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    Purposes Of The Term Paper

    Every study has some certain objectives for which the study is made. So identifying the objectives

    is very much important. My objectives of preparing this report are:

    To identify basically what the sampling is. To gather the practical knowledge by surveying this


    To identify the various sampling methods for problem solving.

    To find out the advantages and disadvantages of these methods

    Showing some business related problems on sampling.

    Methodology Of The Term Paper

    Basically there was no organized and definite methodology in preparing this term paper. All the

    given data are from class lecture and book reference. While facing problem, we (GL and AGL)

    rushed to our dear able course teacher to get suggestion. Then on the basis of his guidelines we

    prepared a submitted on due time for the sweetest favor of our honorable course teacher

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    Executive Summary

    The term paper is a fruit of six months semester. Its value is as fulfillment of course. We got this

    opportunity from our honorable course teacher Md. Mehedi Hasan Sikdar. The main objective of

    this term paper is to know details about Sampling and its application in business. We have tried to

    make it as favor of our course teacher. At the present time sampling is an important matter or part

    of statistics. A statistic is a characteristic of a sample. Statistics use the word population to refer

    not only to people but to all items that have been chosen for study. Statistics use the word sample

    to describe a portion chosen from the population. Finally we hope that this term paper will help us

    as business studying in future in our business field as well as daily life.

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    Surprised! Dont be surprised. Not only for this data sheet, but also for millions of data sheet; one

    technique or way is appropriate in reaching a decision, if anyone goes through this way. This way

    is Sampling. All of us know well that this popular method is used by our mothers or sisters while

    they testing their boiling rice by taking two or three pieces of rice among the whole bowl of rice.

    Avoiding thissampling technique, if they test all the boiling rice one by one then a huge time will

    be consumed for this reason and it is unnecessary. Basically to determine the characteristics of the

    whole, we have tosample only a portion.

    Sampling Terms

    10158230 41256342 25369855 12365056 12353585 12365056 41023688 12035524

    12365056 42536252 12365589 32065820 12301475 32065820 12365056 10236744

    32065820 12356952 12301475 47523666 12365056 11023658 32065820 12365056

    12365056 12365056 12365056 10236557 32065820 14126585 12301475 32065820

    66568464 32065820 32065820 12365056 10236552 41023688 12365056 41023688

    55675364 12524585 10236552 32065820 41236554 12365056 32065820 12365056

    75645288 12365285 41023555 45648521 12354447 32065820 10236552 32065820

    77566333 41265888 41236685 10235522 41023557 10236557 41035400 10236557

    12345636 20310236 10236542 12036952 02123252 20758569 42103256 02012536

    25836925 25332455 21220452 02214522 10236985 52302598 58236512 21546254

    78945612 20147852 20123656 02145223 85596965 12302536 20987456 12036985

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    In order to clarify the concept sample and sampling technique, it is necessary to define and discuss

    certain relevant terms:


    Statistical investigations usually aim at the assessment of general magnitude and the

    study of variation with respect to certain characteristics of the individuals belonging to a group.

    Such a group of individuals under study is known as population. Like all the farmers, students,

    teachers, birds, total agricultural land etc. may constitute a population.


    A sample is small representative fraction of a population. For example, in order to investigate

    certain characteristics of all the farmers of the country, some farmers are selected to collect the

    necessary data; selected farmers constitute a sample of the population of the farmers. A smallquantity of blood, not the whole, is collected for testing; the blood is a sample where the total

    quantity of blood of person is the population.

    Sample size:

    The number of elements selected for a sample is known assample size. A sample of size less than

    30 is termed as small sample size and that having 30 or more elements is termed as a large sample



    If data are collected on all the elements of a population, the process is known as census like

    population census, agriculture census etc. detailed information on all the citizens of a country are

    collected usually in every ten years through population census. Bangladesh Bureau of Statistics

    (BBS) is the government organization who conducts the population census, agricultural census,

    and other nation-wide record.

