
'21070
(44)2
N "" " V 10 = V21071936 = n/171 = 13.076 (c) Assumed Mean Mediod
Solution.
Calculation of Standard Deviation Steps
1. Calculate the deviations of the observations from an assumed mean (X  A). Denote these deviations by d and make the total of deviations.
2. Square the deviations and denote the total LiP
3. Apply the following formula :
a =
d = XA
N
Here,
When the mean is in fraction, this method is used to simphfy the calculations.
25 20 400
50 +5 25
45 0 0
30 15 225
70 +25 625
42 3 9
36 9 81
48 +3 9
34 11 121
60 +15 225
N = 10 Id Id'
= 10 = 1720
Now we get. Here,
a =
/ N
N
1720, N = 10, ZJ =  10
a =
1720 flO^
10
10
= VI72(1)2 = Vm = 13.076
Measures of Dispersion 265
Illustration 19. From the following information, find standard deviation of x and y variables :
Ix = 235, Ey = 250
Ix^ = 6750, = 6840
N = 10
Solution.
: : V X
ox = y N v m) ay = = \ N InJ
6750 10lio J 6840 p50Y 10 I 10 J
 ^675(23.5)2 = ^/675552.25 = 22.75 = 11.079 = V684(25)= ^684625 = V59 = 7.68 .
(B) Discrete Series
Standard deviation can be calculated by any of the following methods :
(a) Actual mean method (b) Direct method
(c) Assumed mean method (d) Step deviation method
(a) Actual Mean Method
Illustration 20. Calculate Standard deviation of the following data :
: 4 5 6 7 8 9 10
Frequency : 6 12 15 28 20 14 5
Solution.
Calculation of Standard Deviation
Size Frequency f . /X X X fx^
4 6 24 3.06 9.3636 56.1816
5 12 60 2.06 4.2436 50.9232
6 15 90 1.06 1.1236 16.8540
7 28 196 0.06 0.0036 0.1008
8 20 160 +0.94 0.8836 17.6720
9 14 126 +1.94 3.7636 52.6904
10* 5 ■ 50 . +2.94 8.6436 43.2180
N = 100 •LfX = 706 Ifx' = 237.6400
266
jj^p^ _ ^tMistics for EconomicsXI
(x'ltrr„
(X X) and denote these deviations by .r
A by Ae respecve frequences and n,ake
3. Apply the following formula :
a =
N IfX
Here,
N
X =
^fX = 706 and N = 100 706
X =
Applying formula, we get
100
= 7.06
c =
Here,
N
Zfx'^ = 237.64 and N = 100 [237J4 r_
= 1.541
{b) Direct Method
direar^od" of 'he data given in I„us„at,on „ by
Solution.
Calculation of Standard Deviation
(c) ass
X f X2
4 5 6 7 8 9 10 6 12 15 28 20 14 , . 5 24 60 90 196 160 126 50 16 25 36 49 64 81 100—__ ^ _ 96 300 540 1372 1280 1134 fDO i
N = 100 IfX = 706 ^X^^"5222 f
Measures of Dispersion Steps :
1. Calculate mean of the series, i.e., X •
2. Obtain the sum after mukiplying f and X (frequency and size), i.e., Z/X.
3. Calculate the square of values (X^)
4. Multiply frequency (/) to X^ and get the total, i.e., ZfX"
5. Apply the following formula :
267
a =
Z/X^
N
(X)'
\
Ifx^ fz/x^ N J
N
m 7.06
N 100
Now we get,
Here, Z/X^ = 5222, N = 100
Substituting the values
a =
IfX"
N
(Xf =
5221
100
(7.06)2
= V52.2249.8436 = ^2.3764 = 1541 .. Standard Deviation = 1.541
(c) Assumed Mean Method
Illustration 22. Calculate the standard deviation of the data given in Illustration 20 by
assumed mean method. Solution.
Calculation of Standard Deviation
Size Frequency Y  7 ^ ' ' —•——^—
4 6 3 18 54
5 12 2 24 48
6 15 1 15 15
7 28 0 0 0
8 20 +1 +20 20
9 14 +2 +28 56
10 5 +3 +15 45
N = 100 ■ E/a = 6 Ifd^ = 238
268
Statistics for EconomicsXI
Steps :
r . Take the dev,ario„s of s,ze from an assnmed mean and denote these delations 2. Multiply these deviations by the respective frequencies and calculate the total Zfd
where.
Now, we get
d= {X A)
W fXfdf
N
m = 238, Yfd = 6, 100
Here, _
Substituting the values.
Ifd N
0 =
238
100
= A/2.38(0.0^ = V2.3764
jlOO
= a/^380.0036 a = 1.541
and — « 
id) Step Deviation Method
^ Jllustradon 23. Calculate Arithmet.c Mean and Standard Deviation for the following
Value Frequency Solution.
140 1
145 4
150 15
155 30
160 36
165 24
170
175 2
Values
^i^l^l^io^ A^^ Deviation
140
145 150 155 160 165 170 175
Frequency
1 4 15 30 36 24
N = 120
X ~ iss d
10 5 0 +5 +10 + 15 +20
d'
3 2 1 0 +1 +2 +3 +4
fd'
3 8 15 0
+36 +48 +24 +8
m' = 90
fd
9 16 15 0 36 96 72
__32
= 276~
269
Measures of Dispersion Steps :
1. Take the deviations of values from an assumed mean and denote these deviations
by (d).
2. Divide these deviations by common factor and obtain step deviations, i.e., d'.
3. Multiply step deviations by the respective frequencies and calculate the total Zfd'.
4. Calculate the squares of the step deviations (J"), multiply these squared deviations by respective frequencies (in other words fd' x d' = fd') and obtain the total Zfd'.
5. Apply the following formula :
where,
Now, we get Arithmetic Mean
Here,
a =
d' =
V N I N
C
C
and C = Common factor
xC
N
A = Assumed mean = 155, Ifd' 9^ = 5, N = 120 90
X = 155 +
Standard Deviation
120
= 155 + 3.75 = 158.75
X 5 = 155 + 0.75 X 5
a =
Xfd'^ fLfd'
N
N
C
Here,
Tfd' = 276, Tfd' = 90, N ^ 120 and C = 5
0 =
(27
il20'
f 90 ^ 120
X 5
= V2.3(0.75)2 X 5 = /2.30.5625 x 5
= V1.7375 X 5 = 1.318 X 5 = 6.59
(C) Continuous series
For calculating standard deviation in continuous series any of the following methods may be applied :
{a) Actual mean method (b) Direct method
(c) Assumed mean method (d) Step deviation method
270
{a) Actual Mean Method
j„ . Statistics for EconomicsXI
calXr«t jalT"^ * formula ,s used ,o
V N
where, x = {X  X)
24. Find .he Mean and S«ndard deviation from .he following dis.rihu.ion :
No. of Students , 4 V
« 2 1
Solution.
Calcularion of Mean and Standard Deviation
Marks X
48 812 1216
No. of Students if)
1 1
N= IS
Midpoints (m)
1 6 10 14
fm
8 48 20 14
Ifm = 90
(m ~ X)
X
A 0 +4 +8
16 0 16 64
fx^
64 0 32 64
Ifx' = 160
Steps :
1. Calculate the actual mean of the series, x
dpomts from the mean,  x). Denote these
~ * the respective frequencies and ohtam
Se ^Sr dS ^^ ^^^ ^^re root to calculate the
Apply the following formula :
a =
N
Mean
Here,
^  N ~15 = 6 marks S/w = 90, N = 15
Measures of Dispersion Standard Deviation
271
a =
If'
N
Here,
Zfx^ = 160, N = 15
0 = J^ = M666 = 3.265
.. Standard Deviation = 3.265 marks.
Note. This method is rarely applied in practice becausc in case the actual mean is in fraction, the calculations becomes complicated and take lot of time.
(b) Direct Method
Illustration 25. Calculate the standard deviation of the data given in Illustration 24 by direct method. Solution.
Calculation of Standard deviation
Marks No. of Students  Midpoints fm^
X (f) (m) fm w'
04 4 1 8 4 16
48 8 6 48 36 288
812 2 10 20 100 200
1216 1 14 14 196 196
N = 15 ■Lfm = 90 E/m = 700
Steps :
1. Calculate actual mean of the Series X.
2. Obtain the total after multiplying f and m, i.e., Zfrn.
3. Calculate square of midpoints, i.e., m^.
4. Multiply frequency to m^ and get the sum, i.e., Zfm^.
5. Apply the following formula :
0 =
i
Ifm^ (Ifm
N
N
{Xf
= V N • ■ We get, N = 15, Zfrn = 90, Zfrn' = 700
2^2
Substituting the values
Statistics for EconomicsXI
15
9o^ 15
= ^ yfiOM ^ 3.265
Standard Deviation = 3.625 marks.
(c) Assumed Mean Method
d J!"" ^^^ I^ation of the followmg frequency
5055 45 50 40^5 3540 3035 2530
29 31 47 51 70
Age in Years No. of Labourers Solution.
Calculation of Standard Deviation
Age in years . , x^ cc No. of labourers Jf> Midpoints (m) m  42.5 d fd fd^
4550 4045 3540 3035 2530 22 29 31 47 51 70 52.5 47.5 42.5 37.5 32.5 27.5 +10 +5 0 5 10 15 220 145 0 235 510 1050 2200 725 0 1175 5100 15750
N = 250 Ifd = 1430 Ifd^ = 24950
Svwt tr'""  and denote these
2. Multiply these devattons by the respective frequencies and calculate the total, m \
4. Apply the following formula : i
a =
where,
d= (XA)
. N
Measures of Dispersion Now we get.
273
a =
lLfd^ fLfd^
. N VN )
Here, = 24950, Zfd = 1430, N = 250
Substituting tbe values.
a =
24950 250
r1430
= ^99.8(5.72)2
250
= V99.832.718 = V67.082 = 8.19
Standard Deviation = 8.19 years
(d) Step Deviation Method
This method is mostly used in practice.
Illustration 27. Find out the Standard Deviation of the frequency distribution given in Illustration 26 by step deviation method.
Solution.
Calculation of Standard Deviation
Age in years X No. of labourers f Midpoints m m  4.25 4 m 4.25 5 d'id' fd"
5055 22 52.5 +10 +2 +44 88
4550 29 47.5 +5 +1 +29 29
4045 31 42.5 0 0 0 0
3540 47 37.5 5 1 A7 47
3035 51 32.5 10 2 102 204
2530 70 27.5 15 3 210 630
N = 250 Ifd'=  286 lA'^ = 998
Steps
1. Take the deviations of midpoints from an assumed mean and denote these deviations by d.
2. Divide these deviations by common factor and obtain step deviations, i.e., d'.
3. Multiply step deviations by the respective frequencies and calculate the total Jlfd'.
4. Calculate the squares of the step deviations {d'^Y, multiply these squared deviations by respective frequencies (in other words fd' x d' = fd') and obtain the total I^fd'.
274
5. Apply the following formula :
Statistics for EconomicsXI
where, and
a =
d' =
X C
I C
C = Common factor
Now
we get.
a =
Ifd
r2
N
Zfd'
{ N
\2
X C
where, ^fjn = 993, ^fj^ ^ _ N = 250 and C = 5
Substituting the values,
a =
998 250
286^ I 250
5 = 73.992(1.144)2 X 5
= V39921.309) X 5 = VI^ x 5 = 1.638 X 5 = 8.19
.. Standard Deviation = 8.19 years. Illustration 28. Find the Standard Deviation of the height of 100 students.
Less than 62.5 Less than 65.5 Less than 68.5 Less than 71.5 Less than 74.5
5 23 65 92 100
Solution. Convert cumulative frequency into class interval.
Calculation of Standard Deviation
Height (in inches) X Frequency Midpoints im) ni67 3 (d) fd' fd"
59.562.5 62.565.5 65.568.5 68.571.5 71.574.5 5 235 =18 65  23 = 42 92  65 = 27 100  92 = 861 64 67 70 73 2 1 0 +1 +2 10 18 0 +27 +16 20 18 0 27 32
N = 100 Ifd' = +15 m'^ = 97
Measures of Dispersion
Applying the formula, we get
275
a =
<
Ifd'^ flfd'^
N
N
X C
where, = 97, Zfd' = 15, N = 100, C = 3
Substituting the values.
o =
97
100
100,
X 3 = VO.970.0225 x 3
= ^/0.9475 X 3 = 0.9733 x 3 = 2.92
Standard Deviation. = 2.92 inches. Illustration 29. Calculate Mean Standard Deviation and mean deviation about mean
Marks Students
More than 20 50
More than 40 47
More than 80 41
More than 100 21
More than 120 9
Solution. (Convert cumulative frequency into class interval)
Calculation of Mean and Standard Deviation
Marks X Frequency (f) Midpoints (m) wjyo 10 (dl fd' fd^
2040 50  47 = 3 30 6 18 108
4080 47  41 = 6 60 3 18 54
80100 41  21 = 20 90 0 0 0
100120 21  9 = 12 110 +2 24 48
120140 90 = 9 130 +4 36 144
1 • N = 50 Ifd' = 24Lfd'^ = 354
Mean :
Applying formula, we get
XA.^.C
where.
A 90, Lfd' = 24, N = 50, C = 10
276
j„ . Statistics for EconomicsXI
 24
^ " ^^ ^ X 10 = 90 + 4.8 = 94.8
Mean = 94.8 Marks
Standard Deviation
Applying the formula, we get
a =
N
where.
I N
X C
= 354, Zfd' = 24,N = 50, C = 10
a =
SO
v50y
xlO
= V^xlO = 2.617 X 10 = 26.17 Standard Deviation = 26.17 Marks
Deviation from Mean
Marks X
2040 4080 80100 100120 120140
Frequency (ft
3 6 20 12 9
Midpoints m
N = 50
Mean Deviation
Applying the formula, we get
30 60 90 110 130
■■m. ~ 94.8 ID!
64.8 34.8 4.8 15.2 35.2
M.D. =
where.
N
If IDI = 998.4 and N = 50
\A r, 998.4
MD = = 19.968
••• Mean Deviation = 19.968
/"IDI
194.4 208.8 96 182.4 316.8
2/IDI = 998.4
3. <
mea
(b
Measures of Dispersion 111
Let us try the same question by assumed mean method (Assumed Mean = 90)
t  . Marks ■ ■ ■■ f m m  90 /lai
h ^ \d\
20A0 3 30 60 180
4080 6 60 30 180
80100 20 90 0 0
100120 12 110 20 240
120140 9 130 40 360
\d\ = 960
Applying formula, we get
M.D. =
I/IJI+(XA)(I/BIM) N
where, IM = 960, X = 94.8, A = 90,1/B = 3 + 6 + 20 = 29 and S/A = 12 + 9 = 21
960 + (94.890)(2921) 50
960 + (4.8)(8) 960 + 38.4
50 998.4
50
50
= 19.968
Mean Deviation = 19.968
3. Other Measures from Standard Deviation
Various measures are calculated from standard deviation. Some of the important measures are as under :
(a) Coefficient of Standard Deviation : A relative measures of standard deviation is calculated to compare the variability in two or more than two series which is called 'coefficient of standard deviation'. This relative measurement is called by dividing standard deviation by arithmetic mean of the data
Symbolically,
Coefficients of S.D. = ^
X
Here, a = Standard deviation and X = Arithmetic mean
(b) Coefficient of Variation : This relative measurement is developed by Karl Pearson and is most popularly used to measure relative variation of two or more than two series. It shows the relationship between the standard deviation and the arithmetic mean expressed in terms of percentage. This measure is used to compare uniformity, consistency and variability in two different series. The series having greater coefficient of variation, it is called to be less uniform, less homogeneous, less
m
278
j„ . Statistics for EconomicsXI
consistent or less stable (in other words, it has higher degree of variability). In the same way, the series having lesser coefficient of variation, it is said to be more uniform, more homogeneous, more consistent or more stable, (in other words, it has less degree of variability).
Symbolically,
C.V. = ^ X 100
Here, C.V. = Coefficient of Variation, a = Standard deviation and X = Arithmetic mean.
(c) Variance : Variance is the square of standard deviation. Standard deviation and variance are measures of variability and they are closely related. The only difference between the two measurements is that the variance is the average squared deviation from mean and standard deviation is the square root of variance. Symbolically,
Variance = a^ and Standard Deviation = VVariance
Calculation of Variance
In Series of Individual Observation :
Variance (a^) =
N
^ nxxf
N
Here, x = X  X
In Frequency Distribution :
Variance (a^) =
flfd'^ 2'
N [ N J
X a
Here,
d' = , and
C = Common factor
Individual Observations
Illustration 30. A batsman is to be selected for a cricket team. The choice is between X and Y on the basis of their five previous scores which are : ■X : 25 85 40 80 120
y ■ 50 70 65 45 80
[a) Calculate coefficient of standard deviation, variance and coefficient of variation.
[b) Which batsman should be selected if we want
(/) a higher run scorer (ii) a more reliable batsman in the team.
Measures of Dispersion Solution, (a)
Batsman X
Arithmetic Mean
JX
X =
N
EX = 350, N = 5 350
X = = 70 Average score = 70 Runs
Standard Deviation
a =
N
Ix' = 5750 and N = 5
a =
5750
= V1150
= 33.91 Runs
Coefficient of Standard Deviation
Coefficient of S.D. = ^
Coeff. of S.D. =
a = 33.91 and X = 70 33.91
70 = 0.484
Variance
E(XX)2 Ijc^
N
N
I(X XV = = 5750
Batsman Y
Y =
N
EY = 310, N = 5 310
Y =
= 62
Average score = 62 Runs
a =
N
Ey2 z: 830 and N = 5
oy
= 12.88 Runs
Coefficient of S.D. = ^
279
Batsman (XX)^ Batsman (yYf
Scores XX Scores X Y
X * Y
25 45 2025 50 12 ■ 144
85 +15 225 70 +8 64
40 30 900 65 +3 9
80 +10 100 45 17 289
120 +50 2500 80 +18 324
EX = 350 Ex^ = 5750 EY = 310 = 830
Coeff. of S.D. =
a = 12.88 and Y = 62 12.88
62 = 0.207
_ E(YY)2 ^^ ^ N N
E (Y Yf = ly = 830
280
Here, X  X = x
5750
Statistics for EconomicsXI
ax^ =
=1150 Runs
Coefficient of Variation
„  ax
c.v.^ =• Y 100
o = 33.91 and X = 70
33.91 C.V = X 100
= 48.44%
Here, y  y ^ ^
, 830 ay = —
= 166 Runs
C.V.^ = ^ X 100
(J, = 12.88 and Y = 62
„„ 12.88 CV. = X 100
= 20.77%
ib) H) Batsman X should be selected as a
(70 ™„S, . than Ae't^ii tt'o^y ST^r,
1 '""IT of
Discrete Series """ * = 48.44%,.
^^ Jlustration M. Calculate variance and coefficient of variation from the foHowin,
Solution.
Size
4
5
6
7
8 9
10
Frequency
3
7 22 60 85 32
8
Size X
4
5
6
7
8 9
10
of Variation
Frequency
—
3
7 22 60 85 32
8
~NT217~
X~7 d
3 1 1 0 +1 +2 +3
fd
9 14 22 0
+85 +64 +24
m = 128
Cot
estir
fd'
17 28 22 0 85 128 71
a an
Measures of Dispersion Variance
, ,, Ifd^ ri/iif
Variance (o^) = —  J Here, ^ = 362, l.fd = 128 and N = 217
362 YmY
Variance (a^) = ^ (^217,
= 1.668  (0.589)2 = 1.668  0.347 = 1.32
Coefficient of Variation
C.V. = I X 100
Let us calculate a and X, _
o = VVariance = >/02 = 1.15
^ + 0.59 = 7.59
281
N
217
Applying the formula, we get
Here,
C.V = I X 100
o = 1.15 and X = 7.59 C.V = ^ X 100 = 0.1515 X 100 = 15.15%
Continuous Series
itmuous series
niustration 32. To check the quality of two bulbs and their life in burmng hours was
Life (in hrs.) Ho. of bulbs
Brand ABrands
050 15 2
50100 20 8
100150 18 60
150200 25 25
200250 22 5
Total 100 100
(i) Which brand gives higher life? (it) Which brand is more dependable?
282
Solution.
Statistics for Eco.
nomicsXl
Brand A
Coefficient of Variation (Brand A^
CV = ^ X 100 Let us calculate first a and X. Standard Deviation
MLjlfd"
X c
Here,Z/^ = 193,Z/^19,N=100andC=50
CT =
/m 1100
'19_ 100
X 50
Co^ffidem ofJS^ation~(BrLd BJ
= X 100 Let us calculate first a and X. Standard Deviation
a =
l^jEfd'^
X c
a =
100
.100,
x50
Measures of Dispersion
283
= Vl.93(0.19)2
Vl.930.0361x50
= Vl.8939x50 = 1.376 X 50 = 68.8 hrs.
Arithmetic Mean
X = A + X C
N
Here, A = 125, I^d' = l9,h!= 100 and C = 50
19
= Vo.61(0.23)2
 V0.610.0529x50
>/0.5571x50
0.746 X 50 = 37.32 hrs.
X = 125 +
X 50
Arithmetic Mean
Ifd'
X = A + X C
N
Here, A = 125, Ifd' = 23, 100 and C = 50
23
100
= 125 + (0.19) X 50 = 125 + 9.5 = 134.5 hrs. Applying the formula, now we get
C.V. = ^ X 100
.A.
where, a = 68.8 and X = 134.5 68 8
C.V = X 100 = 51.15%.
134.5
X = 125 +
X 50
100
= 125 + (0.23) X 50 = 125 + 11.5 = 136.5 hrs. Applying the formula, now we get
where.
C.V. = ^ X 100
.A.
a = 37.32 and X = 136.5 37.32
C.V. =
136.5
X 100 = 27.34%.
(/■) Since the average life of bulbs of brand B (136.5 hrs) is greater than that of brand
A (134.5 hrs), therefore the bulbs of brand B give a higher life. (ii) Since C.V. of bulbs of brand B (27.34%) is less than that of brand A (51.15%),
therefore the bulbs of B are more dependable. Illustration 33. The number of employees, wages per employee and the variance of wages per employee for two factories are given below :
No. of Employees
Average wage per employee per day (Rs) Variance of wages per employee per day (Rs)
(a) In which factory is there greater variation in the distribution of wages per employee?
(b) Suppose in factory B, the wages of an employee are wrongly noted as Rs 120 instead of Rs 100. What would be the corrected variance for factory B?
Factory A Factory B
50 100
120 85
9 16
284
Solution.
(«) Calculation of Coefficient of Variation : Factory A
Statistics for EconomicsXI
C.V. = . X 100 Here, x = 120 and a =
= il? 100 = 2.5%
Factory B
C.V. = J X 100
Here X = 8S and a = ^
CV. = ^ X 100 = 4.7%
W &rrecti„8 Mean and Variation :
— zx
For Factory B .
