v“Statistics may be rightly called the science of averages and their

estimates.” – A.L.BOWLEY & A.L. BODDINGTON v

15.1 Introduction

We know that statistics deals with data collected for specific

purposes. We can make decisions about the data by

analysing and interpreting it. In earlier classes, we have

studied methods of representing data graphically and in

tabular form. This representation reveals certain salient

features or characteristics of the data. We have also studied

the methods of finding a representative value for the given

data. This value is called the measure of central tendency.

Recall mean (arithmetic mean), median and mode are three

measures of central tendency. A measure of central

tendency gives us a rough idea where data points are

centred. But, in order to make better interpretation from the

data, we should also have an idea how the data are scattered or how much they are

bunched around a measure of central tendency.

Consider now the runs scored by two batsmen in their last ten matches as follows:

Batsman A : 30, 91, 0, 64, 42, 80, 30, 5, 117, 71

Batsman B : 53, 46, 48, 50, 53, 53, 58, 60, 57, 52

Clearly, the mean and median of the data are

Batsman A Batsman B

Mean 53 53

Median 53 53

Recall that, we calculate the mean of a data (denoted by

x

) by dividing the sum

of the observations by the number of observations, i.e.,

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STATISTICS

Karl Pearson

(1857-1936)

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348 MATHEMATICS

1

1

n

i

i

x x

n

=

=

∑

Also, the median is obtained by first arranging the data in ascending or descending

order and applying the following rule.

If the number of observations is odd, then the median is

th

1

2

n +

observation.

If the number of observations is even, then median is the mean of

th

2

n

and

th

1

2

n

+

observations.

We find that the mean and median of the runs scored by both the batsmen A and

B are same i.e., 53. Can we say that the performance of two players is same? Clearly

No, because the variability in the scores of batsman A is from 0 (minimum) to 117

(maximum). Whereas, the range of the runs scored by batsman B is from 46 to 60.

Let us now plot the above scores as dots on a number line. We find the following

diagrams:

For batsman A

For batsman B

We can see that the dots corresponding to batsman B are close to each other and

are clustering around the measure of central tendency (mean and median), while those

corresponding to batsman A are scattered or more spread out.

Thus, the measures of central tendency are not sufficient to give complete

information about a given data. Variability is another factor which is required to be

studied under statistics. Like ‘measures of central tendency’ we want to have a

single number to describe variability. This single number is called a ‘measure of

dispersion’. In this Chapter, we shall learn some of the important measures of dispersion

and their methods of calculation for ungrouped and grouped data.

Fig 15.1

Fig 15.2

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STATISTICS 349

15.2 Measures of Dispersion

The dispersion or scatter in a data is measured on the basis of the observations and the

types of the measure of central tendency, used there. There are following measures of

dispersion:

(i) Range, (ii) Quartile deviation, (iii) Mean deviation, (iv) Standard deviation.

In this Chapter, we shall study all of these measures of dispersion except the

quartile deviation.

15.3 Range

Recall that, in the example of runs scored by two batsmen A and B, we had some idea

of variability in the scores on the basis of minimum and maximum runs in each series.

To obtain a single number for this, we find the difference of maximum and minimum

values of each series. This difference is called the ‘Range’ of the data.

In case of batsman A, Range = 117 – 0 = 117 and for batsman B, Range = 60 – 46 = 14.

Clearly, Range of A > Range of B.

Therefore, the scores are scattered or dispersed in

case of A while for B these are close to each other.

Thus, Range of a series = Maximum value – Minimum value.

The range of data gives us a rough idea of variability or scatter but does not tell

about the dispersion of the data from a measure of central tendency. For this purpose,

we need some other measure of variability. Clearly, such measure must depend upon

the difference (or deviation) of the values from the central tendency.

The important measures of dispersion, which depend upon the deviations of the

observations from a central tendency are mean deviation and standard deviation. Let

us discuss them in detail.

15.4 Mean Deviation

Recall that the deviation of an observation x from a fixed value ‘a’ is the difference

x – a. In order to find the dispersion of values of x from a central value ‘a’ , we find the

deviations about a. An absolute measure of dispersion is the mean of these deviations.