    Pilot Survey:

    Small scale surveys are sometimes conducted in order to get quick primary information before

    census. Such a survey is known as pilot survey.

    Statistic and Parameter:

    Statistic is characteristics of a sample and parameter is a characteristic of a population.

    Sample Survey

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    Sample survey

    Sample survey is the method by which detailed information on the population characteristics are

    collected on the basis of sample survey.

    Advantages of Sample Survey

    Sample survey is gaining popularity in day by day, and now being considered as a main way for

    decision making. The main advantages are given below:

    Less time, money and labor

    Through sample survey, the survey results can be obtained

    within a short time considerably less cost and labor.

    Accuracy of results

    For a complete enumeration, a large number of skilled manpower

    is not always possible; as a result errors are more probable. As a result the information

    obtained from sample survey is more accurate and reliable.

    Greater scope

    Sometimes, large number of skilled manpower and modern equipments

    required for complete enumeration may not be possible to make available. In such, cases

    sample survey is more suitable than complete enumeration.

    Large population problem

    Sometimes the population may be very large and in such cases

    sample survey is the only way.

    Limitations of Sample Survey

    Sample survey will be more fruitful if:

    (a) The sampling technique is appropriate and data are collected in scientific method.

    (b) The sample size is appropriate.

    Proper care should be taken in planning and conducting the sample survey, otherwise it

    may yield misleading results.

    Lack (if any) of trained and skilled manpower and necessary equipments at the stage of

    data collection and analysis reduces the reliability of the sample survey.

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    If information on all population units

    is required, or if the population is not very large and if there is no scarcity of time, money

    and trained manpower, complete enumeration gives better results than sample survey.

    Principal Steps In A Sample Survey

    The following are the main steps in planning and conducting a sample survey:

    1. Defining the objectives:

    The objectives of the survey should first be clearly defined. Necessary care should be taken

    so that the objectives are commensurate with the available resources in terms of time,

    money and manpower for the availability of the results of the survey.

    2. Defining the Population to be sampled:

    The population from which sample is to be drawn should be clearly defined. For example-

    if we want to select a sample of firms, clear-cit rules should be framed in order to define

    farm regarding its size, shape etc.

    3. Sampling Units:

    For the purpose of sample selection, the population should be divided into sampling units;

    the sampling units must be distinct and non overlapping so that every element of the

    population belongs to one and only one sampling unit. For example, in a socio-economic

    survey for selecting people in a town, the sampling unit might be an individual person, a

    family, or a household.

    4.Sampling Frame/ Sampling list:

    There should be a complete list of population elements from which sample is to be

    selected. Such a list which covers all the population elements is known as sampling frame.

    This sampling frame should be carefully examined to ensure that it is up to date and free

    from defects.

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    5. Collection of Data:

    The objectives of the survey should be kept in view while planning for data collection.

    Only necessary data should be collected and analyzed.

    6. Data Collection Method:

    The commonly used data collection methods from human populations are-

    (a)Interview Method:

    The investigators meet the individual respondents and collects data by interviewing on

    the basis of an interview schedule.

    (b)Mailed Questionnaire Method:

    A structured Questionnaire is prepared and mailed to individual respondents who are

    required to fill it up and send back.

    7. The Schedule or Questionnaire:

    An interview schedule or a questionnaire is prepared or a Questionnaire is prepared or aQuestionnaire requires skill, special technique and experience in the field of study.

    8. Non response:

    Sometimes it may happen that data cannot be collected from all the units in the sample. For

    example- a sample unit may not be available or an interviewer may not contact some

    respondents or some respondents may refuse to furnish information, or a mailed

    questionnaire may be somehow missing.

    9. Sampling Design:

    An appropriate sampling design is a precondition for selecting a representative sample for

    data collection. Nature of the population, variables on which data are to be collected,

    resources like time.

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    10. Administration of the Survey:

    Field workers engaged in the data collection process should be trained in identifying the

    sampling units, recording the information, data collection methods etc. before starting the


    11. Data Processing and Analysis:

    (a)Editing of the Data:

    The schedule or the questionnaire should be thoroughly scrutinized to examine the

    plausibility and consistency of the collected data.