^ = 100 and X= 85
100 x 85 = 8500
It IS not correct ZX
Corrected ZX= 8500  120 . 100 = Rs 8480
Corrected X = ^ = ^ ^
N
Variance = a^
a^ =
Here,
N
100
 (xp
= 16, X = 85 and JV = 100
ZX^
16 =
100
(85)2
T. = ^^00 + 722500 = 724100
It IS not correct ZX^ /^^lUU
Corrected ZX^ = 724100  (120)2 . (100)^
= 724100  14400 + 100000 = 719700 Corrected Variance = ^o^reaed ZX^
 (Corrected X)^
719700 ~ ~Tdr  <84.8)2
bet
= 7197  7191.04 = Rs 5.96
Measures of Dispersion
285
Illustration 34. The sum of 10 values is 100 and the sum of their squares is 1090. Find the coefficient of variation.
Solution. We are given.
N = 10, IX = 100, and XX^ = 1090
Coefficient of variation (C.V.) = ^ x 100
y\.
Apply the following formula to get Mean (X)
ZX
^ = N
10 X = 100
100 10
X =
NX = IX
= 10
Apply the following formula to obtain standard deviation (a) by direct method.
(X)2
02 =
N
1090
 (X)^
 (10)2
10
= 109  100 = 9
= V9 =3
Therefore,
C.V. = — X 100
=  X 100
= 30
Thus, X = 10 and C.V. = 30
Illustration 35. The means and standard deviations of two brands of bulbs are given below:
Brand I Brand II
Mean 800 hours 770 hours
Standard deviation 100 hours 60 hours
Calculate a measure of relative dispersion for two brands and interpret the result.
286
j„ . Statistics for EconomicsXI
Brand I
C.V. = ^ X 100
Given : X = 800 a = 100
100 800
X 100 = 12.5%
Given
Brand II C.V. = ^ X 100
X = 770 a = 60
60 770
X 100 = 7.79 %
Hence, the bnlbs of brand H are more consistem as compared to brand L
4. Mathematical Properties of Standard Deviation
Standard deviation has the following important mathematical properties ■
 11 5
= 3
Here, xX X, i.e., deviations taken from Mean, w/ , " ^ ~ ^^ deviations taken from any value
Measures of Dispersion
287
less than the sum of the squares of deviations calculated from any other value, which is used to calculate standard deviation.
Symbolically,
But the
sum of deviations calculated from Median (ignoring ± signs) is always less than the sum of deviations calculated from mean (ignoring ± signs), which is used to calculate mean deviation.
Symbolically,
ElXMel < ZIXXI
(b) Standard Deviation and Normal Curve : In a normal or symmetrical distribution apart from mean, median and mode are identical, a large proportion of distributions are concentrated around mean. Following are a relationship {i.e., range of spread of items) can be determined on the basis of mean and standard deviation.
Mean ± 1 a
covers 68.27% of the total items.
Mean ± 2 a covers 95.45% of the total items.
Mean ± 3 ct covers 99.73 % of the total items.
This can be observed in the following frequency curve :
PERCENTAGE OF ITEMS INCLUDED UNDER NORMAL CURVE
Illustration 36. Calculate the percentage of cases lying within X + 1 a, X ±1(5, X. ± 3 a from the following data :
Size : 1 2 3 4 5 6 7 8 9 10
Frequency : 8 12 10 28 16 12 10 2 0 2
288
Solution.
Statistics for EconomicsxA
Calculation of Mean and Standard Deviation : Mean : X = A ■ ^^^
Here,
N
A = 5, I.fd =  68 and N = 100 68
X = 5 +
Standard Deviation
100
= 4.32
CT =
Here, ^ 434, Zfd = 68 and N = 100
0 =
/424 100
68>
—00 I____
liwj = V4.24(0.68)2
Calculation (a)
Cases lying ib)
Cases lying (c)
There is no
 y/4.240A624 = VIT^ = 1.943 of percentage of cases :
X ±10 = 4.32 ± 1.94 = 6.26 and 2.38
between 3 and 6 are (10 . 28 . 16 . 12) = 66 out of 100, 66"/o
X ± 2a = 4.32 ± 2 x 1.94 = 4.32 ± 3.88 = 8.20 and 0 44 between 1 and 8 are (100  2) 98 out of 100, i.e., 98%
X ±3o = 4.32 ± 3 X 1.94
= 4.32 ± 5.82 = 1.5 and 10.14 negative value. All the cases lie between 0 to 10, /.e., 100%.
Measures of Dispersion 289
(c) Combined Standard Deviation : Just as combined arithmetic mean can be calculated, if means and number of items in different groups are given, similarly combined standard deviation can be calculated, if standard deviation means and number of items in different groups are given. Combined standard deviation is obtained as follows : {a) Two related groups :
iNiaf + N2O2 + Njd} + Nidj
= ^ N1+N2
Here,
CTjj = Combined standard deviation of two groups
CTj = Standard deviation of first group a^ = Standard deviation of second group
Xj 2 = Combined arithmetic Mean of two groups (b) Three related groups :
'1.2,3
V
NjO^ + N^al + Njof + N^d^ + Nzcff + N^dj
N1+N2+N3
Here, d^= (XX 1,2,3), d^ = (X2 Xi,2.3) and d^ = {X, X,^^,^)
The above formula can be extended to calculate the combined standard deviation of even more groups.
Illustration 37. In sample A, N = 150, X = 120 and S.D. = 20; in sample B, N = 75, X = 126 and S.D. = 22. Calculate Combined Mean and Combined Standard Deviation. Solution. Combined Mean :
V N1X1+N2X2
 N1 + N2
150x120 + 75x126 18000 + 9450 27450
150 + 75
Combined Standard Deviation :
225
225
= 122
Niol + N2al + N^d^ + N2
. N1+N2
d^ = (X,Xi,2) = 120  122 = 2 d^ = (X2Xi,2) = 126  122 = 4
i
150 (20)2 ^ 75(22)2 ^ _ (_2)2 ^ 75(4^2 150 + 75
290
j60odoT363ddT6^^ * 150 + 75
j„ . Statistics for EconomicsXI
[981^ 
~ V~22r " = 20.88
Thus, Combined Mean = :22 and Combined Standard Devation = 20.88
follow^ A™'ISirs.''' """ ained by combming ,he
Distributions
N
A 20
^ 120 C 60
Solution. Combined Arithmetic Mean :
60 50 40
5.D.
8 20 12
H00 + 6000^h2^ _ 9600
200  ^ = 48
Combined Standard Deviation :
20 + 120 + 60
^1.2,3 =
Here,
and
I nY+N^J^
= ~ ^1,2,3) = 60  48 = 12 = (^2 X12 3) = 50  48 = 2 = (X3 X12 3) = 40  48 = 8
=
' 20 + 120^^60
* 200 
[65120 ,_
~ = ^^25.6 = 18.04
Thus, Combined Mean = 48
Combined Standard Deviation = 18.04
291
measures of Dispersion
\d) Change of origin and change of scale : Any constant added or ^"^ttacted (change of origin), then standard deviation of original data and of changed data after addition or Isuhtraction will not change but the mean of new data will change."
Any constant multiplied or divided (change of scale), then mean, standard deviation
and,variance will change of the new changed data.
Illustration 39. Average daily wage of 50 workers of a factory was Rs 200 with standard deviation of Rs 40. Each worker is given a rise of Rs 20. (,) What is the new average daily wage and standard deviation? (ii) Have the wages become more or less uniform?
(Hi) If each worker is given a hike of 10% in wages, how are the Mean and Standard Deviation values affected? Solution. We are given
N = 50, X = 200, a = 40
(i) Change of Origin
Old Series
Since,
X =
IX
N
NX =IX
50 X 200 = 10,000
Mean
X =
N 10,000
50
= Rs 200
Standard Deviation
a =
N
■(X)2
Suppose each worker is paid Rs 200.
X X^ = (200)^ X 50 workers
= 40,000 X 50 = 2,00,0000
a =
'20,00,000
50
= ^40,00040,000
= Ji
Thus, Standard Deviation (a) and variance
(o') = 1. _
New Series
Rise of Rs 20 to each worker to get new series 20 X 50 workers = Rs 1,000 New XX = 10,000 + 1,000 = Rs 11,000
New Mean EX
X =
N 11,000
" 50 Standard Deviation
= Rs 220
a =
N
Each worker given a rise of Rs 20, i.e., 200 + 20 = Rs 220
XX^ = (220)^ X 50 workers
48,400 X 50 = 24,20,000
a =
^4,20,000
50
(220)2
= ^48,40048,400 Thus, Standard Deviation (a) and variance
«y') = 1._
292
« New . new .ancUrd aeva.on .
devanon remain rhe same as rhat ofold series
w^ are required .„ ca.cuia,e eoefficienr „, variation ro decide *e uniform..
Old Scries
Coefficient of Variation
C.V. = ^xlOO
 200 = 0.5%
X 100
New Seri^
Coefficient of Variation
C.V. = xlOO
220 = 0.45%
X 100
orvaSl^^ro^iZCtrtr^   coe^cienr
(ni) If each woricer is given 20% h,ke
Mean affected: Old + Increase in wages ■•• Rs 200 + Rs 20 = Rs 220 New Mean = Rs 220
 ..e. remain .he same as
5 Relation between JMeasures of Dispersion
and Viation, Mean Devation
(a) Q.D.  ^ s.D. (more precisely 0.6745 S.D.)
(b) M.D. _ ^ S.D. (more precisely 0.7979 S.D.)
(c) Q.D. = I m.D. (more precisely 0.8453 m D )
(d) 6 S.D. = 9 Q.D. = 7.5 M.D. Further, in such distributions •
(0 Arithmetic M " ! stnl^TT""''' ^e items.
The above relationships ~ of the items,
to moderately asymmerric'al for c^t^" *»« Vplied
Measures of Dispersion 293
Merits and Demerits and Uses of Standard Deviation
Standard deviation is the most satisfactory and widely used measure of dispersion ■ause of the following merits :
■ Merits
1. Based on every item. Unlike the range and location based measures of dispersion, the standard deviation makes use of all the observations in the set of series. That is, it includes every item of the distribution.
2. Correct mathematical process. The standard deviation is the easiest measure of dispersion to handle algebraically and it is the resuk of correct mathematical process. The deviations are calculated from arithmetic mean which is an ideal average. The deviations are squared, so that automatically become positive. Being used on correct mathematical process, it is amenable to further statistical analysis.
3. Rigidly defined. Standard deviation is a welldefined and definite measure of dispersion. It is rigidly defined and its value is always definite and based on all the observations and the actual signs of deviations are used.
4. Sampling fluctuations. Standard deviation is less affected by the fluctuations of sampling than most other measures of dispersion.
5. Mathematical Properties. It is amenable to algebraic treatment and possesses many mathematical properties. It is the only measure for calculating combined standard deviation of two or more groups. It is on account of the properties that standard deviation is used in many advanced studies.
Demerits
1. Complex in calculation. Standard deviation is not easy to calculate, nor it is easily understood. In many cases it is more cumbersome in its calculation than either quartile deviation or mean deviation.
2. More weights to extreme items. It gives more weight to extreme items and less weight to those which are near to the mean, because the squares of the deviations which are big in size, would be proportionately greater than the squares of the deviations which are comparatively small. Thus, deviation 2 and 8 are ratio of 1 : 4 but their square, i.e., 4 and 64 would be in the ratio of 1 : 16. Howevei; since standard deviation gives greater weight to extreme items, it does not find much favour with economists and businessmen who are more interested in the results of the modal class.
Uses
Despite the drawback the standard deviation is the best measure of dispersion and )uld be used whenever possible. It is widely used in statistics because it possesses most die characteristics of an ideal measure of dispersion, k is a significant measure for aking comparison between variability of two sets of observations to test the significance f various statistical measures of random samples, correlation and regression analysis etc. ' may regard standard deviation as the best and the most powerful measure of lion.
294
(between H0 tn a u u Statistics for EconomicsXl
reutive measures of or variations
Types of Measures of
Dispersion/Variation
Measure iatiori
Range
^InterQuartile Range and Quartile Deviation Mean Deviation Standard Deviation
1
re of Variation/ nation
Coefficient of Range ^ Coefficient of Quartile Deviation ^ Coefficient of Mean Deviation Coefficient of Standard Deviation
(a) Absolute measure • Absol..^^ ^
. the ^mrfda™ ^f ^^ ^^^ as the data,
^ees. If the data are in kg, the measiV^'e nV Jf " ^e in j
d spersxon cannot be used to compare the Matter o ^ absolute
St r —■ 
or the coefficient of the absoluteltaC If d ' "" »» P^tcentZ
coefficent of dispersion or coeST „f « « caS
c«fhcient of range, co^Bc^ro^^^uXtST' asures ar
and coefficient of standard deviation "oo of mean deviation on standard deviation is ca.ed
Thus, C.V.  ®
= f XIOO
measured m the same of var abihty of two or more series wher "uiLts 5 ' " " «  obtained as percenta;«
Measures of Dispersion
295
method (lorenz curve)"
Lorenz Curve
The graphic method of studying dispersion is known as the Lorenz Curve Method. It is named after Dr Max. O. Lorenz who used it for the first time to measure the distribution of weakh and income. Now k is also used for the study of the distribution of profits, wages, turnover etc. In this method of values the frequencies are cumulated and their percentage are calculated. These values are plotted on the graph and curve that is obtained IS called the Lorenz Curve. The greatest defect of this curve is that it does not give a quantitative measure of dispersion. Let us look at the following illustration. Illustration 40. Draw a Lorenz Curve from the following data :
Income (In thousand Rs) No. of Persons in thousands
Group A Group B
20 10 16
40 20 14
60 40 10
100 50 6
180 80 4
Solution.
Income in
(Rs)
lative Income
wm
20 40 60 100 180
_
20 60 120 220 400
Cumulaui/e
Par^Mt
tage sands) hers
_____
No. of Pers6ns (stt thou
5 15 30 55 100
10 20 40 50 80
Steps
1. The size of items (or if classes are given, then midpoints) are made cumulative. Considering last cumulative total as equal to 100 difference cumulative total are converted into percentages.
2. In the same way frequencies are made cumulative. Considering the last cumulative frequency item as equal to 100, all the different cumulative frequencies are converted into percentages.
296
* ! ■
r ■ !.
Percentage of Income
lorenz curve
Curve
5 """ ^T " ^om 0 to 100
ot^^'ltSUl'"" ™ ^ line . as line
Illustration 41. fa ,he foWi„, ab,,™^ " . and . according ro the allt tX" t'^ » 
Profits earned in Rs '000 Area A Area B
6 25 60 84 105 150 170 400 6 11 13 14 15 17 10 14 2 38 52 28 38 26 12 4
Sii inequa
nil
income Afterta No. of,
Afterm
No. of I
Measures of Dispersion 297
Solution. Obtaining Lorenz Curve, we calculate percentage values as under :
Profits (in thousand) Area A Area B
Cumu Cumu No of Cumu Cumu No. of Cumu Cumu
lative lative Com lative lative Com lative lative
Profit Profit Profit panies Num Per panies Num Per.
Percen ber cent ber cent
tage age age
6 6 0.6 6 6 6 2 2 1
25 31 3.1 11 17 17 38 40 20
60 . 91 9.1 13 30 30 52 92 46
84 175 17.5 14 44 44 28 120 60
105 280 28.0 15 59 59 38 158 79
150 430 43.0 17 76 76 26 184 92
170 600 60.0 10 86 86 12 196 98
400 1000 100.0 14 100 100 4 200 100
Since curve B is farthest from the line of equal distribution, it represents greater inequality in area B as compared to area A.
Illustration 42. 9400 Indian households are classified according to their aftertax income as follows :
Aftertax income : 01000 10005000 500010000
No. of households : 1348 4^9 » 1892
Aftertax income : 1000020000 2000040000
No. of households : 1460 490
Drawrh. T . ^t'^ti^tics for EconomicsXI _^alculatioi^^ Values
Aftertax Income
0  1000 1000  5000 5000  10000 10000  20000 20000  40000
Cunmlatii'c Income r> f Cumulative Percentage of Income No. of houses holds Cumulative Number of households Cumulative % of house . holds
Below 1000 Below 5000 Below 10000 Below 20000 Below 40000 IT 12.5 25.0 50.0 100.0 1348 4210 1892 1460 490 1 1348 5558 7450 8910 9400 14.34 59.13 79.25 94.74 100.00
of measures of oispersion
;
i^orenz Curve. We are now m a position to make a
there is more inequahty
299
Measures of Dispersion
comparative study of these measures. It would help us in the selection of an appropriate measure of dispersion which depends on(^) nature of data, (b) the purpose, and [c) object of an investigation.
1. Definite value point of view : All the fourmethods of dispersion are rigidly defined
and their values are definite.
2. Calculation point of view : Range is the easiest and simplest measure because it is the difference between two extreme items. Quartile deviation is superior to the range, as it is not affected too much by the value of extreme items, instead it is calculated from lower and upper quartiles. This is also easy method. However, Mean deviation and Standard deviation requires more calculations which are based on the deviations from average. Lorenz Curve is a visual aid method but it does not give quantitative measure.
3 Based on every item point of view : Range and quartile deviation are not based on
" all the items of series while mean deviation and standard deviation makes use of
all the items. They are based on every item of the distribution. Range is highly affected by the extreme item.
4 Interpretation and application point of view : All the four measures of dispersion are easy to interpret. Range and quartile deviation are useful for general study of variability. Range is useful for quality control, weather forecasting, etc. Quartile deviation is useful when influence of extreme items is minimised as in the study ot social problems. Mean deviation is used by economists and busmess statisticians It is useful in forecasting business cycles and small sample studies. Standard deviation possesses most of the good characteristics of a measure of dispersion Therefore, in sampling and other areas of statistical analysis, it is the most favoured and indispensable measure.
5 Algebraic treatment point of view : Standard deviation is the best measure of dispersion because of correct mathematical processes as compared to range, quartile deviation and mean deviation. It is widely used in statistics, i.e., in making comparison between variability of two or more sets of data, in testing the significance of random samples, in correlation and regression analysis, etc.
Thus, standard deviation satisfies the most essentials of a goqd measure of dispersion.
These essentials are as under :
(i) It should be simple to calculate and easy to understand.
(ii) It should be rigidly defined, i.e., it has precise value.
(iii) It should be based on all the observations of a series.
(iv) It should not be greatly influenced by extreme items.
(v) It should not be affected by fluctuations in sampling.
(vi) It should be usable for statistical calculations for further or higher order analysis.
300
j„ . Statistics for EconomicsXI
of formulae
Range
Absolute Afeasure
Range = L  S
Q^rtile Deviation
Relative Measures
Coefficient of range
LS
L + S
L = largest item S = smallest item
Absolute Measure
Quartile range = Q^ _ q Semiinterquartile range or Quartile Deviation
Q.D. =
2
Relative Measures
Coefficient of Quartile Deviation
_ QlzQl Qs+Qi
Qj = lower quartile Qj = upper quartile
Absolute Measure Individual Observations
M.D. =
N
  N
 ^Ji^
I £> I = Deviations from median
Discrete and Continuous Series
M.D. =
zfjm
N
Shortcut (Assuiiied mean) Mediod
M.D. = lll^hilzAmB^ N
Absolute Measure Individual Observation
Actual Mean Method
a =
N
Direct Method
N ■ ■ y M Assumed Mean Method
I N
a =
' N ' d = XA
.N
Measures of Dispersion
Relative Measure Coefficient of Mean Deviation
Coeff. M.D. =
M.D.
Mean or Median
M.D.
XorMe
R< lative Measure
ent nf Standard Dcv fon
or mean ignoring ± signs
Coeff. S.D. = EX
Relative Measure Coefficient of Variation
a
C.V =  X 100
301
Step Deviation Method
a =
d' =
N
fX~A
{ N
X C
V c
C = Common factor
"CS
Dirt
a =
a
N
IfX N
2
(Xf =
N
' N [ N
Assumed Mean Method
a =
' N
lfd I N
\2
Step Deviation Method
a 
I N [ N J
Variance or a' = ^^XXf ^ N N
X C
a'
^ J^]
N
N
\ i\ J
X e
d  ——, C = Common factor
1 V
t ■■
r ■■
302
Combined Standard Deviation (a) Two related groups :
j„ . Statistics for EconomicsXI
_ Niof + N^cl + + N^dl
Here,.
{b) Three related groups :
, _ + N^ol + Njof + N, Jf + N^dl + N,dl
3
Here, and
d^ = ^3 X^ 2,3
EXERCISES
Questions :
1. Illustrate the meaning of dispersion with examples.
2. What are the essentials of a good measure of dispersion.?
3. {a) Name four commonly used measures of absolute dispersion.
{b) Name the most commonly used measure of relative dispersion. Give formula for calculating it.
4. Why should we measure dispersion.? Do the range and quartile deviation measure dispersion about same value.?
5. A measure of dispersion is a good supplement to the central value in understanding frequency distribution. Comment. ^
6. Which measure of dispersion is the best and how.?
7. Some measures of dispersion depend upon the spread of values whereas some calculate the variation of values from central value.? Do you agree?
8. Define the first and third quartiles. Explain how the quartiles are used to calculate dispersion values.?
9. (a) What do you understand by mean deviation?
qZt'd^LLT"^"""
10. 'Coefficient of variation is a relative measure of dispersion'. Explain
^^ variability? What is the need of calculating a measure
17.
lo.
19.
20. 21.
303
Measures of Dispersion
H2. In what way is standard deviation a better measure of dispersion than mean deviation?
13 What is Standard deviation? Explain the uses of standard deviation.
14. Why is standard deviation considered to be the most popular measure cf dispersion.
Explain. ... . ,
15. What IS coefficient of variation? What purpose does it serve? Also distinguish
between 'variance' and 'coefficient of variation'.
16. Define coefficient of variation? In what situation would you prefer this as a measure
of dispersion.
Make a comparative study of various measures.
"The standard deviation of heights measured m inches will be larger than the standard deviation of heights measured in feet for the same group of individuals. Comment on the validity of the above statement. Otherwise give appropriate explanation of the statement given above.
What is meant by absolute and relative measure of dispersion?
Briefly explain the concept of 'Lorenz Curve'.
Define Lorenz Curve.
22. Write short notes on:
(a) Coefficient of Dispersion (b) Coefficient of Variation
(c) Variance Standard Deviation
(e) Quartile Deviation (f) Lorenz Curve
Problems : Range
1. The daily wages of ten workers are given below. Find out range and its coefficient.
No. of Workers :A B C D E F G H I ] Wages in (Rs) : 175 50 50 55 100 90 125 145 70 60
[Range = 125; Coefficient ot Range = 0.55J
2. Following are the marks obtained by students m Sec. A and Sec. B. Compare the range of marks of students in two sections.
Marks (Section A) : 20 25 28 45 15 30
Marks (Section B) : 45 52 36 42 28 25
[Section A : Range = 30, Coetticiem ot Range = 0.5
Section B : Range = 27, Coefficient of Range = 0.351]
3. Find range and coefficient of range of the following :
(a) Per day earning of seven agricultural labourers in Rs :
60, 72, 36, 85, 35, 52, 72
304
Jan. +1.5 July 0.1
Feb. +2.4 Aug. 0.6
from normal (2002).
Mar. Apr. May
+3.1 1.5 0.4
^ep. Oct. Nov.
15 0.6 1.9
j„ . Statistics for EconomicsXI
June +3.3 Dec. 6.1
4.
