Knowledge of calculation of mean,median & mode. Variability &

standard deviation.

Presented by:- Dr. Aarati vijaykumar 1st year M.D (K.C)

Introduction: Definition of statistics: It is the ‘science of collecting,

classifying, presenting & interpreting data’ relating to any sphere of enquiry.

Having learnt the methods of collection & presentation of data, we have to understand & grasp the application of mathematical techniques involved in analysis & interpretation of the data.

As medicos, we should learn to apply the formulae straight to our problems without worrying how they have been deduced. Application of methods for analysis is quite easy & we should become familiar with them so as to verify our preconceived ideas or to remove doubts which might arise at the first look of figures collected.

“ If a man will begin with certainties, he shall end in doubts’: but if he will be content to begin with doubts, he shall end in certainties.”

- Francis Bacon

Characteristics of frequency distribution is of two types,

1. Measures of central tendency ( Location, Position, Average)2. Measures of dispersion ( Scatterdness, Variability, Spread)

Definition: It refers to a single central number or value that

condenses the mass data & enables us to give an idea about the whole or entire data.

Types:1. Arithmetic Mean2. Median Q2 3. The mode Z

Measures of central tendency

x

Introduction: It is the most commonly used measure of central

tendency. It is also called as ‘Average’.Definition: It is defined as additional or summation of all

individual observations divided by the total number of observation.

Arithmetic mean ( A.M )

Types of series

1. Ungrouped series ( Ungrouped data, Unclassified data, Raw data ) : Includes individual observations without frequency.

2. Grouped series ( Classified data ) : Includes individual observation with frequency & class frequency.

Calculation : 3. Direct method 4. Indirect method

Merits of Arithmetic Mean1. Easy to understand & to calculate.2. It is correctly or rigidly defined.3. It is based on each & every observation.4. Every set of data has one & only one A.M.5. Used for further mathematical calculations like standard

deviation.

Demerits of Arithmetic Mean1. Affected by extreme values ( either low or high)2. It can not be obtained even if a single value is missing.

Introduction : It is called Q2 because it denotes 2nd quartile or positional

value. It is the 2nd measure of central tendency. Here there are 3 quartile Q1 , Q2 , Q3 which divides the

distribution into 4 parts or equal.

A Q1 Q2 Q3 B.

Median ( Q2 )

Definition : Median divides the distribution into two equal parts i.e.

50% of the distribution is below the median & 50% is above the median.

Q1 = n/4, Q2 = 3 x n/4

Ungrouped data:

When ‘n’ is odd if the total number of observations are even, then arrange the observations either in ascending or descending order & calculate the median by formula.

Q3 = n+1/ 2

Definition : Dictionary meaning of mode is common or fashionable.

Mode is the value which occurs more frequently in a given set of data.

There are 3 types Type 1 Ex: Selection of mode : Observation having the highest

repetition. 10,11,12,26,20,40,20,10,12,10Mode = 10

MODE ( Z )

Type 2 : Selection of mode: Observation containing highest frequency.

Ex: Number of children per family. No.of children/Family No.of families 0 13 1 24 2 25 3 13 4 14

25 is highest frequency so ‘2’ is mode. Type 3: Class containing highest frequency.

Merits of Mode:1. Easy to calculate & understand.2. Not affected by extreme value.3. Mode can be found by both qualitative & quantitative data.

Demerits of Mode:1. Some times no mode or more then one mode in a given set

of distribution.2. Not used for further mathematical calculation.3. Not commonly used.

Examples of Ungrouped series :1. Direct method

= ∑x/n x = Individual observation n = Number of observationEx: Systolic BP of the patients, calculate mean, mode & median. 1. 110mmHg x1

2. 100mmHg x2

3. 150mmHg x3

4. 140mmHg x4

5. 140mmHg x5

6. 120mmHg x6

x

Mean ( Average ) : = ∑x/n ∑ = Summation n = Number of samples x = Individual observation. ∑x = x1+ x2+ x3 + x4 + x5 + x6

= 760/6 = 126.6mmHg

Mode : Most repeated number in the data: 140mmHg

Median : 100, 110, 120, 140, 140,150 = 120+140 = 260/2 = 130mmHg

x

Step deviation method of calculation mean : Ex: Height of the school children's given below find out the

mean. 1. 148cm x1

2. 143cm x2

3. 160cm x3

4. 152cm x4 5. 157cm x5

6. 150cm x6 7. 155cm x7

Working origin ( w ) = 150cm

Formula : = ∑ ( x – w ) / n

148 -150 = -2 143 -150 = -7 160 - 150 = 10 152 - 150 = 2 157 - 150 = 7 150 -150 = 0 155 -150 = 5 = 15/7 = 2.1 = w + = 150 + 2.1 = 152.2

x

xx

Find mean days of confinement after delivery in the following?