    (b)Tabulation and Presentation of Data:

    The method of tabulation will depend upon the data size. For large scale survey,

    mechanical tabulation and presentation will be quicker and economical.

    (c) Statistical Analysis:

    Many computer packages are available; analysis of data may be done using a suitable


    (d)Reporting and Conclusion:

    A report containing the details of the different stages of the survey should be prepared.

    Errors in Sample Survey

    Two types of errors may be involved in the collection, organization and analysis of data:

    Sampling Error and

    No sampling Error

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    1. Sampling Error:

    This is originated in the sampling process and arises because only a part of the whole population is used

    to estimate the population parameters and to draw inference about the population. The sources ofsampling errors are-

    Faulty selection of the sample. Use of a defective sampling technique introduces

    some bias.

    Substitution: Sometimes investigates deliberately substitute a convenient member of

    the population for a difficult sampling unit.

    Faulty demarcation of sampling units: This type of bias is particularly significant inarea surveys, such as agricultural experiments.

    Improper use of statistics for parameter estimation: Increase in the sample size

    usually reduces the sampling error. In many situations this reduction in sampling

    error is inversely proportional to the square root of the sample size. (Illustration in

    figure below):

    Sample size

    2. Non-sampling Error


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    Non-sampling Error mainly arise at different post sampling stages

    (e.g. observation ascertainment and processing of the data). This error may be present in both complete

    enumeration and sample survey, while sampling error occurs only in sample survey. In complete

    enumeration, non sample error is the only source of error. It is difficult to ascertain the sources of non

    sampling error. However, some possible sources of non sampling errors are given below:

    Recording error

    Objectives of the survey, methods, data collection, and processing,

    analysis etc. should be properly defined. Inadequate data specification and inconsistent

    data entry, faculty recording and management of data etc. cause non sampling errors.

    Lack of training and skilled people

    Inadequate skill of data, the data collectors and supervisors may also

    cause non sampling errors.

    Response error

    A respondent, not clearly understanding a quest on, may, furnish

    wrong information and thus causing non-sampling error.

    Wrong information

    Wrong information may be furnished for personal interest of the

    respondent. Many respondents hesitate to disclose the actual income and expenditure.


    Overstatement of the respondent regarding his education, profession,socio-economic status etc. thus causing non-sampling error.

    Non response

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    Non sampling error also caused by non response. The respondent may

    not be reached in spite of repeated attempts or he may be unable to answer all the

    questions or he may decline to answer some of the questions.

    Sampling Methods

    While selecting samples from a population, special care should be taken so as to ensure the

    [presence of population characteristics in the sample. This sampling method depends on the nature

    of the data and the type of the enquiry. Basically sampling methods may be classified into twomain groups:

    1. Probability sampling/ Random sampling/

    Chance / Jance sampling

    1. Non-Probability sampling/

    Non-Random sampling

    a) Simple Random Sampling a) Judgment Sampling

    b) Stratified Random Sampling b) Purposive/ Deliberate Samplingc) Systematic Random Sampling c) Quota Sampling

    d) Cluster Random Sampling d) Convenience Sampling

    e) Multiple Random Sampling

    f) Area Random Sampling

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    Simple Random Sampling

    It is a scientific sampling method based on probability theory. In this method each population unit

    has the same probability for being included in the sample or it aloud each possible sample to have

    an equal probability of being picked and each item in the entire population to have an equal chance

    of being included in a sample.

    Advantages and Disadvantages of Simple Random Sampling:


    In simple random sampling each population element has equal probability to be

    included in the sample.

    Investigators personal bias is completely eliminated in this sampling process.

    Most representative sample can be obtained if the population is homogeneous.


    For very large population the process of numbering the population elements may bedifficult and complicated.

    If the population is not homogeneous, sample drawn in this method may not be


    If the sample units are geographically far away from each other, data collection often

    becomes expensive and troublesome.

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    It is time consuming.