[(a) Range = 50, Coeff. of Range L 0 42
No. of Workers :
50 2
70 8
80 12
90
7
100 4
120 3
130 8
150 6
5.
If o .
X f
10 4
15 12
6.
Fir.^ M , i^vduge = ^u; L.oel
Find the range and coefficient of range of the following • Age m years : cm . & •
Frequency :
20 30 40 50
7 3 5 2
[Range = 40; Coefficient of Range = 0.67]
510 10
1015 15
7.
1520 2025
20 5 [Range = 20, Coefficient of Range = 0.671
Frequency :
15 2
610 8
1115 15
8.
1620 2125 2630 20 10 l^ange = 30; Coefficient of Range = 0.97]
:e
The following table gives the hei^h. f ^ ^^ R^^ge = 0.97
Method. ^ ^^ Calculate dispersion by Rang.
Height (in centimetres)
Below 162 Below 163 Below 164 Below 165 Below 166 Below 167 Below 168 Below 169 Below 170
No. of persons
1 8 19 32 45 58 85 93 100
[Range = 9 cm. Coefficient of Range = 0.
03]
'f
■ ^M
Measures of Dispersion 305
Quartile lc\ialinn
9. Calculate Quartile Deviation and its Coefficient of Rajesh's daily income.
Months : 1 2 3 4 5 6 7 8 9 19 11 12
^ Income (Rs) : 239 250 251 251 257 258 260 261 262 262 273 275
[Q.D. = Rs 55, Coeff. of Q.D. = 0.213] 10. Find the Quartile Deviation and its Coefficient from the following data relating to the daily wages of seven workers : . Daily Wages (in Rs) : 50 90 70
40 80 65 60
[Q.D. = Rs 15, Coefficient of Q.D. = 0.23] ^ Find out Quartile Deviation and Coefficient of Quartile Deviation of the following items :
145 130200 210 198
234 159160 178 257
260 300345 360 390
[Q.D. = 70; Coefficient of Q.D. = 0.304]
12. Find out Quartile Deviation, Interquartile Range and Coefficient of Quartile Deviation of the following series :
Height (in inches) : 58 59 60 61 62 63 64 65 66 No. of Persons : 2 3 6 15 10 5 4 3 1
[Q.D. = 1, Interquartile Range = 2, Coefficient of Q.D. = 0.016]
13. Find out Coefficient of Quartile Deviation from the following data :
X : 10 15 20 25 30 35 40 45
f: 6 17 29 38 25 14 9 1
[Coefficient of Q.D. = 0.2]
14. Calculate Quartile and Coefficient of Quartile Deviation of the following data : Marks : 59 1014 1519 2024 2529 3034 3539 Students : 1 3 8 5 4 2 2
[Qj = 15.906, Q^ = 20, Q^ = 26.687 and Coefficient of Q.D. = 0.25]
15. Calculate lower and upper Quartiles, Quartile Deviation and Coefficient of Quartile Deviation of the following series :
Values : 56 67 78 89 910 1011
Frequency : 5 8 12 15 6 2
[Qj = 6.875, Q3 = 8.733, Coefficient of Q.D. = 0.119]
16. Calculate the Semiinterquartile Range and its Coefficient of the following data : Marks : 010 1020 2030 3040 4050 5060 6070 No. of Students : 48 11 15 12 6 3
[Q.D. = 11.55; Coefficient of Q.D. = 0.337]
17. Find out the Q3, Qj, Quartile Deviation and Coefficient of Quartile Deviation in the following :
Age : 20 30 40 50 60 70 80
No. of Members : 3 61 132 154 140 51 2
I [Q3 = 64.08, Q, = 45.43, Q.D. = 9.33, Coefficient of Q.D. = 0.17]
' i
1
306
Statistics for EconomicsXI
68
M.D. = 12.77] 50 50
18. Calculate the InterquartUe Range for the data given below •
Zency : T T T ^^ ^
6 5 4
[I.Q.R. = 12.9]
Mcaii Deviation
Find Coefflcent of Mean dev,a„on f„n, median • ^
P''^ ms) : 25 28 32 32 36 48 44 45
... Ca,cn,a. Mean Oe™..on from mean and med.an of rS^laTa" ^
^^ 52 49 45 72 57 47
Ca.n,a.e Mean Oe™„o„ f.om « ml"; Llf ;  ^ "
= ^ 4 10 9 15 12 7 9 7
.. o„ .e a™.e devia.on from mel^'X f^ Xtn:
n ■ 12 18 24 30 36 42
Frequency : 4 7 9 18 15 jo 5
24. Calculate Mean Deviation from median : = =
No. of tomatoes per plant : 0 1 9 cj 4 ^
No. of plants . y . ^ , , ^ ^ ^ 7 8 9 10
• 2 ^ 7 11 18 24 12 8 6 4 3
Size of Item : 4 ^ „
Frequency =2 4 I ^^ ^^ 16
_ 3 2 14
[M^D from X = 3.32 and Coefficient of M.D. = 0 342
26 Th. • u, "" ^^ " and Coeff,ciem of M D  0 4051
dev,a,L. ""<1 ""dian and coeffiden, mean
s ^^Mrr r ? r
[M.D. from X = 28.56, Coeffidem of M D = 0 228 M.D. from Me = 28, Coefficient of M.D. = 0.233]
Measures of Dispersion ^^^
127 Calculate Mean Deviation from mean of the following data :
Class : 34 45 56 67 78 89 910 Frequency : 37 22 60 85 32 8
[M.D. = 0.915]
28. Calculate Mean and Mean Deviation and coefficient of M.D. for the following distribution :
Weekly tvages : 2040 4060 6080 80100
Workers : 20 40 30 10
[Mean = 56, M.D. = 15.2 and Coefficient of M.D. = 0.27]
29. Age distribution of hundred life insurance policy holders is as follows :
mlrTstVtrthday : 1719 2025 2635 3640 4150 5155 5660 6170 Number : 9 16 12 26 14 12^ _ ^
Calculate Mean deviation from the median age. [M.D. = 10.73]
30. Find Mean Deviation from median of the marks secured by 100 students in a classtest as given below:
Marks : 6063 6366 6669 6972 7275
No. of students : 5 18 42 27 8
[M.D. = 2.26]
31. Using Mean Deviation from median of the income group of 5 and 7 members given below, compare which of the group has more variability?
Group A : Group B :
4000 3000
4200 4400 4600 4800 4000 4200 4400 4600 4800 5800
[Group A : M.D. = 1240, Coeff M.D. = 0.054 Group B : M.D. = 571.4, Coeff. M.D. = 0.13 Group B has greater variation]
Standard Deviation
32. Calculate the Standard Deviation of wage earner's daily earnings :
Week
Earnings (in Rs)
1
54
2 62
3
63
4
65
5 68
6
71
7 8 9 10
73 78 82 84 [X =70, S.D. = 9.01]
33. Calculate Standard Deviation of the following two series. Which series has more variability:
A : 58 59 60 65 66 B : 56 87 89 46 93
52 75 31 46 48 65 44 54 78 68 [A: X = 56, S.D. = 11.7, C.V. = 20.89% B: X = 68, S.D. = 17.1, C.V = 25.14% Series B has more variability]
308
WaSiSM
wem mtM
j„ . Statistics for EconomicsXI
Income (Rs) ; iqo
120 ISO ^ 140 120 150
[X = 133.7S, S.D. = 22.88]
35.
36.
: (B) Following are given X variable •
55, 49, 67, 89, 44, 59, 57 (a) Calculate
(«■) the arithmetic mean («) the standard deviation («/') the mean deviation Also calculate 0) Z(X  55)^ («) 2 IX  Median! (c) Examine, if
(«)ZXX>z IXMedian!
No. of families
166 552 580 433 268 148
7 8
37.
9 10 11 12 77 41 20 8 6 1
Calculate Mean and Standard Deviation fron. . n ^^ " = 176]
Marks (Above) ■ Q jq the followmg data :
No. : 150 HO 100 80 80 70 30 ^
Calculate Mean and vana.ce f.r the ~
2
X
2 5
4 16
5 21
6 18
7 13
8 10
9 4
38.
10
3
[X = 5.5; a^ = 3.99]
2530 16
3035 8
39.
WKMM wm&m
Ml
35^0 3
"^^^o Vanance ..
Frequency 2 , ^^25
■ 7 13 21
Calculate the Coefficient of V^w r . ^^ " = 7.95, 02 ^
200 workers in a flc.™ lowing distribution of the wagt 2
^OT : 4049 en
r T
fX= 74.45,S.D. = ,5.921.CV...2,.38%J
4:
43
44.
Measures of Dispersion
309
40. The following are the scores made by two batsmen A and B in a series of innings:
: 12 115 6 73 7 19 119 36 84 29 % \; B : 47 12 76 42 4 51 37 48 13 0 ; Who is better as a rungetter? Who is more consistent?
[A : X = 50, S.D. = 41.83, C.V. = 83.66% ■B : X = 33, S.D. = 23.37, C.V. = 70.82%]
(A is better as rungetter and B is more consistent,)
41. The index number of prices of cotton and coal shares in 1998 were as under:
Month : Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec. Index Number of Prices :
Cotton : 188 178 173 164 172 183 184 185 211 217 232 240 Coal : 131 130 130 129 129 129 127 127 130 137 140 142 Which of these two shares do you consider more variable in prices :
[Cotton : X = 193.9, S.D. = 23.80, C.V. = 12.27%
Coal : X = 131.75, S.D. = 4.815, C.V. = 3.65% Cotton shares are more variable in prices]
42. Calculate the arithmetic mean and standard deviation and variancefrom the following distribution :
Class : 05 510 1015 1520 2025 2530 3035 3540
Frequency : 2 5 7 13 21 16 8 3
[X = 21.9, S.D. = 7.99, a^ = 63.97]
43. Calculate arithmetic mean and standard deviation and variance from the foSbwing series:
Marks : 7080 6070 5060 4050 3040 2030
No. of Students : 7 11 22 0 15 5
[X = 51.67, S.D. = 15.13, a^ = 228.92] ,44. The following tables gives the age distribution of students in a school in 2001 and 2002. Calculate Coefficient of Variation for both the groups.
Age :. 17 18 19 20 21 22 23 24 25
2001 : 1 3 8 12 14 14 5 3 2
2002 : 6 22 34 40 32 20 16 9 3
[2001 : X = 21.5, S.D. = 1.703, C.V = 7.92%
2002 : X = 20.91, S.D. = 1.846, C.V. = 8.82%]
You are given the following data about height of boys and girls : •
Boys Girls
72 , 38
Number
Average height (in inches) Variance of distribution (in inches)
■tl
68 9
61 4
310
46.
j„ . Statistics for EconomicsXI
(a) Calculate Coefficient of Variation.
(b) Decide whose height is more variable.
[{a) Boys : C.V. = 4.41%, Girls • C V  1 77°/1 r i
Ufe No of Years : 02 24 46 68 810
No of Refrigerators Model A Model B
1012
^ 16 13 7 s 4
of Wh,chide, has
[Model A : X =5.12, C.V. = 54 9% Model B : X = 6.16, C.V = 36.2%
Neck circumference ■. 12 0 12'^ i^n
(in inches) ^^.O 14.5 15.0 15.5 16.0
No of Students : 5 20 30 43 60 56 37 16 .
crkenon Mean .3 standard deviation. ' ^^^ ^^e
[X = 14.01", a = 0.87". The largest size of collar = 16.62".
48 T, u A r smallest size of collar = 11 4"1
I ^^^ Its : ^
Life ( 000 miles) : 2025 2530 3035 35^0
orand X o c
» 15 12 IS
Brand Y 6 in
20 32 30
Which brand has a greater variation.?
[C.V. (X) = 21.82%; C.V. (Y) = 16 101°/ y U
.o. the data he.ow state lih se'r.s'.rcon'.^l'^
4045 13 12
4550 9 0
Variable Series A Series B
1020 2030 30^0 4050 5060 6070 1832 40 22 18
22 40_ 32 18 10
[Series A : _X = 42.14, S.D. . 14.06,.C.V = 33.36%
50 Th. ( u ■ u, B : X = 37.86, S.D. = 14.06, C.V = 3714%1
50. The following table gives the distribution of wages m the two branches of a falry^
: : — 300350
" 93 157 105 82 500
Measures of Dispersion 311
gr Find the mean and standard deviation for the two branches for the wages separately. F [a) Which branch pays higher average wages?
(b) Which branch has greater variability in wages in relation to the average wages?
(c) What is the average monthly wage for the factory as a whole?
id) What is the variance of wages of all the workers in the two branches—A and B taken together?
[Branch A : Mean = Rs 225, S.D. = Rs 66.20, C.V. = 29.42% Branch B : Mean = Rs 230, S.D. = Rs 62.15, C.V. = 27.02% {a) Branch B pays higher average monthly wages.
(b) Branch A has greater variability.
(c) Combined Mean = Rs 226.67 (d) Combined Variance = Rs 4215]
51. What percentage of frequencies are there in the range of X ± 3 S.D.
X : 60.5 70.5 80.5 90.5 100.5 110.5 120.5 130.5 140.5 f ■ 3 21 78 . 182 305 209 81 21 5
[X = 100.95, S.D. = 13; Range = 99.12%]
52. Goals scored by two teams A and B in football matches were as follows : No. of Goals in a match : 0 1 2 3 4 No. of matches : A : 17 9 8 5 4
B : 17 9 6 5 3
Find the team which is more consistent in its performance.
[Coeff. of variation Team A = 123.6% Coeff. of Variation Team B = 109.0% Thus, Team B is more consistent]
53. Find mean and the standard deviation of the following two groups taken together:
Group Mumb^ : ^. ; Mean : , ^ . SOX :  
A 113 159 22.4
B 121 149 20.0
54.
[Combined .: X^ z = 153.83, a^ , = 21.8]
The number examined, the mean weight and standard deviation in each group of examination by two medical examiners are given below. Calculate mean and standard deviation of both the groups taken together.
a
iW
Medical Exami?ier iiilS^^ MiMm Examined IS^^WMlilli ' y Mean : ^ Weight Standard Deviation
A B 50 60 113 1206.5 8.2
[Combined : X^ ^ = 116;82, Oj , = 8.25]
312
^ The following data gives arithmef ^^ Subsrauh \r.: r77~~~r~——B'^up.
No. of Men
A B
C
{in Rs)
50 100 120
61.0 70.0 80.5
Standard Deviation (in Rs)
8.0 9.0 10.0
c^ A fCombined : X, , ,  7:5 „
56. A sample of 35 value, h.c o ' ~ ' 2.3 = 11.9]
<=0. a group of 50 male worths .he „ '' """ ' =
wages are Rs 63 and Rs 9 respSively F?" of their weeidy
R^ 54 and Rs 6 tespeaively Fi"d If" "0 female workers, th^fare
group of 90 workers. ' '•"> ^^dard deviation for a comLed
Coefficient ofvariation of two series are 5S«/ ' = ",, = 91
and What are their ml, TJ" ""'"'on
f the coefficient of variation of X series is J4 6»/ 'f f" * = 22.608]
nteans are fOf.a and respe':,!:^^^.^ l^dTZ:^
No. Of Persons COOO) ^^ 60 iqO i^q
Class A
Class B ]' 20 40 50 80
^  li It ll  « « .5 85 „ ' ^   « 40
61.
xi/Sty^PC
58.
59.
60.
 [rniuvaiues) No. of students (Eco.) No. of students (Statis.)
Chapter 11
MEASURES OF CORRELATION
T. 2.
3.
4.
5.
6.
Introduction
Correlation and Causation Kinds of Correlation Degree of Correlation Methods of Studying Correlation Scatter Diagram
Karl Pearson's Coefficient of Correlation (£) Spearman's Rank Correlation List of Formulae
iduction
In the previous chapters we have discussed measures of central tendency (Mean, Median and Mode), partitional values (Quartiles) and measures of dispersion (Range,' Quartde Deviation, Mean Deviation, Standard Deviation and Lorenz Curve. These are all relating to the description and analysis of single variable only This type of statistical analysis is called 'univariate analysis'. Now, we will deal with problems involving association in two variables. We find that in social as well as natural sciences, where more than one interdependent variables are involved, change in one variable brings change in others. For instance, in Biology we know that weight of a person increases with height in Geometry we know the circumference of a circle depends on the radius, in Economics prices vary with supply, cost of industrial production varies with the cost of raw materialsagricultural production depends on the rainfall etc. The relationship between variables is measured by correlation analysis. Thus, 'the term correlation (or covariation) indicates the . relationship between two such variables in which change in the values of one variable, the values of the other variable also change.' This statistical analysis of such data is called bivariate analysis
Other Definitions
According to Croxton and Cowden, "When the relationship is of d quantitative nature, the appropriate statistical tool for developing and measuring the relationship and expressing it in a brief formula is known correlation."
According to L.R. Connor, "If two or more quantities vary in sympathy so that movements in one tend to be accompanied by corresponding movements in other(s) then they are said to be correlated."
t
314
H.
^relation and causation
j„ . Statistics for EconomicsXI
1. Cause and effect : There is a cause and effect relationship between two variables shon w,ves and many .h„„ starred husbands may havt K' ^ r;
Measures of Correlation 315
be correlation between price and demand so that in general whenever there is an increase in price the demand falls, and viceversa. But this does not mean that whenever there is a rise in price the demand must fall. It is possible that with the rise in price the demand may also go up. This is on account of the fact that in economic and social sciences various factors affect the data simultaneously and it is difficult almost impossible to study the effects of these factors separately. Thus, correlation measures covariation, not causation. It measures the direction and intensity of relationship among variables.
s^ds of correlation
On the basis of nature of relationship between the variables correlation may be :
(1) Positive and Negative Correlation
(2) .Linear and Curvilinear Correlation
(3) Simple, Multiple and Partial Correlation
1. Positive and Negative Correlation
When both the variables change in one direction, that is when both increase or decrease the relationship between the two variables is called positive or direct. But when the change is in opposite directions that is one is increasing and the other is decreasing, the correlation is negative or inverse. For determining the direction of change average values are taken. For example :
(I) Positive Correlation (il) Negative Correlation
(a) . (b) (a) (b)
Both variables Both variables One variable One variable
increasing decreasing increasing, the decreasing, the
other decreasing other increasing
X Y X Y X Y X Y
10 100 70 . 147 15 125 75 110
20 150 60 140 30 110 60 180
30 160 40 135 35 90 40 190
40 190 30 130 40 80 30 200
50 200 15 120 45 75 20 240
60 255 10 90 50 60 10 250
We find that in I (a) the values of X series are increasing so also of the Y series. In I (b) values of X and Y are decreasing. Thus, they are both instance and positive correlation. On the other hand, in II {a) the values of X are increasing and the values of Y are decreasing, similarly in II (b) the values of X are decreasing and the values of Y are increasing. Thus, hey are both examples of negative correlation.
316
Examples : (Positive Correlation) '^"""""'csXI
1 Age of husband and age of wife.
easeinheatlllT^^^nofr,.
1 Demand of a commodity mav ^
Increase in the number tfTe ^ "": " 3 Sale of woollen garments ant ly ir ""
Yield of crops and Price. " Correlation
u the ratio of chance h^n
of^rrotstr ear
line. If tbTv:ri:trfmS™=™™bles. Their^SaSnshbt'f "P™ """ncy
"f nonhnear "ttnTw are ^'phed, rhe
bear a constant rari^ ('""'''near), the amount of chale 3. Staple, Multiple and Partial Correlation
relationship betwln • T "^"Me or p '' ^^rables are
IS a study of relaf,V,„,l. 7 ""her variables from ,1. u , " ™riables
influencing valbles b '' variables Sfllcit ^ '
of tainfalf""stant. For example, 3;,':'' "f other
correlation. " a certain consrant^Jm::
M
Measures of Correlation
317
of correlation
The relationship between two values can be determined by the quantitative value of coefficient of correlation which is obtained by calculations.
Perfect Correlation : Perfect correlation is that where changes in two related variables are exactly proportional. If equal proportional changes are in the same direction, there is perfect positive correlation betWeen the two values described as +1; and if equal proportional changes are in the reverse direction, there is perfect negative correlation, described as  1. For example, the circumference of a circle increases in the equal proportionate ratio with the increase in the equal proportionate ratio in the length of its diameter; the amount of electricity bill increase in a perfectly definite ratio with an increase in the number of unit consumed, the volume of a gas varies inversely with the pressure at constant temperature etc.
DEGREE OF CORRELATION
ive
Zero Correlation : The value of the coefficient of correlation may be zero. It means that there is zero correlation. It does not mean the absence of any type of relation between the two variables. Two valued are uncorrelated. However; other type of relation may be there. There is no linear relationship between them.
Limited Degree of Correlation : In social science, the variables may be correlated, but an increase in one variable need not always be accompanied by a corresponding or equal increase (or decrease) in the other variable. Correlation is said to be limited positive when there are unequal changes in the two variables in the same direction; and correlation is limited negative when there are unequal changes in the reverse direction. The limited degree of combination can be high (between ± .75 to 1); moderate (± .25 to .75) or low
i j Ii
318
(between H0 tn a u u Statistics for EconomicsXl
Karl Pearson's foLt '"P""'^"'^ degree of eorrelarion according ,o
Dejpee of Correlation
Perfect Correlation
Very high degree of Correlation
Sufficiently high degree of Correlation
Moderate degree of Correlation
Only the possibility of a Correlation
Possibly no Correlation
Zero Correlation (Uncorrelated)
Positive
+ 1
+ 9 or more from + .75 to + .9 from + .6 to +.75 from + .3 ro +.6 Less than +.3 0
Negative
 1
 .9 or more from  .75 to  .9 from  .6 to .75 from  .3 to .6 Less than .3 0
IhhoD;
"on. Somell^ are :
S W Coefficient of Correlation
{c) Spearman's Rank Correlation
scatter diagram
tdea about the presence of
measuring Xvariable on bokolTL^^apb paper. The chart is prepared by pomt for each pair of observation oTx and y JZTi u'''' ^^ P^^^ 
plotted m the shape of points. The cluster of ooin^ U ^^^^^ ^ata are
the scatter diagram. When the plottedTo „te^ "" P^P^^ called
we know that there is some correlation Seen tl ^^^dupward or downwardthe correlat^n is positive, when it ir^ov^nw^d ^  "Pward
 scatter diagrams given below : ^"rreiation is negative. Let us study
r=+l
Perfect Positive Correlation
(a)
High Degree ot Positive Correlation (b)
Low Degree of Positive Correlation (0
Measures of Correlation
Perfect Negative Correlation (d)
319
High Dgree of Negative Correlation

Low Degree of Negative Correlation
I w
V! i
r= 0
No Conelation (9)
Fig. 1
Figure [a), (b) and (c) show an upward trend—they show positive correlation. Figure (d), (e) and show a downward trend—they show negative correlat!on. Howe\ er, there are differences among (a), (b) and (c) and similar differences among {d), (e) and (/).