Mean = ∑fx/n , ∑f = n

= 137/18 = 7.61

Examples of gruoped series:

Days of confinement x

No. of patients grouped f

Total days of each group fx

6 5 30

7 4 28

8 4 32

9 3 27

10 2 20

18 137

Definition: Measures of variability describes the spread or scatterdness

of the individual observation around the central tendency.

Significance :1. Gives complete idea/picture of data2. Helps in comparison of distribution.3. Useful for further calculations4. Gives idea about the reliability of average value.

Measures of variability/Dispersion

Methods of dispersion

1. Range ( R )2. Inter quartile range ( IQR )3. Quartile deviation / Semi inter quartile range4. Mean deviation / Average deviation (MD)5. Standard deviation (SD)

Range : Definition: Is defined as the difference between the highest & lowest values in a set

of data. R = H – LEx: Weight of an adult person 50 -100kg

Merits: Easy to calculate & understand Has got a well defined formula gives first hand information about variation

Demerits: It is not based on all the values Affected by extreme value

Definition: It is the interval between the value of upper quartile

( the value above which 25% observation falls) & lower quartile ( the values which fall below the 25% ).

So the measures gives us the range of middle 50% of observation & it is very helpful when the observations are not homogenous & extreme in nature. It is the superior measure over the range in such conditions.

Inter Quartile Range

Ex: Weight of the persons 1. 40kg 2. 45kg 3. 50kg 4. 55kg Q1 5. 60kg 6. 65kg 7. 70kg 8. 75kg 9. 80kg Q3 10. 85kg IQR = 55kg – 80kg 11. 90kg 12. 95kg

Merits of IQR: Easy & simple to understand Easy to calculate Not affected by extreme values

Demerits of IQR : It is a positional value which is based on two

quartile Based on first & last values

Definition :

It is an average amount of scatter of the items in a distribution from any measures of the central tendency by ignoring the mathematical signs.

Formula: M.D = ∑ |x – | / n

Mean deviation/ Average deviation

x

Example: Average marks obtained in 5 internals by a student.

x x - 25 25- 22 = 3 15 15- 22 = -7 25 25-22 = 3 25 25-22 = 3 20 20- 22 = -2

= ∑x/n = 110/5 = 22

x

x

M.D = ∑ |x – | / n = 18/5 M.D = 3.6%Co-efficient of average/Mean deviation: CAD = MD/Mean × 100 = 3.6/22 × 100 = 180/11 = 16.36%

x

Introduction: It is most widely used, best method of calculating

deviation. Though in AD it takes into consideration of all the

observation & it ignores the mathematical signs, but SD overcomes this problem by squaring the deviation.

Definition:

SD is the square root of summation of square of deviation of given set of observation from the AM divided by the total number of observation.

Standard deviation

Formula : Ungrouped series Standard deviation = ∑( x- )2 / n n ˃ 30 Grouped series Standard deviation = ∑f (x - )2 / n

n ˂ 30 Where, ∑ – is Summation of,x – is Individual observation, – is Arithmetic mean,n – is Total number of observation

x

x

x

Average marks obtained in 5 internals by a student

S.D = ∑ ( X - ) 2 / n = 80/5 = 16 = 4

Marks obtained x

x - ( x - )2

25 25 – 22 = 3 9

15 15 – 22 = -7 49

25 25 – 22 = 3 9

25 25 – 22 = 3 9

20 20 – 22 = -2 4

= 110 = 80

x x

x

Co – efficient of SD = SD/ Mean x 100 = 4 / 22 x 100 = 400 / 22 = 18.1 %Significance of SD : Based on all observations. Best method of calculation without ignoring mathematical

signs. Useful for further statistical calculations. (i.e. Test of

Significance etc.) Useful for calculation of standard error. Lesser the standard deviation, better the estimation of

population mean.

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