    Stratified Random Sampling

    If the population is not homogeneous in respect of the characteristic under study, a simple random

    sample may not properly represent the population. In such cases, the whole population is divided

    into a number of more or less homogeneous subdivisions, these subdivisions are called strata.

    From each of these subdivisions, separate random selections of elements are made of constitute a

    sample. This method of sampling is known as stratified random sampling.The strata should be

    such that-

    1. Elements included in each stratum should be as far as possible of homogeneous nature

    2. Elements of different strata should be as far as possible of different nature.

    In Stratified Random Sampling, properstratification as well as selection of appropriate number of

    sample elements from each stratum is very important. The condition of characteristic under study

    based on which stratification is done is known as stratification factor. Occupation, income,

    education, age, sex, economic condition, social status, geographical area etc. are usually the basis

    on which stratification are done.

    Advantages and Disadvantages of Stratified Random Sampling:


    Sample units are selected from different strata of the population on the basis of relative

    importance, so the sample drawn in this method is more representative compared to the

    sample obtained by other methods.

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    Administration of stratified random sampling is more convenient than simple random


    Sampling unit selection is less expensive and less time consuming in stratified random

    sampling compared to simple random sampling.

    Supervision is comparatively easier in stratified random sampling.


    Stratum selection sometimes may become complicated. Improper stratification leads to

    reduce the reliability of the collected information.

    It is not easy to determine the sample components of different strata without previous


    Sampling is not possible if sizes of the different strata are not known.

    Determination of number of sample units:

    In stratified random sampling, there are two different methods for determining the number of

    choosable sample units from each stratum:

    (i) Proportional allocation

    (ii) Optimum allocation

    Proportional allocation:

    If the number of sample units for each stratum is determined to the same

    ratio, the method is known asProportion allocation; 10% sample unit selection from each stratum

    is an example.

    Optimum allocation:

    Basic principle of this method is to select ample component for each stratum

    such that variance of the estimate becomes the minimum. This method is very useful if there exists

    wide difference in the standard deviations of different strata. This is popularly known as Neymans

    Optimum allocation.

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    Systematic Random Sampling

    This is a mixed sampling procedure. In this method, only one sample unit is selected randomly and

    other units are selected following specific system. From a population of N units (numbered1,2,

    .., N) a sample of size n is drawn such that :

    N = nk k =

    Where k is an integer, usually known as sampling interval. The first unit of the sample is selected

    at random. Then other units are selected systematically one after another with a regular interval of

    k units. If the serial number of the selected first unit is i (i

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    Advantages and Disadvantages of Systematic Random Sampling:


    This method is comparatively easier than both simple random sampling and stratified

    random sampling.

    It requires less time and labor.

    If the frame is randomly arranged, this method is more efficient than simple random


    Proper implementation and supervision is easier in comparison with other methods.


    StartSerial Positions of the sample units





    1 1+k 1 +2k 1+3k..1+ (n-1) k

    2 2+k 2 +2k 2+3k..2+ (n-1) k

    3 3+k 3 +2k 3+3k..3+ (n-1) k

    k 2k 3k 4knk

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    The main demerit of this method is that it is not a random sampling method in the true


    If the correct and complete sampling frame is not known, sampling in this method is not


    If population size is not a multiple of the sample size-

    (i) Resulting sample may not be of the required size.

    (ii) Sample mean will not be an unbiased estimate of the population mean.

    Problematic for unbiased estimation.

    Application of Systematic Random Sampling:

    In quality control (steam line production)



    Market research

    Health studies (largely used by WHO, UNICEF)

    Crop distribution

    Cluster Sampling

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    In this method the population is divided into a required number of mutually exclusive groups or

    classes; these groups or classes are known as clusters. Then some clusters are randomly selected

    and data are collected from all the units included in these selected clusters.

    In cluster sample it is necessary to consider the following issues:

    The clusters should be small as far as possible and should be consistent with the budget and

    constraints of the survey.

    The clusters should be more or less of equal size.