We find from the plottings on the scatter diagrams that there is a certain similarity among (a) and {d), (b) and (e) and (c) and (/). In (a) and (d) the plotted points are almost in a straight lines—this indicates perfect correlation. In [b) and (e) the plotted points are not in a straight line but if we draw a straight line in the middle of their points (regression line) we will find, the points are near about the line. This kind of scatter diagram shows high degree correlation. In (c) and (f) if we draw a similar line (regression line), we will find that the plotted points are very much scattered around the line—^not as near as in the case of {b) and (e). This kind of scattered diagram shows low degree correlation. Finally, diagram {g) shows such a vast scatter of points that it is impossible to see any trend— this shows no correlation or zero correlation.
Illustration 1. From the following pairs of value of variables X and Y draw a scatter diagram and interpret the result.
8 9 10 11 12 13 14 15 54 48 42 36 30 24 18 12
5
72
6 66
7 60
X : 4 Y : 78 Solution.
We note that X = 4 and Y = 78 as given first X and Y values. We may plot this as point (X, Y) on graph paper, where X = 4 and Y = 78. We measure 4 on Xaxis and 78
lik
,1.
j„ . Statistics for EconomicsXI
scatter diagram
Scale : 0.5 cm = 2 on Xaxis
B
64 56
48 40 32 24 16 8 0
320
coordinates of
measure 5 along the x'axis and 72 alongT axis and so on for all d,e given X and y Xl
from the above scatter diagram we can decide Aat the variables X an" Y "e corre ated. The points take the shape of li^e
hen tC r 'r "" Weei X aid
Rate of Change
It is slope of the straight line rwhirh depends on an angle that the str^lghT Hnt
makes with the Xaxis and is equal to j^Zj
rate of change
showing almost equal change
t » .__ —
— i • — — 
1
• 
4. U —1
0 2 4 6 8
10 12 14 16 IS
showing more than proportionate
an<
change
I'fl::
m
in
a (i
Measures of Correlation
321
showing less than proportionate change
showing no change
nonlinear relationship
Fig. 3
We know when the plotted points show some upward trend, the correlation is positive and when there is downward trend, the correlation is negative.
(/•) If the straight line makes an angle of 45° with the Xaxis, the change is exactly in the same proportion as the change in the value of X [Fig. {a) and (b)].
Hi) If the angle that the straight line makes with the Xaxis is greater than 45° the
change in the value Y is more than proportionate to the change in the value of X [Fig. (c) and (d)].
(iii) If the angle that the straight line makes with Xaxis is less than 45", the change in value Y is less than proportionate to the change in the value X [Fig. (e) and (/)]
(w) If there is no angle and it is a straight line parallel to Xaxis, it shows that value Y does not change at all [Fig.
(v) Linear correlation exists when the ratio of change between two variables is uniform
The relationship is described by the straight line. In case of nonlinear relationship
(curvilinear) the amount of change in one variable does not bear a constant ratio
to the amount of change in the other variable. Such relationship will form a curve on graph [Fig. (h)].
322
Ir
!
m
» f
Statistics for EconomicsXI
Merits and Demerits of Scatter Diagram Merits :
1. It is very easy to draw a scatter diagram.
5. fa case of linear relationship between x lni y lT T ^ donate change in th^ .a,„e t Jcha^^r,: t'^nTT" "
t?™ r^atX™ — ■'"own ,„ n„„er,ca,
Whether YLTes XorTcau^ —^^ ^oes not tell,
^hen ,t . not possible to draw a scatter
pearson's coefficient of correlation
X fgr^BS variable
(18671936).ItisthemostLerused me^^^^^^^^^ and statistician Karl Pearson correlation of coefficient. This is atrcaHedX^^^^
represented by r. It is based on arithmel^ a„d f't'!, ^^ ^
of Correlation (r) of two variables « ob aiZ l jfjS^^T, ^^ffftent
corresponding deviations of the various 7eZ of the products of the
by the product of their standard devZ^Zan JZ ^T
Symbolically, <^evtattons and the number of pairs of observations.
r =
Ixy
Here,
Nxaxxay
x=(X X); y={Y Y)
<yx = Standard deviation of X Series ie
' ' " \ N
Measures of Correlation
ay = Standard deviation of Y Series, i.e.
N = Number of pairs of observations r = Coefficient of correlation The above formula can be rewritten as under :
Txy
323
N
r =
N.ax.ay
The above formula is based on the study of covariance between two series. The covariance between two series is written as follows :
N N
r =
Exy 1 1
—y. — X — N
Zxy,
ox oy 1
^y X 1 )c 1 = ^y
" if " if
r =
Ixy
yjlx^xZy^
or
I(XX).(YY)
^Jiixxfylmyf
Applying the Karl Pearson's formula Coefficient of Correlation is calculated by following methods :
(a) Actual Mean Method
(b) Direct Method
(c) Assumed Mean Method
(d) Step Deviation Method
(a) Actual Mean Method
Illustration 2. Calculate Product moment of correlation from the following data and interpret the result.
Serial No. of Students : 1 2 3 4 5 6 7 8 9 10
Marks in Mathematics : 15 18 21 24 27 30 36 39 42 48
Marks in Statistics : 25 25 27 27 31 33 35 41 41 45
Solution. Karl Person's coefficient of correlation is also called Product Moment of Correlation.
I
324
II
t
n ^ > 
:
i*'
Statistics for EconomicsXI Cdcula^on of Coefficient of Correlation
15 18 21 24 27 30 36 39 42 48
ZX = 300
15 12 9 6 3 0 +6 +9 +12 +18
225 25
144 25
81 27
36 27
9 31
0 33
36 35
81 41
144 41
324 45
= 1080 2Y = 330
Steps :
1. Calculate arithmetic means of X and Y series 7. Apply the following formula :
r =
Ixy
Here, ^ = (X  X) and y = (Y _ y)
Let us calculate arithmetic means of X and Y series :
300
X =
Now we get.
Y =
r =
N ZX
10
330 N
Ixy
= 30
= 33
Measures of Correlation
Here, Zxy = 708, Zx^ =1080 and ^ 439
708 708 708
325
r =
= 0.98
>/l080x480 >/518400 720 Hence, there is high degree of positive correlation.
Illustration 3. Calculate Karl Pearson's coefficient of correlation between birth rate and death rate from the following data :
Year Birth rate Death rate
1931 24 15
1941 26 20
1951 32 22
1961 . 33 24
1971 35 27
1981 30 24
Solution.
Calculation of Coefficient of Correlation
Actual Mean Method :
EX
X =
180
N
30 ; Y =
Applying formula, we get
Ixy
EY _ m N " 6
= 22
r =
Here,
yJlx^xLy^ Ixy = 81, Zx^ = 90 and = 86 81 81 81
r =
V90x86 V7740 87.9772 Hence, there is high degree of positive correlation.
= 0.920
Birth Death XX Y y
rate rate
X ■ y xy
24 15 6 36 7 \ 49 42
26 20 4 16 2 4. 8
32 22 +2 4 b 0 0
33 24 +3 9 +2 . 4 6
35 27 +5 25 +5 '25 25
30 24 0 0 +2 4 0
EX = 180 EY = 132 Ex^ = 0 Ex = 90 Ey = 0 If = 86 Exy = 81
Pi'
ti \vi
I;,
326
Statistics for EconomicsXI
X data compute product moment correlation between
No of items Arithmetic Mean Square of deviation from Mean
X series 15 25 136
Y series 15 18 138
c 138
Summation of product of deviations of X v o r means = 122. "cviauons ot A and Y series from their respective
Solution. Regarding deviations of the values in X anH v t . means, we are given the following Morr^^^^ ""
= 136, ^ 138^ ^^ ^ ^^2
Applying formula
Now, we get
r =
r =
Ixy
122
122
122
= 0.891
>^36x138 V18768 136.996 Hence, there is high degree positive correlation between X and Y
their arithmetic means be 420. Ld the coSf^fl^^^^^^^^^ IXe^^S
Solution. Given N = 50, cx = 4.5, ay = 3.5 and Zxy = 420 Applying formula.
r =
Ixy
Now, we get
N X ax X ay
r =
420
420
= 0.533
50x4.5x3.5 787.5 ^rl^'^Z ^ cl Y.
^^ation 6. If the covariance between X and Y variables is3
X and Y are respectively 13 8 and 16 4 Vir.A .u t +12.3 and variances of
between them. " ^he Karl Pearson's coefficient of correlation
Solution. We are given.
Covariance of X and Y = ^  l ? ^ Variance of X (ax^) = 13.8
= y/l3:s = 3.71
Measures of Correlation
Variance of Y (a/) = 16.4
oy = Vl^ = 4.05
327
Applying formula, Now, we get
r =
N.ax.ay
Exy 1 1
—X — x — N ax ay
r = 12.3 X —X ^
3.71 4.05 = 12.3 X 0.27 X 0.25 = 0.83 Hence, there is high degree of positive correlation between X and Y. Illustration 7. Find the standard deviation of X series if coefficient of correlation between two series X and Y is = 0.28 and their covariance is 7.6 and variance of Y series is 81.90.
Solution. Given, coefficient of correlation (r) = 0.28
Covariance of X and Y =
Lxy
= 7.6
Variance of Y (af) = 81.90
ay = V81.90 = 9.05
Applying formula.
Now, we get
r =
lay
N.ax.ay
2jcy 1 1 x — x —
0.28 = 7.6 X
x
N ax ay 1
or
ax 9.05
0.28 X 9.05 ax = 7.6 or 2.534 ax = 7.6 7.6
ax =
2.534
= 2.99 approx 3
Therefore, variance of X (ax^) = (3)^ = 9
Illustration 8. Calculate the number of items for which r = + 0.8, Ixy = 200, standard deviation of Y = 5; and Ix^ = 100, where x and y denotes deviation of items from actual mean.
Solution. Applying formula.
r =
Ixy
yjlx^xly^
or 0.8 =
200
Now, we get
^O.sr = or 0.64 =
100 xZy
or 0.64 X 100 x = 40000 64 Ey^ = 40000
yJlOOxZy^
40000 100 xZy^
i
^Iv
i > t
f
Ii ;;
328
Statistics for EconomicsXI
V 2 40000 64 =625
Now, = or 5 = 1625
V N V N
or /CX2 625 (5)^ = ^ or 25 = 625 N
or 25 N = 625 'N = 625
Hence, number of items is 25. 25 ~
= 25
(b) Direct Method
J:XYN
r =
i n ,
ll In
N.
In j
n
nJ
• EXY
= p Kf
/ix
* N V N
nixy~zxxy
y/NEX^ iZXf Xy/NZY^ {ZYf
• ^ "   fr— the following
where.
r =
X =
lix' N {Xfx
ZX N , Y = ZY N
N
~(Y)
v\2
r =
X.
1XYN.{X).{Y}
X ^V X—
r =
Measures of Correlation 329
Illustration 9. The data of price and quantity purchased relating to a commodity for 5 months are given below. Calculate the product moment correlation (Karl Pearson's coefficient of correlation) between price and quantity and comment on its sign and magnitude.
Months Price (in Rs) Quantity (in kg)
1 10 5
2 10 6
3 11
4
4
12 3
Solution. Calculation of coefficient of correlation.
• 10 10 11 12 12
2:x = 55
5
6 4 3 2
ZY= 20
100 100 121 144 144
ZX' = 609
25 36 16 9 4
lY' = 90
Steps :
1. Calculate arithmetic means of X and Y series.
2. Square the values of X series and obtain the total, i.e., EX^
3. Square the values of Y series and obtain the total, i.e., EY^
4. Multiply X and Y values and find out the total, i.e., ZXY.
5. Apply the following formula :
5 12 2
xy
50 60 44 36 24
EXY = 214
r =
IXYN.X.Y
VeX^  N(Xf x ^zy^  N(Yf
Let us calculate arithmetic means of X and Y series
X =
EX 55
EY 20
N 5 ~ ^^ ' ^  N 5
Here, ZXY = 214, EX^ = 609, ZY^ = 90, X = 11, Y = 4 and N = 5 Now, we get
2145x11x4
= 4
r =
^6095(11)^ xV905(4)2 2145x44
Ii ■ [i 1
330
214220 V609605x>/9080 6 6
6
Statistics for EconomicsXI
= 0.949
>/4xV10 2x3.162 6.324
Hence, there is high degree of negative correlatiori between price and quantity purchased relating to a commodity of 5 months.
In other words, purchase (demand) decreased due to increase in the price of commodity.
Illustration 10. Draw a scatter diagram and calculate Karl Pearson's coefficient of correlation between X and Y. Interprete the result and comment on their relationship. X : 1 3 4 5 ■ 7
8
: 2 6 8 10 . 14 16
Solution.
^
¥
%
u
scatter diagram
Scale: 1 cm = 1 on Xaxis 1 cm = 2 on Yaxis
 Q
lo
v ■SA
1
in
1V Q
(V) U —A
At
2
u ) ■ 1 J. t ( 1 7 ! <
fx,
Fig. 4
From the above scatter uiagram we can decide that variables X and Y are correlated. The points take the shape of line, and it goes up from left bottom to right top then there IS perfect positive correlation between X and Y.
fon
equ in s con negt
Measures of Correlation 353
331
X y X' XY'
1 2 1 4 2
3 6 9 36 18
4 8 16 64 32
5 10 25 100 50
7 14 49 196 98
8 16 64 256 128
X = 28 ZY = 56 IX' = 164 ZY' = 656 ZXY = 328
Applying formula, we get
r =
N
V
N N
Here, IXY = 328, 2X = 28, lY = 56, IX' = 164, IT = 656 and N = 6
r =
6
(28)^ r
rrf
1164 56
{56r
328261.33
V164 130.666 X V656  522.666
66.67 V33.334 X VI 33.334
66.67 66.67
= +1
5.774x11.547 66.67 r = +1
There is perfect positive correlation by scatter diagram and even by Karl Peaison^ formula, resulting to r = +1.
We observe from the illustration the changes in tw^o values X and Y are exactiv in equal proportion. Y values are exactly double than the corresponding values of X movuig in same direction (upward). In such situation, correlation results to perfect positiJe correlation. If equal proportional changes are in the reverse direction, there is perfect negative correlation (r = 1).
f/ '
fiPl
IN
332
fflcient of c„„e,ado„
Statistics for EconomicsXI I
on their relationship.
between X and Y and
X Y : Solution.
comment
3 9
2 4
1 1
2 4
3 9
3 2 1 1 2 3
IX = 0
l^jgilftjon of Coeffident of Correlation
9 4 1 1 4 9
xy = 28
9 4 1 1 4 9
= 28
81 16 1 1 16 81
ly^ = 196
XY
27 8 1 1 8 27
Zxy= 0
r =
IXY
N
■2 (ZYf
fzy^
N
= 0
yf^x^/65.334 = 0 5.291x8.083"
rhey
Measures of Correlation 333
(c) Assumed Mean Method
When actual mean is not a whole number; but a fraction or the series is large, the calculation by actual mean method and direct method will involve a lot of calculations and time. To avoid such tedious calculations, we can use the assumed meat, method. Correlation coefficient can be obtained by the following formula.
Idxdy
r =
Ux.Zdy N_
W
(Zdyf
N
Illustration 12. Calculate Karl Pearson's coefficient of correlation of the following data of height of fathers in inches (X) and their sons (Y). Interpret the result.
Height of fathers (in inches) : 65 66 57 67 68 69 70 72 Height of sons (in inches) : 67 56 65 68 72 72 69 71 Solution.
Calculation of Coefficient of Correlation
65
66 57
67
68
69
70 72
3 2 11 1 0 +1 +2 +4
Zdx = 10
9 4 121 1 0 1 4 16
Z^^ = 156
67 56 65
68 72 72 69 71
+2 9 0 +3 +7 +7 +4 +6
ldy = 20
4 81 0 9 49 49 16 36
Zdy" = 244
6 +18 0 3 0 +7 +8 +24
Idxdy = 48
Steps :
1. Calculate the deviations of X series from an assumed mean (68) and denote them
by dx and find out the total, i.e., ILdx. 1. Calculate the deviations of Y series from an assumed mean (65) and denote them by dy and find out the total, i.e., Uy.
3. Square the deviations of X series and obtain the total, i.e., Zdx^.
4. Square the deviations of Y series and obtain the total, i.e., I^y^.
5. Multiply t/x and f/y and find out the total,/.e;, Ikixiiy.
6. Applying the following formula, we get
334
Statistics for EconomicsXI
Here, Zdxdy = 48, ZJx = 10, I^y = 20, N = 8, W = 156, ZJ^a ^ ,44 Now, we get r = j
8
48 + 25 >/i5612.5XV24450
73 73
73
11.97x13.92 ~ U^ ""
the™: " —l>etwee„ height of fathers and
We can simplify the above calculations by using log tables : Taking Logarithms
73
Hence,
VM3.5xVm log r . log 73  1 [log 143.5 + log 194]
= 1.8633  i [2.1563 + 2.2878]
= 1.8633  1 [4.4441]
= 1.8633  2.2220 = 0.3587 = 0.3587 (+1)  1 + (1  0.3587) = Antilog T.6413 r = 0.4378 = 0.438
{d) Step Deviation Method convenient common factor to redZ „ T , "" by
u Stq
, • 335
Measures of Correlation
I unaffected by the change of origin and change of scale of X and Y. After changing these deviations, we apply the same formula of assumed mean method.
Illustration 13. The data on price and supply relating to a commodity for 7 months are given below :^ ^
".s) : 40 lo^ .o .o
Supply (in kg) : 400 200 500 1000 400 1100 1200
Calculate product moment of correlation between price and quantity and comment on its sign and magnitude. Solution.
Calculation of Coefficient of Correlation
X60 X60 dx' (kg) y700 dy^ drdy
.Price (Rs) 20 I /W 100 t*jf
X dr Y ■ dy
20 40 2 4 400 300 3 Q 6 :
40 20 1 1 200 500 5 25 s
60 0 0 0 500 200 2 4 o;
80 +20 +1 1 1000 +300 +3 9 3
100 +40 +2 4 400 300 3 9 6
120 +60 +3 9 lioo +400 +4 16 12
140 +80 +4 16 1200 +500 +5 25 20
ZX = 560 Tdx = 7 Zdx^ = 35 ZY = 4800 Zdy = l Idf = 97 lAxdy= 40
Steps :
1. Calculate the deviations of X series from an assumed mean and divide them by common factor. Denote them by dx and find out the total, i.e., Idx.
2. Calculate the deviations" of Y series from an assumed mean and divide them by common factor. Denote them by dy and find out the total, i.e., Uy.
3. Square the step deviations of X series and obtain the total, i.e., Idx\
4. Square the step deviations of Y series and obtain the total, i.e., Zdf.
5. Multiply dx and dy and find out the total, i.e., Idxdy.
6. Applying formula, we get
(Zdx).CZdy)
N_
Zdx.dy
r =
[Zdxf N
Zdy^
(Zdy)^ N
336
Here, Idx.dy = 40, Zdx = 7, Uy = _
r =
401^
'7
Statistics for EconomicsXI 1; N = 7, W = 35 and Ldy^ = 97
^_  —
41
y/Bxy/96857
41
41
= + 0.787
_ 5.29x9.84 52.05
commodity. ner words, supply mcreases due to mcrease in the price of
Change of Scale in the Calculation of r
valu" of ~ ff ^^ ^^^
we take the values as T 2 ani s thrva L T Z ^^ ^ ^00 and
values of Yseries are 1 i 2.4 and 3 7 lev ci\ 1 T J'l^^^^
these values to he 12,24 ^n,^''  —
The following example would illustrate the poim •
Illustration 14.^ Calculate^coeffic^^^^^
SolL^W^rmultSTLes^lOa^r."^^^ ^^^^^
values of X and Y would L ^ ^ ^y ^o that the
X : 1 ■
^ = 5 6 t ' ' '
; 10 11 J3
Calcularion of Coefficient of Correlation
Measures of Correlation 359
Applying formula
337
"Ldxdy —
r =
(ldx)(Uy) N
Here, Uxdy = 46, Ux = 0
Idy = 0, EJx^ = 28, Zdy^ = 76, N = 7
Now, we get
+46
r =
(OHO) .7
f^xf^
+ 46
sBmm
LIBRARY
= 0.997
V28x47
Hence, there is high degree of positive correlation.
If the original values of X and Y were used the result would still be the same and r would be +0.997.
Assumptions of Karl Pearson's CoefiScient of Correlation
Pearsonian coefficient is based on following assumptions :
1. Linear relationship : If two variables are plotted on a scatter diagram, it is assumed that the plotted points will form a straight line. So there is a linear relationship between the variables.
2. Normality : The correlated variables are affected by a large number of independent causes, which form a normal distribution. Variables like indices of price and supply, ages of husbands and wives, heights of fathers and sons, price and demand are affected by such forces the normal distribution is formed.
3. Causal relationship : Correlation is only meaningful, if there is a cause and effect relationship between the force, affecting the distribution of items in two series. Itis meaningless, if there is no such relationship. There is no relationship between rice and wheat, because the factors that affect these variables are not common. Similarly, the weight of an individual during the last ten years may show an upward trend and his income during this period may also show similar tendency but there cannot be any correlation between the two series because the forces affecting the two series are entirely unconnected with each other. The calculated coefficient of correlation of such series is usually termed as ''nonsense or spurious^ correlation.
4. Proper grouping : It will be a better correlation analysis if there is an equal number of pairs.
5. Error of measurement: If the error of measurement is reduced to the minimum the coefficient of correlation is more reliable.
Ji ■
" i
338
Mathematical Properties of th^ r Statistics for EconomicsXI
Th^ ( 11 Coefficient of Correlation
are muIdpUed or divided by so™ ^rant ^ f'"" * ^ri' consr^r . subtracted or added frorivl®'f^^'/v""®" "" '^at a " and 14). of X and Y series. (See Illustration
3. The converse of the dieorem r=n i.
and need not necessarily be indepe^detit Uncor^L ^ """elated variables and y Stnyly implies the absence of Itoear rZI^ \ ™™bles X
however, be related in some other fo™ between them. They may,
MerU^^d Demerits of Karl Pearson's Coeffident Demerits
3 process is time consuming.
• relaZ'Sp'™ between the variables, whether such
4. Correlation lies between ± 1 Thi ^ i
otherwise it may be misinterpreted. ' interpretation.
^Rww^ ____________
under consideration
Measures of Correlation ^^^
Charles Edward spearman, a British psychologist developed a formula in 1904 which consists in obtaining the correlation coefficient between ranks of N individuals in the two attributes under study called coefficient of correlation, by rank differences. It is the Product Moment Correlation between the ranks.
This method is applicable only to individual observations rather than frequency distribution. The result we get from this method is only approximate one, because under ranking method original values are not taken into account.
After assigning ranks to the various items, the differences of corresponding rank vaiues are calculated and following formula is used :
rfe = 1 
N^N
where,
rk = Coefficient of rank correlation ZD' = the total of squares of the differences of corresponding ranks N = the number of pairs of observations Like Karl Pearsons, the value or rk lies between +1 and 1. If rk = +1, then there is complete agreement in the order of ranks and the direction of the rank is also the same. When rk = 1, then there is complete disagreement in order of ranks and they are in opposite direction. Let us examine by following example :
1
2 3
1
2 3
0 0 0
0 0 0
niy = 0
rk = \
= 1 
ZD^ N^N
6x0
3^3
=10=1
Perfect positive correlation, i.e., there is a complete agreement.
1 •2
3
3 2 1
2 0 2
4
0 4
= 8
rk = l
= 1 
6ZD^ N^N
6x8
3^3
= 1  2 = 1
Perfect negative correlation, i.e., there is complete disagreement.
The problems are of three types of calculation of rank correlation :
When Ranks are given, (fc) When Ranks are not given, (c) When Ranks are equal or repeated.
Ir i
(If:
340
(a) When ranks are given :
Illustration 15. In a hah
Entry Judge I Judge II
Calculate the rank correlation coeLent!" Solution.
j„ . Statistics for EconomicsXI
A B C D E
1 2 3 4 5
2 3 1 6 4
F 6 5
G // I / K
7 8 9 10 11
8 7 10 11 9
^^^ff^^ffident of Correlatic
A
B
C
D
E
F
G
H
I
J
K
N= 11
1 2
3
4
5
6
7
8 9
10 11
2 1
3 1
1 +2
6 2
4 +1
5 +1
8 1
7 +1
10 1
11 1
9 +2
Steps :
1 1 4 4 1 1 1 1 1 1 4
ZD'= 20.
1. Calculate the difference of two ranks /e Rn u
2. Square these differences and find o^ 'Z tok]
3. Apply the following formula :
rk=l
Now we get.
N^N
n'n
where, ZD^ = 20 and N = H
Measures of Correlation 363
341
rk=l
6x20 11^11
= 1 
120 1320
= 1  0.091 = 0.909
Hence, there is high degree positive correlation, i.e., two judges are agreeing to the degree of 0.909. It indicates that judges have fairly strong likes and dislikes so far as ranking of the babies are concerned.
{b) When ranks are not given :
Illustration 16. From the following marks obtained by 10 students in Statistics and Economics, calculate Spearman's coefficient of rank correlation.
Statistics : 36 56 20 65 42 33 44 53 15 60 Economics : 50 35 70 25 58 75 60 45 89 38 Solution.
Calculation of Rank Coeffident of Correlation
36 4 50 5 1 ' 1
56 8 35 2 +6 36
20 2 70 8 6 36
65 10 25 1 +9 81
42 5 58 6 1 1
33 3 75 1 ? 6 36
44 6 60 7 1 1
53 7 45 4 \ +3 9
15 1 89 10 9 81
60 9 38 3 +6 36
N= 10 = 318
Steps
1. Assigns ranks to given data. Ranks can be given by allotting the biggest item the first rank, the next to its second rank and so on or smallest item the first, next to its second rank and so. on. Any one of the above method of ranking must be followed in case of both the variables.
2. Find the difference of two ranks (i.e., R^  R^) and denote these differences by D.
3. Square these differences and find out the total TB^.
4. Apply the formula.
342
Applying the formula, we get
j„ . Statistics for EconomicsXI
rk = 1
Here,
. N'N ^D = 318 and N = 10
rk = 1 
^x318
= 1 
1908 990
th uu = 1 1.927 .0.927.
2^x^226(55). (55)
iiOx
22603025
V3850  3025 x 73850^30^ 765
^^ X ^JS25 =  0.927, which is same as before
343
Measures of Correlation
Blustration 17. Ten entries are submitted for a competition. Tbree judges study eacb
Ranks given by :
Calculate the appropriate rank correlations to help you to answer the following questions :
(a) Which pair of judges agree the most?
(b) Which pair of judges disagree the most? Solution.
Entry No. : 1 2 3 4 5 6 7 8 9 10
Judge A : 9 3 7 5 1 6 2 4 10 8
Judge B : 9 1 10 4 3 8 5 2 7 6
Judge C : 6 3 8 7 2 4 1 5 9 10
Calculation of Rank Coefficient of Correlation
 ■A • a
1 9 9 6
2 3 1 3
3 7 10 8
4 5 4 7
5 1 3 2
6 6 . 8 4
7 2 5 1
8 4. 2 5
9 10 7 9
10 8 6 10
N = 10
 IjF^
0 0 +3 9 +3 9
+2 4 0 0 2 4
3 9 1 1 +2 4
+1 1 2 4 ,3 9
2 4 1 \  1 +1 1
2 4 +2 . • 4 +4 16
3 9 +1 1 +4 16
+2 4 1 " 1. 3 9
+3 9 +1 1 ' 2 4
+2 4 2 '4 4 16
SD^ = 48 ID' = 26 ID  88
Applying the following formula, we get
rk = l
N'  N
rk (between Judges A and B) rk = 1 
6x48
10" 10
= +0.71
rk (between Judges A and C)
rk = I 
6x26
10^10 = +0.8425
= I _ ^ = 1 _ 0.29 290
= 1^=1 0.1575
990
344
rk (between Judges B and C)
Statistics for EconomicsXI
rk = l
6x88 10^10
^ ^e ^e nearest approach
(*) ^mce is n^nimun. of the pair of judges ^ and C. therefore, they disagree the
in and" ^^rr/t^S"^nts
in ranks in t»o subjects obtained W L of 1 ' <liff"ence
of 7. Find the corrit coefficiem'jrnLl:!::^™^" "" ^ ^^d Solution.
rk=l
N^N
Substituting the values in above formula, we get
0.5 = 1 
(10)^10 = 1  0.5 = 0.5
990
6 ED2 = 0.5 x 990
2£>2 ^ 05x990 6
= 82.5
Corrected ID^ = 82.5  (3)^ + (7)2
= 82.5  9 + 49 = 122.5 6x122.5
Corrected rk = I 
= 1 
(10)^10 735
990
rk = + 0.2576
(c) When ranks are equal or repeated
13
13
24
15
20
19
Measures of Correlation ■ Solution.
Calculation of Rank Coefficient of Correlation
345
X Y
48 13 8 5.5 +2.5 6.25
33 13 6 5.5 +0.5 0.25
40 24 7 10 3.0 9.00
9 6 1 2.5  1.5 2.25
16 15 3.5 7 3.5 12.25
16 4 3.5 1 +2.5 6.25
65 20 10 9 +1.0 1.00
25 9 5 4 +1.0 1.00
15 6 2 2.5 0.5 0.25 _
57 19 9 8 +1.0 i.oo"
N = 10 ZD^ = 39.5
Steps :
1. Assign the ranks to given data. When two or more items are of equal value, they are assigned average ranks. For example, in X series value 16 repeated twice and
they are each ranked ^^ =3.5 and in Y series value 13 are given the rank ^^ = 5.5 and so on.
2. Obtaining ID^ apply the formula. When equal ranks are assigned to same of the
entries and adjustment is made in the formula of rank correlation, i.e., adding —
{m^ m) to the value of SD^ Here, m represents for number of times whose ranks are repeated. In case, there are more than one such group of values with common rank 1/12 (m^  m), is added as many times the number of such groups. The adjusted formula is as under :
..3
rk = 1
ID^ + ^(m^ m) +...
N^N
Now we get, rk
6{39.5 + 2) + 2) + 2)} = 1 ~ in3„n
ta :
= 1 
= 1 
6(39.5 + 1.5)
990
= 1 
10^0 6x41.0
990
246
= 1  0.246 = 0.754
990 rfe = 0.754
Hence, there is high degree of positive correlation.
346
j„ . Statistics for EconomicsXI
Merits and Demerits of the Rank Method Merits :
M^hT " »  compared ,o Karl Pearson.
De^r;'  degree of correlation,
grouped frequency dis^ibnrion (bivtiaSs^l^n' " '
OF FORMULAlt
1. Karl Pearson's Coefficient of Correlation (a) Actual Mean Method
r = ^y _ Zxy J_ J N.ax.ay N ^ ax ^
ay
(b) Direct Method
4
N N
Ixy
xZ/
ZXYN
r =
In
N
N.
' N
In
I N
.N
_N
N
N
N1XYEX.IY
zxyn.x.y
I^^ ± —1\.
Measures of Correlation
(c) Assumed Mean Method and Step Deviation Method
r =
347
N
N
Explanation of Symbols
r = Karl Pearson's Coefficient of Correlation. X = (X  X), deviations taken from actual mean of X series. y = (Y  Y), deviations taken from actual mean of Y series. ax = Standard deviation of X series. ay = Standard deviation of Y series. ZX = Sum of the values of X series ZY = Sum of the values of Y series ZX^ = Sum of square of the values of X series ZY^ = Sum of square of the values of Y series ZXY = Multiplying X and Y values and obtaining the total N = No. of pairs of observations
dx = {X  A), deviations taken from assumed mean of X series.
dy = (Y  A) deviations taken from assumed mean of Y series.
dxdy = Multiplying the deviations taken from assumed mean of X series with the deviations taken from assumed mean of Y series.
2. Spearman's Rank Coefficient of Correlation
rk = \
= 1 
N^N
N^N
rk = Spearman's rank correlation. D = Difference of ranks. N = Number of pairs of observations. m = Number of times the value repeated.
348
exercises
j„ . Statistics for EconomicsXI
4.
5.
6. 7.
10.
11
12.
13.
14.
Questions ; ~ "
fea What is meant by correlaHon? wi,^^ t » Distrnguish betv^e;^^;!^^::.^^ T™ ^^ —
Distinguish between : '^oelation from coeffidem of variation.
(a) Positive and negatives correlation
ib) Lmear and nonli„ear correlation,' and
ic) Simple, partial and multiple correlation Give three examples of perfect correlation. '
^e^pl^r between X and y is .ro, does it mean that variable
d^r Smir born and export over last
correlation): correlations (positive, negaL or no J
(/) Sale of woollen garments and the day temperature
dCflllrrtfou^^^^^^^^^^^ /J" 'he po,„« .He scaler
Distinguish between covariance and variance.
ia) Define Spearman's rank correlation.
ib) What are the limits of rk
! ' ^aSl.^/^ we .legate product moment
r ..correlation be equal to the val^^^? ^^ ^ d Y, will this
8.
9.
2 6
3 5
5 7
6 8
8 12
9 11
349
Me^swres of Correlation
15 What are the advantages of Spearman's rank correlation over Karl Pearson's correlation coefficient? Explain the method of calculatmg Spearman s rank
correlation coefficient.
16. (a) How is Karl Pearson's coefficient of correlation defined?
(b) What are the limits of the correlation r?
(c) If r = +1 or r = 1. What kind of relationship exists between X and Y?
17. Write short notes on :
(a) Spurious correlation, and
[b) Positive and negative correlation.
Problems :
1. Give the following pairs of value of variables of capital em^ployed and^profit^. Capital employed (in crores of Rs) (X) Profit (in lacs of Rs) (Y)
(a) Make a scatter diagram.
ib) Do you think that there is any correlation between profit and capital employed.
Is it positive or negative? Is it high or low? (c) By graphic inspection, draw an estimating hne.
2 Plot the following data as a scatter diagram and comment over the result : ' X ■ 11 10 15 13 10 16 13 8 17 14
y : 6 7 9 9 7 . 11 9 6 12 11
3 Following are the heights and weights of 10 students in a class. Draw a scatter ■ diagram and indicate whether the correlation is positive or negative.
Height im inches) : 72 60 63 66 70, 75 58 78 72 62
Weight (m kg) : 65 54 55 61 60 54 50 63 65 50
4. Construct the scatter diagram of the data given below and interpret it
Average value (in Lakhs Rupees)
Year Cotton (import) Cloth (export)
: 1990 : 47 1991 641992 100 1993 971994 126 1995 203 1996 171
: 70 85 100 103 111 139 133
{expuri/.
Draw a scatter diagram for the data given below and interpret it
X : 10 20 30 40
y . 32 20 " 24 36
6, Draw a scatter diagram of the following data :
50 40
60 28
70 48
80 44
X Y
15
7
18
10
30
17
27 16
25 12
23 13
30 9
f!
350 •
 , Statistics for EconomicsXI
X Series
Y Series
Arithmetic Mean Square of deviations from Arithmetic Mean
■ J__
Summation of products of deviatioZl^lTZTV'^
means = 122 ^ ^nd Y series from their respective
Number of points of values = 15
15 25
18 25
39 41
24 27 30 36 27 31 33 35
Calculate product moment of correlation between X and Y ■
 ^^  S 2 2  
42 41
48 45
[r = 0.98]
91 95 49 40
SliL" Z r ''' whea, and ^
;  « .^o 550
16 17 20 19 19 20 25 27
11. Calculate Karl Pearson's coefficient nf i l ^ '' = 01621]
the following 10 firms : ^^ correlation between the sales and expenses of
Firms 12 3
■ ™    J J
Expenses . n 13 14 ..
(/« Rs '000) ^ 15 14 13 j3
12. Ten students got the following percentage of b ■ c • f'" = Ser,al No. ; 1 ^ 'J' ^^ Statistics and Mathematics 
Statistics : gO 60 51 76 58 J J ' ^ 10
Mathematics 45 yj ^^ 62 64 72 56 58
. Calculate product moment correlation ^^ ^^ ^^ ^^ ^^ 60
13. Calculate correlation coeffident between X the nnn,K . • ^^ = " 3ri Y, the number of rain coats solrl in
ot rainy days per month
Interpret the results. ^ ^ certain shop for 12 months.
^ =Vl4 8 18 10 22 9 3 ,
^ 11 20 12 15 73'' ^^
^ 3 4 7 10 11 29
[r =  0.67]
Measures of Correlation
351
^ f The deviations from their means of two senes (X and Y) are given below . Ky . 4 3 2 10+1+2+3+4
: j 3 4 0+4+1 .2 2 1
< Calculate Karl Pearson's coefficient of correlation and interpret the resu.t. ^^ ^ ^^
15 Find the product moment correlation of the following data : ' X • 1 2 3 4 56 7
Y ; 9 8 10 12 11 13 14
8 16
16.
9 15
[r = +0.931
Calculate the correlation coefficient of the marks obtained by 12 students m Mathematics and Statistics and interpret it.
Students
Marks (in Maths) Marks (in Statis.)
A 50 22
B 54 25
C 56 34
D 59 28
£ 60 26
F 62 30
G 61 32
H 65 30
7 7 K ^ 67 71 71 74 28 34 36 40
[r = + 0.783]
67
68
68 72
69 VO
71 73 69 70 [r = 0.47]
17. The height of fathers and sons are given below : Height of fathers (in inches) : 65 66 67 Height of sons (in inches) : 67 68 64
Calculate Karl Pearson's coefficient of correlation. , A
18 Find Karl Pearson's coefficient of correlation from the following index numbers and
'f;'" «''« ''
Costofltvtng : 98 99 [r = +0.85]
19. Find the product moment correlation between sales, and expenses of the following 10 firms.
Firms Sales Expenses
1 50 11
2. 50 13
3
55 14
4
60 16
5
65 16
6
65 15
7 65 15
8 60 14
9 60 13
20.
10 50 13
[r = +0.797]
Calculate the coefficient of correlation for the following ages of husbands and wives in years at the time of their marriage.
Age of husbands : 23 27 28 28 29 30 31 Ageofu^ives : 18 20 22 27 21 29 27
33 35 36 29 28 29 [r = +0.82]
21 Find suitable coefficient of correlation for the following data : .^iFertUizers used ^ tons) : 15 18 20 24 30^ 35^
Productivity (in tons) ■ ^^
40 50 150 160 (r = +0.99]
352 22.
23.
P , Statistics for EconomicsXI
« age of ca„ and annua, matoenance
Age of cars (years) ; 2 4
Annual maintenance ■. I6OO 1 Ton lonn ^ ^ 10 12 Co^MmRs) 1800 1900 1700 2100 2000
Calculate coefficient of correlation K,. p , ^^ = +0.836]
population and death rate ^ ^^^^^^ between the density of
C 400 14
Citites Density Death rate
A 200 10
B 500 16
D
700 20
E 600 17
24.
F 300 13
[r = +0.988]
Total of the deviation of X = 170 Total of the deviation of Y = 20 Total of squares of deviation of X = 2264 lotal of the squares of deviation of Y  8288
series : Arithmetic Average = 65
•• Standard deviation = 23 33 r series : Arithmetic Average = 66 Standard deviation =14 9
Rauk Correlation ir = +0.78]
26. Calculate Spearman' r , „on from the following data :
^ •• 15 10 I '' 20 25 40
^ 25 16 12 8
27. The following are the marks obtained fout nf mm u t'" = +0143] emp oyment interview held by two Ldependem  ^ '
coefficient of correlation. ^n^ependem judges separately Calculate the rank
Candidates ■■ A n n r^
''  " n
Judge X Judge Y
20 22
20 15
14
10
8
11
12
13 9 [r = 0.721]
353
Measures of Correlation
28 Two judges in a beauty competition rank the 12 entries as follows
X : 1 2 3 4 5 6 7 8 9 10
y. 12 96 10 354782
Calculate rank coefficient of correlation.
29 Calculate rank coefficient of correlation of the followmg data : ' X : 80 78 75 75 68 67 60
Y : 12 13 14 14 14 16 15 ^^ _ ^^^^
30. Twelve entries were submitted in a flower show competition. They were ranked by
two judges as under : ^ , 7 8 9 10 11
^ 7 8 2 1 9 3 12 11 4 10 6
futeB : ^ : 5 3 11 2 12 10 5 9 7 Calculate Spearman's rank correlation. 31 Calculate the coefficient of rank correlation from the following data X ■ 48 33 40 9 16 16 65 25 15 y . 13 13 24 6 15 4 20 9 6
11 12 11 1 [r = 0.454]
59 17
12 5 8
[r = +0.86]
57 19
[r = +0.73]
32. Calculate rank coefficient of correlation between years of service and efficiency rating.
Persons
Years of Service Efficiency rating
A 24 66
B 30 51
C 12 84
D
25 66
E 29 45
F 19 81
G 16 72
H 10
97
I J
11 7 92 70
[r = 0.78]
33. From the following data calculate coefficient of correlation by the method of rank
95 70 60 80 81 150 115 110 140 142
X Y
75 120
68 134
50 100
[r = +0.93]
1. 2.
3.
4.
5.
6.
7.
8. 9.
10. 11.
Chapter 12
IWTRDOCUTION TO INDEX NUMBERS
Introduction Definition
Types of Index Numbers Problems in Construction of Index Numbere Methods of Constructing Index Numbers Consumer Price Index (CPI) Index of industrial Production (IIP) General Uses of Index Numbere Inflation and Index Numbers Limitations of Index Numbers List of Formulae
li'ftSWf
INTRODUCTICm
■ have r'r' we
Quantities; in measures of rp r ionab j^^^^^ Po'ional values :
Deviation, Mean Deviation and tondat Devia,l"t""
two variables. From the above tookAe s™ ' association of
chapter we will learn how to obtS sum jrr"^ ""T? "is
variables. "" """"nary measures of change in a group of related
ail .^l^tZSieThlt:rtTp^t I P^.ces of
price of some commodities may'alat » 'T ™°''it,es,
illustration : ^ ^^^ ^^^^ "s examine from the following
Commodity Prices of ve^ 2000 ?etable oil and tea
Vegetable oil (per litre) Rs Tea (per kg in Rs) 40 100 ^yjyjj 80 150
ti c
Introduction to Index Numbers 355
We can measure the change in the prices of vegetable oil and tea in two ways :
(a) Actual Difference
(b) Relative Change (Price Relative)
(a} Actual difference. The actual difference in price is the difference between the current year price and the base year price.
Actual difference = Current year price  Base year price
Current year : 2005 Base year : 2000
Difference in : Vegetable oil (per litre) Tea (per kg)
We find that the rate of vegetable oil is increased by Rs 40 and of tea by Rs 50 from the year 2000 to 2005. From this, it appears that the increase in price of tea is more than the increase in price of vegetable oil. (b) Relative change (price relative). The relative change in prices is the actual difference in prices relative to the original price. From the above example :
80  40 = Rs 40 150  100 = Rs 50
Relative change =
Actual difference
Base year Price Relative change in vegetiable oil :
8040
or 1
Current year Price Base year Price
For vegitable
For Tea
40
150100 100
= 1
or
1 
= 0.5 or 1 
40 150 100
= 1
= 0.5
This change can also be expressed in percentage : For vegetable oil : 1 x 100 = 100% And for tea : 0.5 x 100 = 50%
The ratio of prices in two years is called price relative which is a pure number and this price relative for a single commodity even may be called an index number of that commodity.
However, if we calculate the rise in percentage taking 2000 as the base year, we. find that the rise is 100% of vegetable oil and 50% in case of tea.
Symbolically,
Current Year Price Base Year Price
iL Po
100
100
356
P^  price of the current year (2005) Pa = price of the base year (2000)
Statistics for EconomicsXI
Vegetable oil : x 100 = 200
Tea
40
m 100
= 150
M Increase of vegetable o,l is 100°/Th f . ™ " "" :
(b) The relative comparison o o^e T ^000 = 100.
Thus, change in pncrrrll ^"at of tea. actual difference in prices. important than just the
As measurement of veeerahip nil .v. iv " i of measurement, their absZe duffel l ""
, "" "Ot be combined. But relative
changes or price relattons are pure numbers namely for vegetable „„
Rs 80
Rs150
R740 ^^^ tea
Rs 100 be combined to obtam the arithmetic mean of price relatives,
I.e.,
80 150
+
40 100
2 + 1.5
= 1.75
it c!:  "seeher ,s ,.,5 . ,00 = 1.5. Now,
mcreasedby27%.,nthe!ame„ay cltsi'XL''"" "f  i
hat Dearness Allowance of Governlm eij^ »
".dex goes up. Similarly, we come acroTs rdex ^b '"T"''production, sales, export, prices, wages «c Thev ar7 f , "Sncultural and industrial economy. Index numbers are the bafo^rlX^^VrrnX:'™''''''"
■
definition
orhif Jechats^—trd r Tllnr IrT " "r »
commodtty to another. Tley are usually IXJIZI^Z^
Introduction to Index Numbers 357
According to Spigel, "An index number is a statistical measure designed to show changes in variables or a group of related variables with respect to time, geographic location of other characteristics."
According to Croxton and Cowden, "Index numbers are devices for measuring differences in the magnitude of a group of related variables".
Thus, the characteristics of index numbers can be heighhghted as under :
1. Index numbers are expressed in terms of percentages so as to show the extent of relative change. However, percentage sign (%) is never used.
2. Index numbers are relative or comparative measurement of group of items. They compare changes taking place over time or between places or like categoriesschools, persons, hospitals etc.
3. Index number are called SpeciaUsed type of averages in the senses that they help us in comparing change in series which are in different units. Averages like mean, median and mode can be used to compare only those series which are expressed in the same unit.