    The clusters should be more or less homogeneous in respect of the characteristics under



    City blogs



    Cluster is possible without having complete and up-to-date lists.


    Loss of decision


    Analysis problem

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    Mathematical Problems on Sampling


    Draw possible samples of size 2 from a population of size 5 having elements 1, 2, 3, 4, 5 and thus

    show that sample mean is an unbiased estimate of the population mean.

    Solution: From a population of size 5, probable number of samples of size 2 is ( ) =10.

    Probable samples and the corresponding sample means will be:

    Mean of the sample means is = 1/10 (1.5 + 2.0 + 3.0 + 2.5 + 3.0 + 3.5 + 3.5 + 4.0 + 4.5) = 3.0

    Population mean = 1/5 (1 + 2 + 3 + 4 + 5) = 3.0

    .;. Mean of the sample means and the population mean are equal.



    1,2 1,3 1,4 1,5 2,3 2,4 2,5 3,4 3,5 4,5



    1.5 2.0 2.5 3.0 2.5 3.0 3.5 3.5 4.0 4.5



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    Problem -2:

    Suppose you have a population whose elements are 3,4,5,6,7. Draw all possible samples of size 2

    and prove that the mean of these is equal to the mean of the population.


    From a population of size 6, probable number of samples of size 2 is ( ) =10.

    Probable samples and the corresponding sample means will be:

    Mean of the sample means is = 1/10 (3.5+ 4.0+ 4.5+ 5.0 + 4.5 + 5.0 + 5.5 + 5.5 + 6.0+6.5) = 5.0

    Population mean = 1/5 (3 + 4 + 5 + 6 + 7) = 5.0

    .;. Mean of the sample means and the population mean are equal.



    3,4 3,5 3,6 3,7 4,5 4,6 4,7 5,6 5,7 6,7



    3.5 4.0 4.5 5.0 4.5 5.0 5.5 5.5 6.0 6.5



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    Problem -3:

    There are three people living in a certain city. They are one educated, one uneducated, one literate.

    In a certain month, their income was as follows:

    Educated tk.1500

    Uneducated- tk. 800

    Literate- tk. 1000

    For a 10% stratified sampling based on their education find the composition of the sample for each



    Here, N=1500+800+1000=3300

    N= 3300x 10%= 330

    The population is to be stratified on the basis of education as shown below:

    Education Type

    Income Taka Total

    Educated 1500 x 10% = 150

    Uneducated 800 x 10% = 80

    Literate 1000 x 10% = 100

    All = 330

    The sample will be size 330 of which 150 are educated mans income, 80 are uneducated mans

    income, 100 are literate mans income.

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    Problem -4:


    Suppose you have a population whose elements are 2,3,4,5,6 Draw all possible samples of size 2

    and prove that the mean of these is equal to the mean of the population.

    Solution: From a population of size 6, probable number of samples of size 2 is ( ) =10.

    Probable samples and the corresponding sample means will be:

    Mean of the sample means is = 1/10 (2.5 + 3 + 3.5 + 4.0+ 3.5+ 4.0 + 4.5 + 4.5 + 5 + 5.5) = 4.0

    Population mean = 1/5 (2 + 3 + 4 + 5 + 6) = 4.0

    .;. Mean of the sample means and the population mean are equal (4).



    2,3 2,4 2,5 2,6 3,4 3,5 3,6 4,5 4,6 5,6



    2.5 3.0 3.5 4 3.5 4 4.5 4.5 5 5.5



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    Problem -5:

    A certain farmer has production of three types rice; Amon 60 bags, Iri 40 bags, Boro 50 bags.

    Immediately this farmer has to sell this production in local market so he needs a sample of 20%

    rice bags among these types. So draw a sample of 20% rice bag types.


    We have N= 150: N1= 60: N2= 40 and N3= 50.