4. The technique of index numbers is utilised in measuring changes in magnitude which are not capable of direct measurement due to composite and complex character of the phenomenon. Examples of such phenomena or magnitudes are price level, 'cost of living', prices of specified list of commodities, volume of production in different sectors of an industry, production of various agricultural crops, 'business or economic activity' etc. Changes in business activity in a country are not capable of direct measurement but it is possible to study relative changes in business activity by studying the variation in the values of same such factors which affect business activity and which are capable of direct measurement.
ttpes of index numbers
The usefulness of any index number lies in the types of questions it can answer. Each index number is designed for particular purpose, and it is the purpose that determines its method of construction
There are various kinds of index numbers. In economics and business, they can broadly classified as under :
1. Wholesale Price Index (WPI)
2. Consumer Price Index (CPI) or Cost of Living Index
3. Index number of Industrial Production (IIP)
4. Index number of Agricultural Production (L\P)
5. Sensex
1. Wholesale price index (WPI) is used to measure the general price level where we are required to obtain the wholesale prices of industrial, agricultural and other products from wholesale market. It does not include the items pertaining to services like repairing charges, barber charges etc. WPI is used to eliminate the effect of
358
Statistics for EconomicsXI
the Priva" str " "he public and
4. number of Agriculn.ral Production (lAP) is used to study the rise and fall of the yteld of pnncpal crops from one period to other period
5. Sensex is a useful guide for the investors in the stock marter If rh. • ■
appropnate time for mvestment. The rise in sensex at the highest level reflects the base''??S'valut'oft°hT'°™ Bombay Stock Exchange Sensitive todex with 197879 as
iiuinoer will replace wholesale price index. Producers Price Index /PPT^
Introduction to Index Numbers
359
in construction of index numbers
Following are the important problems which must be well defined for the construction of index numbers :
1. Purpose : Every index number has its own particular uses and hmitations. The first and foremost problem in the construction of index numbers is in regard to the objective or the purpose for which they
are required. It is important to know what is to be measured and how these measures are used. If the purpose is to measure the general price level, then wholesale price index number is used. If the purpose is to measure cost of living of middle class families, working class (labour) or agricultural workers, in a particular region or city, then consumer price mdex number is used. If the object is to measure relative change in industrial production, then index number of industrial production is to be used.
2. Selection of base period : When comparison is to be made between different time periods or different places, some point of reference is to be decided. This is called base. In the above illustration about prices of vegetable oil and tea, we have taken year 2000 as the base year and 2005 as current year for our calculations' of index numbers. The base is assigned the value of 100%.
For making the comparison over a period of time it must be remembered :
(a) The base period should not be either too short or too long : It should be neither less than a month nor more than a year from calculations' point of view.
(b) The base period should not be too near or too far : This is because people usually prefer to compare present conditions with conditions in base or reference period that is not too far back time. If the base period is too far the comparison becomes meaningless. Due to introduction of new commodities, change in habits, taste, fashion, in economy many commodities may go out of use. In such situation it becomes necessary to shift the base period.
(c) The base period should be normal and representative period : Base period should be free from all sorts of abnormalities and random or irregular fluctuations like earthquakes, wars, floods, famines, labour strikes, lockouts, economic boom and depression.
(d) It should be a period for which actual data is available
Fixed base and chain base : If the period of comparison is kept fixed for all current years, it is called fixed base period. However sometimes chain base method is used, in which the changes in the prices for any given year are compared with prices in the preceding year and not with the fixed year. Naturally, the chain base method gives a better picture than what is obtained by fixed base method. However, much would depend upon the purpose of constructing the index.
3. Selection of items : Collection of data is a special problem in constructing index numbers, since there is a large variety of goods and prices. Care also must be taken that data from unrelated commodities or periods are not grouped together for the calculation of price index. If the number of the commodities is too large, a choice
360
Statistics for EconomicsXI
of some representative items has to be made. On the other hand, inclusion of too few Items would make the index number unrepresentative of he ™
mcrdfal, '' ^^^^ calculations, it is nof^"s bk «
include all commodities in construction of index numbers
be considered :
(«) Commodities selected should be relevant and representative of the group according to the purpose of mdex number. For example, in const uction of wholesale price mdex number to know the general price level, we sZw nclude wholesale prices of some major mdustrial and agricultural ^olmoS and Other goods and services. In the same way for coLtruction ofZsle
^^ — whichsm^nt toT W ^^^ ^^^be neither too
ot data. Ihere is always a chance of getting spurious or misleading results if the
w ^^ r  ^be changes m mdustrfalToI
we must collect the prices relating to production of various goods of factories For
nlrh^'. ' ^^ ^^^ consideration, w! can re y onT^^^^^^^
published market reports by business houses. Chamber of Commerce aS
L utilised " Pl also
I
Introduction to Index Numbers
361
6. Choice of an average : For constructing an index number any average such as mean, median, mode, geometric mean and harmonic mean can be used Frorn the practical point of view median and mode are unsuitable because of their being Latic. The geometric mean and harmonic mean are difficuh to calculate hence; arithmetic mean is used. Though with the development of the use of electronic computers, the use of geometric mean is also becoming popular.
7. System of weighting : In order to allow each commodity to have reasonable influence on the index it is advisable to use a suitable weighting system. Unweighted index numbers are those where all commodities are given equal importance. But in most cases different commodities are given different degrees of importance, therefore, weights are assigned to the various items.
The method of weighting used would depend on the purpose of index. Weighting may be done according to : (a) Value or quantity produced, (b) Va ue of quantity consumed, and (c) Value or quantity sold. When the quantity is the basis of weight it is called quantity weighting and when the value is the basis, it is called value weighting. Weight may ,be either implicit (arbitrary) or explicit (actual).
8. Choice of method : There are various methods of calculating index numbers such as the aggregative method or the price relative method. Various methods have been proposed for calculation of weighted index number such as Laspeyre's method, Paasche's method, Dorbish and Bowley's method, Marshall Edgeworth's method, Kelley's method and Fisher's method. Fisher's method is considered as ideal for constructing index numbers. No single formula can be said to be appropriate for all types of index numbers and as such the choice of a formula wi 1 have to be made taking into account the object of index numbei; the data available and the resources at the disposal of the person or organisation constructing the mdex
number. »
constructing indei
The methods of construction of index numbers are given below :
Methods
_
Price
362
Statistics for EconomicsXI
Unweighted Index Numbers
I. Simple Aggregative of Actual Price Method
This is the simplest method of calculating index numbers. In this method, total of the current year prices for the various commodities is divided by the total of base year prices and the quotient is multiplied by 100.
Symbolically, Price Index
Po^ =
Quantity Index
^01 =
^Po
X 100
100
where.
Poi = Current year price index number ^01 = Current year quantity index number Zp^ = Total of current year prices for various commodities ^Po = Total of base year prices for various commodities
= Total of current year quantities for various commodities = Total of base year quantities for various commodities
Illustration 1. Calculate price index number for 2005 taking 1995 as the base year trom the following data by simple aggregative method.
Commodities
Prices in 1995 (in Rs)
Prices in 2005 (in Rs)
Solution.
I
A B C D E
100 80 160 220 40
140 120 180 240 40
Construction of Price Index Number
Commodities Price in 199S (Rs) Price in 2005 (Rs)
A 100 140
B 80 120
C 160 180
D 220 240
E 40 40
Zp^ = 600 Xpj = 720
It:
exi
Introduction to Index Numbers
363
Steps
1 Aggregate the current year prices of various commodities (Ep,)
2. ISrelate the base year prices of various commodities W
3. Apply the following formula :
Po, =
= ^ X 100
Zpo
Here,
p = Price index number of the current year (2005) Zp = Total of current year prices for various commodities Zp  Total of base year prices for various commodities
Now, we get
= ^xlOO = —X 100 = 120
thus, the pncr^.^.;::
Number for 2004 wirh base 1995 from Ae^followmg • ^ ^
Commodities ■■ ^ ^^^ jj 5 2
80 60 20 10 6
1995 Quantity (kg) 2004 Quantity (kg)
Calculation of Quantity Index Number
Commodities
A B C D E
Total
Quantity in 1995 (in kg)
(qp)
___ _
40 10
5 2
■Quantity in 2004 (in kg)
Iq^}
Uo = 107
80 60 20 10 6
24, = 176
Applying formula, we get
^^  100
^01 E^o
Here,
101
= Quantity index no. of 2004, Zq, = 107, and Zq, = 176
_ IZi X 100 = 164.48
^01 107
extent of 64.48% as compared to 1995.
364
j„ . Statistics for EconomicsXI
Illustration 3. Compute index numbers for the vears 1996 innn ( u r ., data (Base Year 1995). ^ 1996 to 2000 from the following
Year : 1995 Price : 10 Solution.
1996 14
1997 16
1998 20
1999 22
2000 24
Calculation of Index Numbers (Base year 1995)
Price Index Numbers ^ ^xlQO Pff
1995 10 100
1996 14 ~ X 100 = 140 10
1997 16 16 jQ X 100 = 160
1998 20 20 jQ x 100 = 200
1999 22 24 10 22 ^ X 100 = 220
2000 Hprp h —24 c ^u 24 Y^ X 100 = 240
» 1 'I'* ^.ixv, wiiixi^iiL ycai
Po = Price of the base year
Limitations
1. No weight is given to the relative importance of items
2. Index IS influenced by the items with the large unit prices
II. Simple Average of Price Relatives Method
X 100. Index number by this method is the arithmetie mean or median or geometric'
365
Introduction to Index Numbers
Illustration 4. Construct pure index number for 2005 taking 2000 as the base year from the following data by simple average of price relative method.
A B C D L
100 80 160 220 40
140 120 180 240 40
" Solution. 'A price relative is the price of the current period expressed as a percentage of the price at the base period'.
Construction of Price Index Number
Commodities Price in 2000 (in Rs) Price in 2005 (in Rs)
Prices in 2000 Prices in 2005 Price Relatives
Commodities   'a : : SL X 100 ; :: Pov , ^
A 100 140 ^^^ X 100  140 100
B 80 120 120 „ " X 100 = 150 80
C 160 180
D 220 240 X 100 = 109.1 220
E 40 40 "^^x 100  100 40
Total • Z^x 100 F 611.6 Po
Steps :
1. Calculate the price relatives of current year
LxlOO
2. Aggregate the calculated price relatives.
3. Divide the total of price relatives by number of commodities.
4. Apply the following formula :
>1
Po^ =
Po
xlOO
N
Here, = Price index number of current year (2005)
366
j„ . Statistics for EconomicsXI
^ X 100 = Price relatives of current year N = Number of commodities
Now we get, p^^ =
^xlOO {Po
611.6
= 122.32
N ~ s
m th^nrVt' T" 2005 is 122.32. In other words, there is net increase
m the prices of commodities m the year 2005 to the extent of 22.32% as compared to
Merits
1. Index number is not influenced by extreme items. Equal importance is given to all
LfxC? ItvlliS*
Limitations
1. The relatives calculation are assumed to have equal importance. This assumption may not be always correct. ^
2. There is a problem of selecting a proper appropriate average. Weighted Index Numbers
weil?*''''^ '' 1" appropriate
weights are assigned to various commodities to reflect their relative importance in the
I. Weighted Aggregative Method
Weights are assigned to the various items. There are various methods of assigning
merk r J Tn Laspeyre's method, Paasche's method, Dorbish and Bowley's
method, Marshall Edgeworth's method, Kelley's method and Fisher's method Fisher's
constructing index numbers. According to syllabus of Class XI, we are discussing here Laspeyre's and Paasche's method of constructing index
Laspeyre's Method Paasche's Method
Price Index fo: = X 100 Zpolo Quantity Index = X 100 ZqoPo Price Index = v^^ X 100 Quantity Index = X 100 ZqoPi
for EconomicsXI j Introduction to Index Numbers
jthere is net increase 2% as compared to
^nce is given to all fquoted or absolute
!. This assumption
^od appropriate pportance in the
of assigning ndex numbers r and Rowley's hod. Fisher's ' syllabus of ucting index
' "" " ' ^ ''■'a 'O ger quantity index n
denoted as ,, and' the'^na^; „ ^t™ ylJlt^^^.T'''' ''' Laspeyres metltod is very widely used" Ti, ha The
which are not changed front one^yert^n « S„ Te r^ T ^ V'
Value Index Number =
Base year values
V = ^lilyinn •^01 xioo
or
=
XlOO
where
XVo
Poi = Price index number ^01 = Quantity index number Vgj = Value index number Pi = Current year price pQ = Base year price
= Current year quantity = Base year quantity V, = Current year value (Zp^q^) Vp = Base year value {Lp q )
Commodities
A B C D
1996 Base Year
Price
10 8 6 4
Quantity
30 15 20 10
2005 Current Year
Price
12 10
6 6
Quantity
50 25 30 20
368
Solution.
Construction of Price Index Numbers
Statistics for EconomicsXI
Year (ZOOS) 1
Quantity It PSt Ptlo Pitr
50 25 30 20 300 120 120 40 500 200 180 80 360 150 120 60 600 250 180 120
= 580 = 960 = 690 = 1150
where
1, = quantity of current year (200J)
Po = price of base year (1996)
. ° quantity of base year (19961
(A) Laspeyre's Method. In this merhoH K
Steps. (Price Index nuntM " ^ ""^hts.
otrC.'™' contntodtties „.th base year weights and
oti^i.^^ "" —« wtth base year weights and
TptfL^t' ' * "'ent by ,00. The above steps give us
Laspeyre's Price Index Number Symbolically,
X 100
690
= J^ X 100 = 118.96
Thus the price index number of 2005 is 1 IS v pnces of commodities m the year 2005 to the ex,^^ urease m the Laspeyre's Quantity Index : 1896% as compared to 1996.
= »xlOO ^oPo
960
^01 =
580
XlOO = 165.52
JOU
Thus, the quantity index number of 2005 is 165
,.ant.ty of commodittes n, the year ^OO^l'^^^irroH^I^t™
Introduction to Index Numbers 391
(B) Paasche's Method : In this method current year quantities are. taken as weights. Steps. (Price Index Number)
1. Mukiply current year prices of various commodities with current year weights and obtain Spj^j.
2. Multiply the base year prices of various commodities with the current year weights and obtain
3. Divide by JLp^q^ and multiply the quotient of 100. Symbolically,
Paasche's Method :
p 3,^x100
1150 960
X 100 = 119.79
Thus, the price index number of 2005 is 119.79. In other words, there is net increase in prices of commodities in the year 2005 to the extent of 19.79% as compared to 1996.
Paasche's Quantity Index :
_ = ^fiPL X 100 ZqoPr
1150 690
xlOO = 166.67
Thus, the quantity index number of 2005 is 166.67. In other words, there is net increase in quantity of commodities in the year 2005 to the extent of 66.67% as compared to 1996.
Value Index Number :
Zpo^l
1150 580
X 100 = 198.28
Thus, the value index number of 2005 is 198.28. In other words, there is net increase in value of commodities in the year 2005 to the extent of 98.28% as compared to 1996.
n. Weighted Average of Price Relatives Method
Illustration 6. Calculate weighted average of price relative index number of prices for 2005 on the basis of 2004 from the following data :
370
Commodities
A B
C D E
Solution.
Weights
20 12 8 4 6
j„ . Statistics for EconomicsXI
Price 2004
20 15 10 5 4
Price 2005
Commodities
A B C D E
Weights
(w) %
20 12 8 4 6
Steps :
of Price Index Numbers
35 18 11 5 5
Price 2004
20 15 10 5 4
Price 2005
35 18 11 5 5
Value weights
(PolJ [V]
400 180 80 20 24
Pi
^xlOO ypo
fp]
IV = 704
175 120 110 100 125
[PV]
70000 ■21600 8800 2000 3000
IPV = 105400
Calculate the price relatives of the current year fexiool ■
Item of the period for which the in^ "Jexpressmg each
of the same item in the tse pL^" ca4>ated as a percentage
p. calculated pereentages for each ttem hy value weights,
nr fPl/l
or [PV]
Po, =
— XlOO \Po
^Poqo ZV  75J = i^y./I
n .ere .s a net mcrease
^^'./l /o as compared to 2004.
or
^^ _ 105400
= 149.71
tr in
n
CO
im
fin coi pu] are
Introduction to Index Numbers
371
somer price index (cpi)
The wholesale price index numbers measure the changes in the general level of prices and they fail to reflect the effect of the increase or decrease of prices on the cost of living of different classes or group of people in a society. Consumer price index numbers are also called (/) Cost of living index number, or («) Retail price index number, or (ttt) Price of living index numbers. Consumer price index numbers are designed to measure the average change over time in the price paid by ultimate consumer for a specified quantity of goods and services. They measure the change in the cost of living of a particular section of society due to change in the retail price. A change in the price level affects the cost of living of different classes of people differently. The general index number fails to reveal this. So there is the need to construct consumer price index. People consume different types of commodities. People's consumption habit is also different from person to person, place to place and class to class, i.e., richer class, middle class and poor class.
Construction of Consumer Price Index
The following are the steps in construction of consumer price index : (1) Determination of the class of people : Consumer price index numbers are constructed separately for different classes of people or groups or sections of the society, e.g., urban wage earners, agricultural labourers, industrial workers, government employees etc.'and also for different geographical areas like town, city, rural area, urban area, hilly area and so on. The group has to be clearly defined. When we talk of government employees, then we have to decide about the low paid or high paid government employees as their consumption pattern differs. The class for which the index number has to be constructed must be as far as possible homogeneous from the point of view of income and habits. The major groups of consumers for whom the consumer price index numbers have been constructed in India are : (/■) the industrial workers, (ii) the urban nonmanual workers, and (Hi) the agricultural labourers In India, the consumer price index for industrial workers is by far the most popular index. This is constructed on monthly basis with lag of one month. CPI measures changes in retail prices of goods and services covering 260 items of consumption from 70 centres. The base year of CPI (IW) is 1982. The CPI for industrial workers is increasingly considered as the appropriate indicator of general inflation, which shows the most accurate impact of price rise on the cost of living of common people.
(2) Conducting family budged enquiry : Family budget enquiry is held with a view to find out how much an average family of this group spends on different items of consumption. The quantity of the commodities consumed, as also prices at which they are purchased are noted down. The enquiry is done on a random sample basis. Some famihes are selected from the total number by lottery method, and their family budgets are
372
Statistics for EconomicsXI
Jt''™^ ""  groups;
(/■) Food («) Clothing (Hi) Fuel and Lighting
(iv) House Rent
(v) Miscellaneous
the "is is very in,pona„t in
to shop and W 17,■
cCect tetai, priee. The pHncpies ^tSroTtr;^^^
1. Pnce must relate to a fixed list of items for a fixed quality.
2. Retad price must be the price which is given by the consumer.
3. M drscount U given to all customers, it can be taken into account.
In ttcTt'oft ° T""' """ " """ OP'"price
questiomtaires. First we musfSaToecw" >e"ts or mail
must be instructed properly anTK listTtem ,'^^^^ "
be conducted by "cU pLngC^thllhSm?
for rhfferenr classes <ff Peopl^Tltrnd^^r^ ^r^^^ Methods of Construction
(t) Famdy Budget Method or Weighted Relatives :
IWR
Consumer Price Index =
IW
Here,
R = ^x 100 for each item
Po.
W = Weights
Introduction to Index Numbers
(ii) Aggregative Expenditure Method or Aggregative Method :
Consumer price Index = ^^ x 100
Zpo^o
This is based upon Laspeyre's method. According to this method, the various items are given weights on the basis of quantity consumed in the base year.
If the calculated cost of living index number is more than 100, it means a higher cost of living, necessitating an upward adjustment in the wages and salaries of employees The rise of wages or salaries is equal to the amount of percentage it exceeds 100. If the calculated index number is less than 100, it means the cost of hving has decline by the balancing percentage between 100 and calculated index number.
Illustration 7. An enquiry into the budgets of the middle class families in a certain city gave following information. What is the cost of living index of 2004 as compared with 1995. Calculate by :
(i) Family budget method, and
(ii) Aggregative expenditure method.
Expenses on items
Price (in Rs) 2004
Price (in Rs) 1995
Solution.
(i) Constructing Cost of Living\Index Number
(Family Budget Method)
Food Fuel Clothing Rent Misc.
35% 10% 20% 15% 20%
1500 250 750 300 400
1400 200 500 200 250
Expenses on item Weights (%) (W) Price (in Rs) 1995 (Po) Price (in Rs) 2004 (P.) Price Relative  A (R) Weighted Relatives (WR)
Food 35 1400 1500 107.14 3749.9
Fuel 10 200  250 125 1250
Clothing 20 500 750 150 3000
Rent 15 200 300 150 2250
Misc. 20 250 400 160 3200
ZW = 100 IWR = 13449.9
Cost of living Index for 2004
CPI =
LWR 13449.9
= 134.499
ZW 100
Thus, there is increase of 34.5% in prices of 2004 with that of 1995.
374
Expenses on items
Food Fuel
Clothing
Rent
Misc.
Weights
35 lb 20 15 20
(ii) Aggregate Expenditure Method
Statistics for EconomicsXI
Price (in Rs) 1995 Po
1400 200 500 200 250
Consumer Price Index for 2004
Price (in Rs) 2004 Pi P<Ao Pi'Jo
1500 • 250 750 300 40049000 2000 10000 3000 5000 52500 2500 15000 4500 8000
^Po^o = 69000 = 82500
CPI  ^Pi'^o 82500
100
" « ^ 100 = 119.565
items are 75, 10, 5, 6, and 4 rewrtvdv Pr^r ""^^ts of these
hving for 2005 with 1980 as JS hSe ' """ber for eost of
Items Price in 1980 Price in 2005
Food Clothing Fuel and lighting House rent Misc. 100 20 15 30 35 200 25 20 40 65
Solution.
_^Consfructing Cost of Living Index Number
Items Weights W Price 1980 (Pol Price 200S (PJ Price: Relative R 100 Weighted Relatives (WR)
Food Clothing Fuel and lighting House rent Misc. 75 10 5 6 4 100 20 15 30 35 200 25 20 40 65 200 125 133.33 133.33 185.71 15000 1250 666.65 799.98 742.84
ZW = 100 IWR = 18459.47
Introduction to Index Numbers Cost of living Index for 2005
CPI =
375
IWR IW
18459.47 100
= 184.594
Thus, there is increase of 84.6% in prices of 2005 with that of 1980„ Illustration 9. The consumer price index for June 2005 was 125. The food index was 120 and that of other items 135. What is the percentage of the total weight given to food? Solution.
Items Index Weights WI
(I) : m
Food 120 w, 120 Wj
Other items 135 w. 135 W,
IW = 100 IWj == 120 Wj + 135 W^
Let the total weight = 100, W^ = Food and W^ = other items Hence, 100 = Wj +
We are given consumer price Index =125
IWR
.(1)
CPI =
125 =
IW
120 Wi +135 W2 . 100
or
12500 = 120 Wj + 135 W^
•(2)
Now, solving equation (1) and (2), .
100 = 12500 =
We get 13500 =
12500 =
... X (135)
Wj + W^ 120 Wj + 135 W^
135 Wj + 135 W^ 120 W, + 135 W^
1000 = 15 Wj W, = iff^ = 66.67
Now using equation (1), we get 100 = Wj + 100 = 66.67
W, = 100  66.67 = 33.33 Hence, percentage of total weighs given to food = 66.67% and for other items = 33.33%
M.
376
Verification
Statistics for EconomicsXI
Items
Food Other items
Index
(I)
120 135
Weights (W)
66.67 33.33
IW = 100
W.
8000 4500
IW, = 12500
CPI =
IWI IW 12500 100
= 125
Consumer Price Index No. is 125 as given in question. Uses of Consumer Price Index
Consume, price index is'called J prici^dX^fri^Ze '
{a) Purchase Power of Money = _ ^_
Consumer Price Index
(fe) Real wages =
Consumer Price Index
Suppose, the consumer price index was 400 in 7(\[\a n^ u 100 m 200001. Then a rupee in 2004^5 wol be ^^uTto ' " ""
100 400
= 0.25
F I 11 £ purchasmg power of rupee in 200405
Rs 32T:Ich w" tr P" nonth was
100 and for 2004.0J was AmwTlnA A Consumer pnce mdex for 200001 was by rise of h,s wages^T^elrl^Telltagt""" """"""
Real wage = Money Wages
Consumer Power Index
For 2000 01 : ^ X 100 = Rs 3250 For 200405 : ^ X 100 = 1250
Introduction to Index Numbers
377
However the monthly money wage was raised from Rs 3250 to 5000 in 200405. The worker has not gained. In fact his real wage has gone down. The real wage of the worker is Rs 1250 in 200405 as compared Rs 3250 in 200001.
Example 12. If the salary of a person in the base year is Rs 4000 per annum and the current year salary is Rs 6000, by how much should his salary rise to maintain the same standard of living if the CPI is 400.
Solution. We are given :
Year ^ Salary CPI
Base Current Rs 4000 Rs 6000 100 400
When 100 is the CPI of Base year, bis salary is Rs 4000 400 CPI of current year, his salary should be
400x4000
100
= Rs 16,000
Hence, his salary rise should be of Rs 10,000 (16,000  6000 = Rs 10,000) in current year to maintain the same standard of living.
2. The government (central or state) and many big industrial and business units use consumer price index numbers to regulate the Dearness Allowance (D.A.) or grant of bonus to employees. This compensates them for increased cost of living due to price rise. They are used by the government for the formulation of price policy, wage policy and general economic policies.
3. If the prices of some important essential commodities (like wheat, rice, sugar, cloth, etc.) increase, due to shortages, the government may decide to provide them through fair price shops or rationing.
4. Costs of living index numbers are used for deflating value series in national accounts.
5. Consumer price index numbers are used widely in wage negotiations and wage contracts. They are used for automatic adjustment (increase) of wages corresponding to a unit increase in the consumer price index.
mdex of industrial production (iip)
Index numbers of industrial production are fairly common these days. They tell about the relative increase or decrease in the level of industrial production in a country in relation to the level of production in the base year. They are the best measures of economic progress in any country. These indices can be constructed by studying variations in the level of industrial output. As such the first step in the construction of such index numbers is to find the level of output of various industries of the country. It should be remembered that these index numbers throw light on changes in the quantum of production, not in
^ ^ Statistics for EconomicsXJ
values. If the variations in the value of output are to be studied, data about the value of mdustnal output have to be used for the purpose of constructing such index numbers. Thus, mdices of mdustrial production are constructed either by studying changes in the quantum of production or its value.
Index of Industrial Production in India
A number of index numbers of industrial production are compiled in India by official and nonofficial agencies. The general index of industrial production is the most popular among these. In India, Index of Industrial Production is published by Central Statistical Organisation (CSO), Industrial Statistics Wing. The old series of Index of Industrial
.'aatlT''. . ' but now new series is pubHshed with the base year
199394. An abstract from Economic Survey 200506 is as under :
General Index of Industrial Production Base 199394 = 100
199394 199899 199900 200001 200102 200203 200304200405
100 145.2 154.9 162.6 167.0 176.6 189.0 204.8
Usually important data about production are collected under following major heads:
I. Mining Industries : Coal (inc. lignite). Petroleum, crude (offshore and onshore) Iron ore.
II. MetaUurgical Industries : Hot metal (inc. pig iron), crude steel, semifinished steel, steel castings, aluminum, bister copper.
III. Mechanical Engineering Industries : Machine tools, cotton textile machinery, cement machinery, railway wagons, automobiles, (commercial vehicles, cars, jeeps, land rovers), power driven pumps, diesel engines, earth moving equipment, bicycles sewing machines, agricultural tractors. '
IV. Electrical Engineering Industries : Power transformers, electric motors, electric fans, electrical lamps, radio receivers, aluminum conductors.
V. Chemical and Allied Industries : Nitrogenous fertilizer (N), phosphatic fertihzer (P^Oj), soda ash, caustic soda, paper and paper bond, automobile tyres, bicycle tyres, cement, petroleum refinery products, penicillin, streptomycin, chloramphenical powdei; vitamin A.
VI. TextUe Industries : Jute textiles, cloth (cotton cloth), mixed/blended cloth, spuny and filament yarn, staple fibre etc.
VII. Food Industries : Sugar, tea, coffee, vanaspati, salt.
VIII. Electricity Generated : Related to utility.
IX. Miscellaneous : Glass, soap etc.
Usually important data about production are collected under above major heads.
Introduction to Index Numbers
379
The data relating to the production of the above mentioned industries are cq^lected either monthly, quarterly or yearly. The production of the base year is taken as 1^0 and the current year's production is expressed as a percentage of the base year's production. These percentages are multiplied by the relative weights assigned to various industries. Weights are usually assigned on the basis of the relative importance of different industries. The relative importance of industries is usually decided on the basis of capital invested, the gross value of productions, turnover, net output etc. Many other criteria of relative importance can also be laid down. Usually weights in an index number of industrial production are based on the values of net output of different industries. The weighted arithmetic average or geometric mean of the relatives give the index number of industrial production. Such index numbers can be constructed both for gross output as well as net output.
The following table shows broad industrial grouping and their weights.
Broad groupings Weight in % Index No. in May 200S
Mining and quarrying 10.47 155.2
Manufacturing 79.36 222.7
Electricity 10.17
General Index 213.0
From the above table, we find that the growth performances of broad Industrial categories differ.
Method of Constructing Index of Industrial Production
Formula : Using simple arithmetic :
Index No. of Industrial Production (IIP) = where.
iL Uo
W
IW
q^ = Current year Quantity produced q^ ^ Base year Quantity produced
W = Relative importance of different outputs. Illustration 13. Construct Index of Industrial Production for 2004 from the following information.
Industry Output (Units)
1996 2004 Weights
1. Mining 120 160 20
2. Textile 80 110 25
3. Mechanical Engineering 70 90 15
4. Chemical 80 70 25
5. Electrical 90 120 15
380
. ^ , , Statistics for EconomicsXI
Solution. of Index Number of Industrial Production
In.Iusln
I ■ Milling
2. Textile
3. Mechanical Engineering
4. Chemical
5. Electrical
1996 la nm W ^xlOO % / \ 'SL ! w
120 80 70 80 90160 110 90 70 120 20 25 15 25 15 133.33 137.50 128.57 87.50 133.332666.66 3437.50 1928.55 2187.50 1999.99
__ XW = 100 12220.20
Industrial Production Index =
X iL W
ZW
12220.20
100
= 122.20
22.20% increase of industrial production in 2004.
general uses of index numbers
ItTor^b™^^ can be summarised as follows :
as barometers to find th^Ta^td^^^^^r T"" ^^ act
barometers which are used'm physlTrmr" ^ L^ke
measure the level of economic andTsmesTac ™ ""^bers barometers' or 'barometers of econorr ^ ^ ^'economic exchange, reserve bank deposits^r"row 1on T ""
activities of a country and these md cesTan t u" business
could act as an economic blromeJer ^ex which
2. To measure comparative chanirfe ■ tu
measure relative temporal or crol^eS^n^^.r''™ ""'"bers is to compared with same base figure Inde^^^^^^^^ ' "^"^^le or a set of variables time to time, among differentXe^and T " comparison of changes from
m the phenomena like prTce £ cost ^^ ^ ^^^ ^^^anges
measured with the help'of ''' ^^^^^^
f ^g complex variable through time or space " 'be
  red ,„
Infme"r„gfc7:a7uf ^ nature into
/
Introduction to Index Numbers
381
3. They help in framing suitable policies : Index numbers are indispensable tools for the management of any government organisation or an individual business concern for efficient planning and formulation of business policies. For example, relative wholesale and retail price index numbers are the output (volume of trade, industrial and agrxultural production etc.) help in economic and business policy making.
It is not in the field of business and economics that index numbers are used as a basis for policy frame but even in disciplines like Sociology and Psychology their utility is immense. For example, sociologists may speak of population indices, psychologists measure intelligence quotients which are essential index numbers comparing a person's intelligence score with that of an average for his or her age. Health authorities prepare indices to display changes in the adequacy of hospital facilities and educational research organisations have devised formulae to measure changes in effectiveness of school systems.
4. lo measure the purchasing power of money : Index numbers are helpful in finding out the intrinsic worth of money as contrasted with its nominal worth. Very often statements are made that purchasing power of the Indian rupee in 2000 is only 20 paise as compared to its purchasing power in 1990. It means that a person who was having an income of Rs 1000 per month in 1990 should have an income of Rs 5000 to maintain the same standard which he was maintaining in 1990. This helps in determining the wage policy of a country.
5. To help in study of trend : Index numbers are very useful in the trend or tendency of a series over a period of time. It is easy to find out the trend of exports, imports, balance of payments, industrial production, prices, national income and variety of other phenomena. It is also useful in forecasting future trends. With the help of index numbers of prices, demand, wages, income etc., a business executive is in a better position to take decisions about whether a new product should be launched or whether there is scope for exploring new markets or whether the existing pricing and production policies need a change.
6. For adjusting National Income : Index number are vfery helpful in deflating (adjusting) national income on the basis of constant prices to enable us to find out whether there is any change in the real income of the people. They are used to adjust the original data for price changes, or to adjust wages for cost of living changes and thus be transformed into real income and nominal sales into real sales through appropriate index numbers.
inflation and index numbers
According to Samuelson— By inflation we mean a time of generally rising prices for goods and factors of production—rising prices of bread, cakes, haircuts, rising wages, rents, etc.
According to Ackley "Inflation can be defined as a persistent and appreciable rise in general level of average prices".
1
382
Statistics for EconomicsXI
inflaS,:. T" T"
to be cateeorised a, / „ Pe'^P'ible and persistent over time
with rismg prices The essence of '' Tl
pnce levef rLg ove ' t me Jnfl^^ " f  PP^X that keeps the
IS expressed as percentarrist ^ P mdex and
or week. The variationsin gen rarprfcel" 1 m ^^^^ ^^^^
numbers. They are : ^ measured by two types of index
(a) The Wholesale Price Index (WPI), and
(b) Consumer Price Index (CPI)
pnce at which a commortv^ oU fn " ® '"'n'"^ P"« 'te
measure the general p^e it ,„ the 17 7 """^ers which
(WPI). „ ,s aL.n L T
movements ,n a comprehensive way h ' n ifdl:ri„?T
commodities in all trade and transact,™ tL movement m prices of
important in modern ecoTo^c " tTa ' ifa/T T P™es is very
activity WPI is the most commo::i;r;;:dt:ar^f
Wholesale Index Number in India
Eco^dvt: Mti^r^^^^^^^^ published every week by the office of
was constructed m 194ny he S <" ^h index number
195253 as base year in mrTre ^ T^uT
Government of India has tl base st 98,I""
base year 199394. ^ published with the
of c^dilX'atT''"  «> "ver large number
Introduction to Index Numbers 405
383
Category
Weifihts %
22.0 14.2 63.8
■No. of items
98 19 318
(a) Primary goods
(b) Fuel power, light and lubricants ^_(c) Manufactured goods
Source : Economi^ Survey ^0506 Inflation and Wholesale Price Index Number
WPI IS the only price index m ndt wS ^^^ ^^^de and transactions,
lag of two weeks I^is due to rhe^ ^^ '' ^^ """ ' ''''''
Table 1
Index Number of Wholesale Prices
Year
Primary Articles
Fuel, Power, Light and Lubricants
Manufactured Goods
AU , Cunnnodirics
Last week of
198990
199091
199192
199293
199394
199495
199596
199697
199798
199899
199900
200001 200102
200203
200304
200405 Average of weeks
200506 OctoberNov. (Provisional)
167 196 225 . 232 259
(Base 198182 = 100)
165 189 214 246 278
175 171.1
190 191.8
214 217.8
231 233.1
254 258.3
121 125 136 142 153 159 162 168 178 181 183
198
199
Base 199394 = 100)
109
115
130
148
153
193
223
231
256
263
290
313 312
117 ■ 116.9
123 122.2
126 128.8
129 134.6
135 141.7
139 150.9
144 159.2
144 161.8
152 172.3
162 180.3
169 189.5
172 197.8
173 198.3
Statistics for EconomicsXI
Uses of Wholesale Price Index Numbers
From the above Table 1 of Wholesale Price Index Number obtained from Economic Survey, 20052006, let us understand the uses of WPI under the following heads:
1. Price trends in India
2. Measuring rate of inflation
3. Forecasting future prices
4. Estimation of demand and supply
5. Determining real changes in aggregates
6. Uses in planning
1. Price trends in India : Ever since independence the price trends in India have varied
between sharp to moderate increases. With the exception of some years of the First
FiveYear Plan, viz., 195253 and 195455 when prices showed a moderate decline, almost
the entire period of over five decades since 195051 has shown persistent rise in prices.
molesale Prices. The rising trend in wholesale prices, as shown by the Wholesale
, bas continued ever since 196061, but it assumed alarming dimensions since
197273 after the first oil shock of 1973 when" OPEC nations affected a manifold rise
in oil prices. OPEC again increased petroleum prices in 1978 that adversely affected our
To^o'?! ^ u ^^^ """^ber of wholesale prices, with
197071 as base 100) increased to 175 in 197475 and further to 256 in 198081 thus
showing two and a half fold .increase in price in just one decade. The base year for WPI
was changed to 198182 = 100 under the new index which rose to 258.3 in 199394
showing another two and a half fold rise in price in a little over one decade The base
year was again changed to 199394 = 100 under the current series of Wholesale Price
'''''' P"ce level between
199394 and 200405. The Wholesale Price Index stood at 189.5 in 200405. Table 1
shows the movement of wholesale prices of various commodity groups since 198687
2 Measuring rate of inflation : WPI is used to measure the rate of inflation. The rate of inflation is useful to know the real value of income, savings and wealth etc. Using WPI of 200304 and 200405 for all the commodities from the table given above, the rate of inflation can be calculated as under :
Rate of inflation =
WPI of current year WPI of previous year
XlOO
100
= [189.5 " L180.3 = 5.1%
or Rate of inflation = WPI of current year  WPI of previous year' XlOO
WPI of Picvious year
"189.5180.3' 180.3 X 100
= 5.1%
Introduction to Index Numbers
385
Thus, the annual inflation rate during 200405 was 5.1% in case of all commodities. One can also calculate inflation rates for different commodities or commodity groups as required for policy purposes.
Economic Survey 2005' Annual pointtopoint inflation rate in terms of
the Wholesale Price Index (WPI) increased from 4.6 per cent at end March 2004 to 5.1 per cent at end March 2005. The year 200506 started with an inflation rate of 5.7 per cent on April 2, 2005, which was followed by a softening trend until August 27, 2005 when it reached a trough of 3.3 per cent. While the rate rose steadily thereafter, it remained below 5 per cent. At 4.5 per cent on January 21, 2006 it was significantly lower than 5.4 per cent recorded a year ago. Average WPI inflation decelerated from 10.6 per cent in the first half of 1990s to 4.7 per cent during 200102 to 200405.
3. Forecasting future prices : From the above time series data of WPI understand that the wholesale price level has increased in 200405 for primary articles by 83%, for fuel power, light and lubricants 191%, for manufactured products 69% and for all commodities by 89.5%. Thus, WPI can be used to forecast the increase in future prices.
4. Estimation of demand and supply : One can use an appropriate model to estimate the future demand and supply as the prices affect both the demand and supply. WPI therefore is useful for analysing and forecasting trade situations by interpreting the present trend in supply and demand conditions.
5. Determining real changes in aggregatives : WPI are useful to determine the real changes in aggregates like, national income, national expenditure, capital formation etc. National income is defined as the value of goods and services produced in a certain year. National income at current prices can be obtained after calculating the value of goods and services according to prices prevailing in the same year.
The real change in national income can be calculated as given below :
Real Change of National Income
WPI of Base year WPI of current year
X National income of current prices
For example, suppose the national income of the country in 2001 on the basis of current year prices amounts to Rs 700 crore which is increased to Rs 780 crore in 2002. Suppose the WPI increased to 150 in the year 2002 as compared to 2001 WPI as 140. The real change in national income can be calculated as :
= 11^x780
150
= Rs 728 crore
Here, the real increase in national income of Rs 28 crore (728  700). while actual monetary increase is Rs 80 crore (780  700).
An increase in the national income at the current prices may be due to :
(a) an increase in the general price level, or
(b) an increase in the real output.
386
j„ . Statistics for EconomicsXI
amotJ^rerpeSra^d'tr^^^^^^^^^ lajches number of projects wh.ch require huge for rhese profecrs in .rs I'a, bVd.e " " " " P"™'™
year due to rising costs of nroiecr t i cannot be same every
projects. The origfnal e«imatKo« ofT' the real cost of such
indicated by Wpf is con rred Ss tfelTe „7 '' ^
to revise the annual cost of projects On this b'^^^^^ " governmem
of funds for various schemLr^
Inflation and Consumer Price Index in India (CPI
r ^J  as services
labourers or nonn^JTurlTZXye^
Workers (CPIIW Base 1982 . 100r£s cha^.e I"dex for Industrial
Changes in cost of living of rur 1 aretfl^^^^^^^^^ ^^ ^tnal workers.
(CPIAL Base 198687 f 100) while CW for TIrt m "" ^S"cultural labourers
does It for urban nonmanulltorkS
measure of price rise of inflation and is used for de^erm nL ^ '' considered a good governmem employees as well as otLHlr T ^ allowance (DA) of
monthly basis and is availabTe after Tw of r " ^ ^^^ ''
and agriculture labourers are publ^sheJ bVLa^^^^^
Organisation published the CpCmber of lib. ^^^ Statistical
because their typical consumptrbtll^o^^^^^^^^^^
groupAslivt trZ^^rJ^TtJ^''  'y — —ty
the food price will have gSam ™ ^^^^^^ the rise in
price increase will not be inflationary Government gives the statement as oil
Table 2
Major Group
1. Food
2. Pan, Supari, tobacco etc.
3. Fuel and lighting
4. Housing
5. Clothing, bedding and foot wear
6. Misc. group
Base 1982 (Weight in %)
Grand Total
57.00 3.15 6.28 8.67 8.67 16.36
100.00
Base 2001 (Weight in %)
86.19 2.27 6.43 15.27 6.58 23.26
100.00
Source . Economic Survey, 200506 (p. 87) '
thus^ea^'res t ST^m ^th'atg Sr of""^ — ^^
that the consumers pa^. The''cotrrC'^'drS
Introduction to Index Numbers
387
commodities and regarded as an index of changes in cost of hving of industrial workers This index shows that there has been a fivefold rise in retail prices and cost of living of industrial workers since 1982. This index has gone up to 548 in Oct. 2006 as against 100 in base year 1982. General Index for Urban NonManual Employees showed over fourfold rise between 1982 and Oct. 2006. Thus, the impact of rising prices on urban nonmanual employees was a little less than that on tbe industrial workers. Prices in rural areas also showed a rising trend but the extent of rise in prices this case was lower than that in urban areas as is indicated by the the general index for agricultural labourers. Table 3 shows the movement of prices in India as shown by the All India Consumer Price Index.
Table 3
All India Consumer Price Index Numbers
Industrial Workers Urban NonManual Agricultural Labourers
(Base 1982 = 100} Employees up to 199495
(Base 198485) Base 196061 = 100 199697 onwards
Last Month = 100)
of Food General (Base 198687 = 100)
Index Index General Index General Index
198687 141 137 115 572
198788 154 149 126 629
198889 168 163 136 708
188990 177 173 145 746
199091 199 193 161 803
199192 230 219 183 958
199293 254 240 202 1076
199394 272 258 216 1114
199495 304 284 237 1204
199596 337 313 259 234
199697 369 342 283 256
199798 388 366 302 264
199899 445 414 337 293
199900 446 428. 352 306
200001 453 444 371 305
200102 466 463 390 309
200203 477 482 405 319
200304 495 500 420 331
200405 506 520 436 339
Oct. 06 538 548 460 356
Source : Economic Survey, 20052006 (p. S63).
High inflation hurts the poor with their incomes not indexed to prices. It also puts pressure on interest rates, and adversely affects both savings and investment. Because of its implications for the poor and its possible destabilizing effects on macro economic stability, containment of inflation is high on the Government agenda.
388
Statistics for EconomicsXI
Causes of Rising Prices (Inflation)
in India, we have to look^tn L tr t understand the various price or inflation
Demand Side Factors
1. Increase in Money Supply
2. Faster Growth in Money Supply than the Growth Rate of National Income
3. Massive Increase of Government Expenditure
4. Deficit Financing
5. Growth of Black Money
6. Increased Wages and Salaries Supply Side Factors
1. Slow Growth of Agriculture
2. Slow Pace of Industrialisation
3. Increase in Petroleum Prices
4. Changes in Administered Prices
5. Wage Price Spiral (i.e., workers demanding higher wages)
limitations of index numbers
4
Introduction to Index Numbers
> list of formulae
1. Unweighted Simple Aggregative Method
Price Index p  ^ x 100
^Po
Quantity Index ^ = ^ x 100
y V.
Value Index V^j = ^ x 100
evq
^ im. X100 Zpo%
2. Unweighted Simple Average of Price Relative Method
X^xlOO VPo J Po. = 
3. Weighted Aggregative Method : Price Index
A. Laspeyre's Method = ^^ x 100
B. Paasche's Method p.. = ^^ x 100
ZPoqo
4. Weighted Average of Price Relative Method
Z ^xlOO x(Poqo) [Po J
Poi =
389
Quantity Index
. ^ ^ X 100
ZqoPo
, 100
^oPi
Consumer Price Index
A. Aggregative Expenditure Method or Aggregati
CPI = ^ X 100
B. Family Budget Method
CPI =
Index of Industrial Product
ZPV Method
:ive
'ZWR " or "XW7"
_ XW .XW.
Industrial Production Index No. =
iL ZW
W
390
j„ . Statistics for EconomicsXI
exercises
Questions :
1. Distinguish between actual difference and relative difference in prices. 2 Define index numbers. Why do we need an index number.?
3. What are the problems in construction of index numbers?
4. Briefly explain the importance of index numbers in the study of Economics.
5. What is index number? Discuss briefly the uses of index numbers.
6. What is the difference between a pure index and quantity index?
7. State the general uses of index numbers.
8. What are the desirable properties of the base period?
9. Distinguish between 'weighted' and 'unweighted' index of prices?
10. Is change in any price reflected in a price index?
11. Distinguish between Laspeyre's method and Paasche's method of constructing index number.
What does consumer price index for industrial workers measure?
Define Consumer Price Index number. Explain the uses of consumer price index numbers.
What are the uses of Wholesale Price Index numbers?
15. Explain Index Numbers of Industrial Production.
16. Distinguish between 'Wholesale Price Index' and 'Consumer Price Index'.
17. Why is it essential to have different CPI for different categories of consumers?
18. Discuss the limitations of index numbers.
19. Can CPI number for urban nonmanual employees represent the changes in cost of hvmg of President of India?
20. What do you mean by inflation? How the wholesale price index numbers are useful for measuring the rate of inflation?
21. Try to list the important items of consumption in your family.
22. Write short notes on :
(a) Base year Index of Industrial Production
c Value Index (J) Consumer Price Index
; , (e) Wholesale Price Index.