    20% random sampling from each stratum gives us

    N1 = N1 x 20/100 = 60 x 20/100 = 12

    N2 = N2 x 20/100 = 40 x 20/100 = 8

    N3 = N3 x 20/100 =50 x 20/100 = 10

    The required sample will be of size

    n= n1+n2+n3= 12+8+10 = 30

    The required sample of size 30 will consist of 12 bags of Amon rice, 8 bags of Iri rice and10 bags

    of Boro rice.

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    Problem: 6

    In a certain locality there are 600 farmers of whom 400 are small farmers, 150 are medium farmers

    and 50 are big farmers. In order to collect data on HYV rice cultivation pattern, a sample of 10%

    farmers is required to be drawn. Also indicate which method is appropriate?


    Here the characteristic under study is the cultivation pattern of HYV rice which is likely to differ

    for different types of farmers (small, medium, big). As a result sample drawn by the method of

    simple random sampling may not be representative for the population. The appropriate sampling

    method in this case is stratified random sampling. The whole population will be divided into three

    strata on the basis of farm size and 10% farmers from each stratum will be randomly selected to

    form the required sample.

    We have N= 600: N1= 400: N2= 150 and N3= 50.

    10% random sampling from each stratum gives us

    N1 = N1 x 10/100 = 400 x 10/ 100 = 40

    N2 = N2 x 10/100 = 150x 10/100 = 15

    N3 = N3 x 10/100 =50 x 10/100 = 5

    The required sample will be of size

    n= n1+n2+n3= 40+25+5=60

    The required sample of size 60 will consist of 40 small farmers, 15 medium farmers and 5 big


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    Problem: 7

    300 farmers, 100 businessmen, 200 factory workers and 100 service holders live in a certain

    locality. The ratio of mid income incumbents is 20 : 80 among the farmers, 70 : 30 among thebusinessmen, 20 : 80 among factory workers and 40 : 60 among the service holders. For a 20%

    stratified random sample for each stratum.


    Here N = 300+ 100+ 200+100 = 700

    n = 700 x 20% = 140

    The population is to be stratified on the basis of profession and income as shown below:

    ProfessionIncome group

    TotalMid- income Low income

    Farmer 300x20/100= 60 300x80/100=240 300

    Businessman 100x70/100= 70 100x30/100= 30 100

    Factory worker 200x20/100= 40 200x80/100=160 200

    Service holder 100x40/100= 40 100x60/100= 60 100

    All 210 490 700

    If Nij is the stratum size corresponding to ith profession and jth income group the corresponding

    sample constituent will be

    nij = Nij x 20%

    That is, nij = Nij x 20/100

    So the composition of the sample will be as follows:


    Income group

    TotalMid- income Low income

    Farmer 60 x20% = 12 240 x 20% = 48 60

    Businessman 70 x 20% = 14 30 x 20% = 6 20

    Factory worker 40 x 20% = 8 160 x 20% = 32 40

    Service holder 40 x 20% = 8 60 x 20% = 12 20

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    All 42 98 140

    The sample will be of size 140 of which 42 and 98 represent the medium and low income groups

    respectively. Among the 42 mid income respondents 12 are farmers, 14 are businessmen, 8 are

    factory workers, the remaining 8 are service holders; and among the 98 low income respondentsthe corresponding numbers will be 48, 6, 32 and 12 respectively.

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    Statistically methods are playing an increasingly important role in almost all field of knowledge.

    The influence of statistics has now spread to agriculture, business, economics, education,

    sociology, communication, political science, geography, medicine, physics, electronics, chemistry

    and many other fields of science and technology. And Sampling is the scientific method of

    representation of the population which helps to draw a valid conclusion. In modern times,

    importance of statistics is felt in every walk of life. Sampling also helps by measuring the selected

    part from the whole population using the statistical techniques for collection of necessary data.


    Class lecture.

    Ahmed Dr. Abdur Rashid, Bhuiya Md. Ali Azgor, Reza Zufikar Ahmed, Hossain Md.

    Zakir;Methods of statistics.

    Islam M. Nurul ;An Introduction to statistics and Probability; Book world.

    Levin Richard I, Rubin David S.; Statistics for Management; 7th edition; Hall of India

    Private Limited.

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