Problems :
1. Construct the Index Number for 2002 with 2001 as base from the following prices of commodities by simple (Unweighted) aggregative method. Commodities : A B CD E
12.
13.
14.
Prices in Rs 2001 Prices in Rs 2002
50 80
40 60
10 . 5 2
20 10 6
[Index Number = 164.48]
Introduction to Index Numbers
391
2. Using the following data and 2002 as the base period, compute simple aggregative price indices for the two fuels.
Item Producers Price     " c ;
V  2000 2001 2002
Coal (Rs) 5 3 4
Crude oil (Rs) 2 3 4
5.
[Index Number : 2001 = 85.71, 2002 = 114.28]
3, Calculate the index number for 2002 with 2001 as base from the following prices of the commodities by simple (unweighted) aggregative method.
Commodity Price Price
and unit ' (2001) (2002)
Butter per kg 20.00 22.00
Milk per Litre 3.00 4.50
Cheese per Tin 18.00 19.80
Bread per kg 2.00 3.80
Eggs per Dozen 4.00 4.50
[Index Number = 116.47]
4, Calculate Quantity Index Numbers from the following data by simple aggregative method taking quantity of 1998 as base.
Commodity Quantity (in tons)
1998 1999 2000 2001 2002
A 0.30 0.33 0.36 0.36 0.39
B 0.25 0.24 0.30 0.32 0.30
C 0.20 0.25 0.28 0.32 0.30
D 2.00 2.40 2.50 2.50 2.60
(Quantity Index No. = 117.1, 125.1, 127.3, 130.5) Calculate index number for 2002 on the base prices for 1991 from the following by average of price relative method.
Items Prices (1991) Prices (2002)
6.
Bricks Timber Plaster Board Sand Cement 10 20 5 2 7
16 21 6 3 14
[Index No. = 147]
Construct the index number for 2000 taking 1990 as base by price relative method using arithmetic mean.
Commodities .A B C D
Price (1990) : 10 20 30 40
Price (2000) : 13 17 60 70
[Index No. = 147.5]
392
(between H0 tn a u u Statistics for EconomicsXl
1993
1994
1995
1996
1997
(in Rs)
75 50 65 60 72
1998
1999
2000 2001 2002
Wholesah Prices (in Rs)
7« 69 75 84 80
wholesale prices in India for second week of Sept. 2002 and L th'e wLr'"^'' ^^^ ""ber of wholesale prLes
Weights Index
Food Article Manufactures Industrial Raw Material SemiManufactures Miscellaneous 31 30 18 17 4473.6 390.2 510.2 403.3 624.4
Q 1 1 ... [WPI = 449.2491
m^hor Tt^ ^^^^ data by weighted aggregative
method using : a Lasoevre's method ^ u aggregative
Commodity Price (2001) Quantity (2001) Price (2002) Quantity (2002)
A B C D 4 3 2 5 20 15 25 10 6 5 3 4 10 23 15 40
in v^r^rr. ^u J . , . L^^^f^j^o . XJ/.//, I'aascncs : 158.99]
aZtitv in? ^"^u price index and quantity index numbers wnli onoi__j 
quantity index numbers with base 2001 and interpret.
2001
Cummodtty
A B C
Pnce
_
4
3
Quantity
2002
Price
s
2 5 2
_
6 2 4
Quantity
3 1
6
[Laspeyre's : Price Index = 76.92, Quantity Index = 143 18Paasche's : Price Index = 69.84, Quantity Index = 130]
Introduction to Index Numbers 415
11. Calculate weighted aggregative of actual price index number and quantity index number from the following data using (/) Laspeyre's Method, and {ii) Paasche's Method. Also calculate value index number and interpret them.
Commodity Base year Current Year
Quantity lbs. Price per lb. Quantity lbs. ' Price per lb.
Bread Meat Tea6 4 0.5 40 paise 45 paise 90 paise 7 5 1.5 30 paise 50 paise 40 paise
[Index Number = (/") 86.02, (ii) 81.25]
Commodity Price Base Year (in Rs) Price Current Year Quantity Base Year (in kg)
A 6.0 8.0 40
B 3.0 3.2 80
C 2.0 3.0 20
[Index Number = 122.3]
13. Prepare consumer price index numbers from the following data for 2000 and 1999 taking 1998 as base.
Group 1998 1999 (Price in Rs) 2000
A 20.00 24.00 21.00
B 1.25 1.50 1.00
C 5.00 8.00 8.00
D 2.00 2.25 2.12
[Index numbers, 1999 = 127.25, 2000 = 107.43] From the data given below construct the consumer price index number
Commodity Price Relatives Weights
Food 250 45
Rent 150 15
Clothing 320 20
Fuel and Lighting 190 5
Miscellaneous 300 15
[Index Number = 253.5]
UNIT 4
DEVELOPING PROJECTS IN EC
Chapter 13
PREPARATION OF A PROJECT REPORT
Introduction
2. Uses of Project Report
3. Consumer Awareness
4. Questionnaire for Dealers , 5.. Productivity Awareness
In the previous Unit 1, we have studied the Meaning of Economics; Scope and Importance of Statistics in Economics; in Unit 2, Collection and Organisation of Data; and in Unit 3, About the Various Statistical Tools. These tools are Very important in our daily life to analyse different economic activities such as consumption, production, distribution, transport in land and foreign trade and different business activities. In this chapter we will learn the method of developing a project report which will help us in understanding the application of statistical tools to analyse the various types of business activities.
Reports are prepared to give information about the development of institution, business, product, government activities etc. For example,
1. Consumer may be interested in knowing the quality, price and uses of product in changing environment and technology, e.g., preference for landline phone or mobile phone,^ detergent powder or detergent cake, fully automatic or semiautomatic washing machine, etc. Such surveys are conducted by manufacturing organisations.
2. Shareholders may be interested to know about the earning of organisation and possibility of getting dividend while holding the shares of the company. Such surveys are conducted by nongovernment organisations, societies, etc.
3. Central/state governments prepare reports for future development in priority areas such as road, power, teleconununication, education, health, etc. For example, for this purpose the government conducts surveys to know about likely requirement of primary health centres and schools for basic
education. Similarly government decides the requirement of power (Mega Watts), roads to construct in the light of changing population of a respective area.
4. Reserve Bank of India plans the opening of new branches of commercial banks, cooperative banks or agricultural banks in the light of increasing credit requirement of population on the basis of survey reports. Chamber of Commerce, namely.
Preparation of a Project Report 395
Federation of Indian Chamber of Commerce and Industry (FICCI), Confederation ofIndian Industry (CII) conduct surveys of abroad to know the business opportunities arising out of economic development of respective nation.
5. In the international context, United Nations Organisation (UNO) plans humanitarian help (food, hfe saving drugs, etc.) in war, drought, earthquakes and such other natural calamities based on survey reports.
In the light of above examples it is very clear that project reports help in understanding the requirements of shareholders, consumers. Central and State Governments, Reserve Bank of India and financial institutions and national and international bodies to plan their activities for future operations. Those organisations who ignore the changing requirement of the consumers or population may fail in achieving their goals and objectives.
uroject
Uses of project report can be highlighted as under :
1. To make aware individual groups about the present environment conditions of business/government, etc.
2. To help in the pohcy formation about the economic and social development of the country.
3. To direct the efforts of organisation in given objectives based on opportunities provided in the changing environment.
4. To pinpoint the weaknesses of organisation so as to overcome such weaknesses.
5. To pay competitive prices for irequired goods by the consumer to take the real value of the price paid to sellers.
6. To invest in those securities that provides higher rate of interest/dividend to shareholders.
7. To exploit opportunities in the national and international markets by trade associations:
8. To provide food, medical help to badly affected areas due to any natural calamities by national, international, social and nongovernment organisations.
9. It helps in conducting research on various issues such as political, social, economical, technological aspects of national and international significance.
^^imers iuvareni
Consumers may be exploited by manufacturers, government agencies, board of directors and national and international agencies, e.g., manufacturers charge higher price, provide poor quality, lesser weight, defective product, etc., to the consumers. The Indian Consumer Protection Act, 1986 has provided various rights to the consumers, such as right to basic needs, safety, choice, information, education, redressal, representation and healthy environment. Any consumer is exploited on this ground can approach to the appropriate authorities to seek compensation or replacement of goods. For this consumers may be made aware about their rights and informed about proper agencies, which they can approach for grievances.
^^^ Statistics for EconotnicsXl
There are five steps in preparing a project report for consumer awareness :
1. Identification of Problem
2. Preparation of Questionnaire
3. Collection of Data
4. Analysis and Interpretation
5. Conclusion
Identification of Problem
We want to know about consumers'/dealers' knowledge about the product of a company manufacturing namely, colour TV., airconditioner, washing machine, refrigerator, car, scooter, computer etc. Let us take the example of airconditioner where we are interested to know from dealers about the performance of airconditioner with respect to price, cooling technology, quality, availability, warranty, after sales service etc. keeping in view other airconditioners' manufacturers product available in the market in competition.
Preparation of Questionnaire
To know more about various aspects of airconditioner in a more systematic manner, we must design a questionnaire covering all the aspects discussed above.
questionnaire for dealers
Name:.....
Address :. Phone No.
Q. 1. Please recall some aircondition brand name :
W ........................................... {ii) ......................
{Hi) ............................................ {iv) ......................
Q. 2. Which brands of AC's do you currently deal in :
(/■) Videocon (//) Carrier
(f) Samsung {viii) Others
{iv) National {vii) Voltas
{Hi) Amtrex {vi) LG
Q 3. Which brand AC would you recommend to the customer? (rank them) (1Best, 8Worst)
(/) Videocon
{iv) National {vii) Voltas
{ii) Carrier {v) Samsung {viii) Others
{Hi) Amtrex {vi) LG
Preparation of a Project Report
Q. 4. Reason behind above recommendation (rank the factors) : (1Best, 8Worst)
(ii) Quahty (Hi) Performance
(v) After sale service (viii) Warranty
397
(vi) Authentic
(i) Price (iv) Availability (vii) Technology
Q. 5. Which is most demanding brand of AC. (mention the name)?
Name .............................................................................................................
Q. 6. Rank the customer conferences while buying an AC. (1Best, 8Worst)
(ii) Quality (Hi) Performance
(v) After sale service (viii) Warranty
(vi) Authentic
(i) Price (iv) Availability (vii) Technology Q. 7. Does the brand name influence the customer?
Yes ............... No................ Some time...................
If No, then what else influences him (specify) :
Q. 8. Which brand of AC has least customer complaints (mention the name)?
Name : ............................................................................................................—
Q. 9. Rank the companies in regularity of supply (1Best, 8Worst) :
(iv) National
(/■) Videocon (ii) Carrier (Hi) Amtrex
(v) Samsung (vi) LG (vii) Voltas
(viii) Others
Q. 10. Which AC company provides the highest margins to their dealer?
Name of the company : .........................................................................................
Q. 11. Which AC company do you feel is the most aggressive in giving discounts and scheme (please specify)?
Name of the company :....................................Specification :......................................
Q. 12. Do you agree that huge advertisement campaigns are the most responsible factors for the changing market scenario and increasing demand?
(i) Agree very strongly (Hi) Agree (v) Disagree (vii) Don't know
Q. 13. Generally what short of problem do you face while doing a sale? Specify : .......................................................................................................
398
j„ . Statistics for EconomicsXI
Q. 14. Which brand of AC in your opinion is having most advance technology?
Name the brand : ...................................
Q. 15. Warranty period offered by the companies (kindly tick) :
Brand name
(i) Videocon
(ii) Carrier
(iii) Amtrex
(iv) National
(v) LG
(vi) Samsung
(vii) Voltas
(viii) Others
Warranty
6 month1 yr.
12 yrs.
23 yrs.
3 yrs. and above
Q. 16. What is the market size of the area you are dealing in'
(i) 01000 Machines —
(ii) 10002000 Machines (iv) 30004000 Machines
(iii) 20003000 Machines (v) 4000 and above
Q. 17. Average No. of units sold per month from your counter.
(Please specify brandwise)
(i) Videocon fl («) Carrier Q (Hi) Amtrex
(v) LG [j^ (vi) Samsung
(viii) Others
(iv) National (vii) Voltas
n 1 Brand name ___ Window —» '/f*"" ^^^ uiaiiu. Split
Lastyr. Projected Last yr. Projected
(/■) Videocon
(ii) Carrier
(iii) Amtrex
(iv) National
(v) LG
(vi) Samsung
(vii) Voltas
(viii) Others — 
Preparation of a Project Report 399
Q. 19. Please give the sales breakup for the month of April, May and June in last three years :
{For the brand which you deal in)
Company name No. nf sales (Monthwtse)
200 i 2004 2005
May Apr. May June Apr. May Jme
(i) Videocon
(ii) Carrier
(Hi) Amtrex
(iv) National
(v) LG
(vi) Samsung
(vii) Voltas
(viii) Others
Q. 20. To get a substantial growth in your present sale which of the following would you prefer?
(/) Having a better brand name
If, name the brand ......................................................................................
{ii) Enhancing your infrastructure and sales persons' team.................................
Collection of Data
The above questionnaire with the help of investigators using sampling method will be filled in by the dealers. The number and geographical areas depend upon our requirement, where we want to position our product, namely, Delhi, Kolkata, Chennai and other capital cities of states.
We can also collect the information from government and industrial publications to know about the growth of airconditioner industry and future government policy in this respect.
Analysis and Interpretation
Data collected through questionnaire will be classified and presented in the form of tables, graphs and diagrams, viz., bar diagrams, piediagram etc. For rigorous analysis
400
Statistics for EconomicsXI
standard deviation and eoeffident oTva^t ^ OnV^K"" ""
also pro,eet fntnre demand tWongh ^ 
Blustration: Table and dtagram (based on hypothetiea. data, are g.ven beiow ,
Table 1
Consumer Awareness about Airconditioners
Atvareness
Brand Present Availability Price
After Sales Service Technology
Conclusion
(U
O <
ai o cr
UJ CL
consumer awareness about airconditioners
Scale : 0.5 cm = 10 percentage on Vaxis
Brand
Present Availability
Price
After sales Services
xxxxxxv xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx
Technology
E3 Videocon □ Amtrex H Samsung EHJ Voltas m Carrier g National
m LG
^ Others
Observation
obse?ve bar diagram, „e
of view of : ^ customers prefer to buy airconditioner from the point
1. Brand : Either Vidiocon or Samsung
2. Present AvatlahtUty : Voltas or other brand
4
Preparation of a Project Report 401
3. Price : LG or Videocon
4. After Sales Service : LG or Videocon
5. Technology : Videocon or LG
Thus, the Airconditioner company will come to know about the brand, present availability, price, after sales service, technology etc. Through this observation the company will be in a position to decide regularity of supply the number of units to be produced and to improve after sales service as per requirements of future consumers.
Analysis
Let us analyse the given data by applying different statistical tools (Mean, Standard deviation and Coefficient of Variation) using the following formulae :
1. Mean :
N
2. Standard Deviation :
a =
nxxf
N
N
3. Coefficient of Variation (C.V.)
100
a
= J "
Table 2
Consumer Awareness about Airconditioners
(Figures in percentages)
Awareness Name of Companies
Videocon Cjirrier Amtrex National Samsung LG Voltas Others
Brand 24 3 8 10 20 18 8 9
Present Availability 12 9 7 11 14 13 17 17
Price 25 10 6 5 12 28 9 5
After Sales Service 18 12 6 6 17 21 12 8
Technology 22 10 7 8 12 17 14 10
IX 101 44 34 40 75 97 60 49
Mean : X % 20.2 8.8 6.8 8 15 19.4 12 9.8
Observation
Considering brand, present availability, price, after sales service and technology, average percentage of customers of Videocon airconditioner is the highest as 20.2% and hence will prefer to buy Videocon airconditioner.
402
Videocm Carrier Af
Jff fX X (X~ X
X Si/^ R
+3.8 14.44 5.8 33.64 +1.2
.8.2 67.24 +0.2 0.04 +0.2
M.8 23.04 +1.2 1.44 0.8
2.2 4.84 +3.2 10.24 0.8
+1.8 3.24 +1.2 1.44 +0.2
Z(X 
XV 112.8 46.8
a 10.62 6.84
C.V 52.57 77.72
Thus, we get
j„ . Statistics for EconomicsXI
Calculation of Standard Deviation and Coefficient of Variation Name of Companies
1.44 0.04 0.64 0.64 0.04
2.8 1.67 24.56
+2 +3 3 2 0
4
9 9 4 0
26 5.099 63.74
+5 1 3 +2 3
  — LG Vbit» t Othm 1
X  1 /X  XX flfX)^ X X jf/'
M
25 1.4 1.96 4 16 0.8 0.64
1 6.4 40.96 +5 25 +7.2 51.84
9 +8.6 73.96 3 9 4.8 23.04
4 +1.6 2.58 0 0 1.8 3.24
9 2.4 5.76 +2 4 +0.2 0.04
48 125.22 54 78.8
6.93 11.19 7.35 8.88
46.2 57.68 61.25 90.61
Name of Companies
Videocon
Carrier
Amtrex
National
Samsung
LG
Voltas
others
Mean
»
X
20.2
Standard devtatton
8.8 6.8 8 15 19.4 12 9.8
10.62 6.84 1.67 5.099 6.93 11.19 7.35 8.88
Coefftcient of Variation
52.57
77.72
24.56
63.74
46.2
57.68
61.25
90.61
Observations
ruJr!'^ of variation is the highest for other brands as 90.61%, hence the
customers will not prefer to buy other branded airconditioners.
Requirement
to gyJhTLT'"''^ dealers/consumers to fill in the questionnaire
raTned\^o' " ^^ ^he information
reoZd Ar Potential dealers/customers, they may be asked to prepare required tables, graphs and diagrams etc.
4. Further, students should be asked to analyse and interpret the data collected by
5. They may also suggest the future course of action for the company.
r
Preparation of a Project Report
403
ivity
Productivity is the ratio between input and output of an organisation. Productivity varies from company to company. For example, X company manufactures a colour T.V. for Rs 10,000, while Y company manufactures the similar T.V. for Rs 11,000. In this case X company is more productive than Y company because X company's manufacturing cost of colour T.V. is less by Rs 1,000. Therefore, we say X company is more productive than Y company. In addition to this productivity, we can also be able to calculate productivity of different factors of production such as labour, capital etc. For example, the cost of labour of X company to manufacture colour T.V. is Rs 3,000 and that of Y company is Rs 2,500. In this case the labour productivity of Y company is better, although overall productivity of X company is better as compared to company Y.
Productivity is determined with internal and external factors. Internal factors are technology, organisation structure, managerial ability, ability of the firm to substitute different inputs, etc. External factors include growth of agriculture and industrial production, price, growth of bank deposits and credit, composition and growth of GDP, structure of foreign trade, savings and capital formation etc.
4
Identification of Problem
We want to know productivity awareness amongst the enterprises of the following economic problems. We can identify the problems like :
[a) Industrial production
{b) National budget
(c) Population growth
{d) Gross national product
(e) Financial assistance by All India Financial Institutions
Collection of Data
Different ministries and departments of Central and State Governments publish regularly current information alongwith statistical data on the number of subjects. This information is quite reliable for related studies. We can collect data about identified problems from Newspapers/Economic Surveys/RBI Bulletin/Government Budget of the State or the Nation/Census Reports/NSS Reports/Annual Survey of Industries/ Labour Gazettes/Agriculture Statistics of India/Indian Trade Journals etc.
Statistical Tables and Analysis
Following are few illustrations for analysis.
404
Illustration 1.
P , Statistics for EconomicsXI
Table 1
Annual Growth Rates of Industrial Production in Major Sectors of Industry
(Base : 199394 = 100)
Period
Weights
199596
199697
199798
199899
199900
200001 200102
200203
200304
200405
200405 (AprilDec.)
200506 (AprilDec.)
Mining and Quarrying
10.47
9.7 1.9
6.9 0.8 1.0
2.8 1.2 5.8 5.2 4.4 5.1
0.4
(In per cent)
Manufacturing Electricity Overall
79.36 10.17 100.0
14.1 8.1 13.0
7.3 4.0 6.1
6.7 6.6 6.7
4.4 6.5 4.1
7.1 7.3 6.7
5.3 4.0 5.0
2.9 3.1 2.7
6.0 3.2 5.7
7.4 5.1 7.0
9.2 5.2 8.4
9.2 6.4 8.6
8.9 4.8 7.8
economic survey : 20052006 (p 132) loofotconr' ^'^^very that commenced from the second quarter of
per cent compared ,o a gro«h of 8.6 per Li ,n Ae^Xondmg^erd'of'zm^^^^^ Impressive performance of the mannfacuring sector which grew at 8 9 n!
mwsmmsm
^ ... iiivcstmen
capacity additions and contributed to this shortage
.vear^mctde" during the current
H) normal business and investment cycles, (//■) lack of domestic and external demand'
Preparation of a Project Report 405
(Hi) lack of reforms in land and labour markets,
(iv) bigh oil prices,
(v) existence of excess capacity in some sectors,
(vi) business cycle,
(vii) infrastructure bottlenecks particularly power, roads and transport,
(viii) continuing high real interest rates.
Illustration 2.
Table 2
Trends in Deficit of Central Government
(As per cent of GDP)
Year Revenue Primary s 1 , Bscat
Deficit befkit y Deficit
199091 3.3 2.8 6.6
199192 2.5 0.7 4.7
199293 2.5 0.6 4.8
199394 3.8 2.2 6.4
199495 3.1 0.4 4.7
199596 2.5 0.0 4.2
199697 2.4 0.2 4.1
199798 3.1 0.5 4.8
199899 3.8 0.7 5.1
199900 3.5 0.7 5.3
200001 4.0 0.9 5.6
200102 4.4 1.5 6.2
200203 4.4 1.1 5.9
200304 3.6 0.0 4.5
200405=^ 2.5 0.6 4.1
(Provisional)
200506 (BE) 2.7 0.5 4.3
1
* Provisional and unaudited as reported by Controller General of Accounts, Department of Expenditure, Ministry of Finance.
Notes: 1. The ratios to GDP for 200506 (BE) are based on CSO's Advance Estimates GDP at current market prices prior to 19992000 based on 199394 series and from 19992000 based on new 19992000 series. 2. The fiscal deficit excludes the transfer of States' share in the small savings collections.
Source : Budget Document, Economic Survey—200506 (page 24).
Anlaysis : The Fiscal Responsibility and Budget Management Act (FRBMA), 2003 continued to provide a strong institutional mechanism for making sustained progress at
406
Statistics for EconomicsXI
demand on «t^tt^lr^^^^^ a proportion of GDP, declined from 6.6 per cen, t itsTsi o " cSrS^^^^^
A^dr^i^ l^rov.;, leading to a marked improvement m the quahty of deficit Th^
available ar,ha, ^  <>' G^P
Budeet for 7005 n^ u.^ ' 7 , ^ ^ P^'" cent, respectively. The
..dj, 5S.S
Requirement
2. TTiey can also be asked ro make presentation of snch problems
ItotTr" " ""rpretation of data of the stable they have
1 t