To find the mean, we must obtain the sum of the deviations. But, we know that a

measure of central tendency lies between the maximum and the minimum values of

the set of observations. Therefore, some of the deviations will be negative and some

positive. Thus, the sum of deviations may vanish. Moreover, the sum of the deviations

from mean (

x

) is zero.

Also Mean of deviations

Sum of deviations 0

0

Number of observations n

= = =

Thus, finding the mean of deviations about mean is not of any use for us, as far

as the measure of dispersion is concerned.

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Remember that, in finding a suitable measure of dispersion, we require the distance

of each value from a central tendency or a fixed number ‘a’. Recall, that the absolute

value of the difference of two numbers gives the distance between the numbers when

represented on a number line. Thus, to find the measure of dispersion from a fixed

number ‘a’ we may take the mean of the absolute values of the deviations from the

central value. This mean is called the ‘mean deviation’. Thus mean deviation about a

central value ‘a’ is the mean of the absolute values of the deviations of the observations

from ‘a’. The mean deviation from ‘a’ is denoted as M.D. (a). Therefore,

M.D.(

a) =

Sum of absolute values of deviations from ' '

Number of observations

a

.

Remark Mean deviation may be obtained from any measure of central tendency.

However, mean deviation from mean and median are commonly used in statistical

studies.

Let us now learn how to calculate mean deviation about mean and mean deviation

about median for various types of data

15.4.1 Mean deviation for ungrouped data Let n observations be x

1

, x

2

, x

3

, ...., x

n

.

The following steps are involved in the calculation of mean deviation about mean or

median:

Step 1 Calculate the measure of central tendency about which we are to find the mean

deviation. Let it be ‘a’.

Step 2 Find the deviation of each x

i

from a, i.e., x

1

– a, x

2

– a, x

3

– a,. . . , x

n

– a

Step 3 Find the absolute values of the deviations, i.e., drop the minus sign (–), if it is

there, i.e.,

axaxaxax

n

−−−− ....,,,,

321

Step 4 Find the mean of the absolute values of the deviations. This mean is the mean

deviation about a, i.e.,

1

( )M.D.

n

i

i

x a

a

n

=

−

=

∑

Thus M.D. (

x

) =

1

1

n

i

i

x x

n

=

−

∑

, where

x

= Mean

and M.D. (M) =

1

1

M

n

i

i

x

n

=

−

∑

, where M = Median

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A

Note

In this Chapter, we shall use the symbol M to denote median unless stated

otherwise.Let us now illustrate the steps of the above method in following examples.

Example 1 Find the mean deviation about the mean for the following data:

6, 7, 10, 12, 13, 4, 8, 12

Solution

We proceed step-wise and get the following:

Step 1 Mean of the given data is

6 7 10 12 13 4 8 12 72

9

8 8

x

+ + + + + + +

= = =

Step 2 The deviations of the respective observations from the mean

,

x

i.e., x

i

–

x

are

6 – 9, 7 – 9, 10 – 9, 12 – 9, 13 – 9, 4 – 9, 8 – 9, 12 – 9,

or –3, –2, 1, 3, 4, –5, –1, 3

Step 3 The absolute values of the deviations, i.e.,

i

x x

−

are

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

Step 4 The required mean deviation about the mean is

M.D.

(

)

x

=

8

1

8

i

i

x x

=

−

∑

=

3 2 1 3 4 5 1 3 22

2 75

8 8

.

+ + + + + + +

= =

A

Note Instead of carrying out the steps every time, we can carry on calculation,

step-wise without referring to steps.

Example 2 Find the mean deviation about the mean for the following data :

12, 3, 18, 17, 4, 9, 17, 19, 20, 15, 8, 17, 2, 3, 16, 11, 3, 1, 0, 5

Solution We have to first find the mean (

x

) of the given data

20

1

1

20

i

i

x x

=

=

∑

=

20

200

= 10

The respective absolute values of the deviations from mean, i.e.,

xx

i

−

are

2, 7, 8, 7, 6, 1, 7, 9, 10, 5, 2, 7, 8, 7, 6, 1, 7, 9, 10, 5

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Therefore

20

1

124

i

i

x x

=

− =

∑

and M.D. (

x

) =

124

20

= 6.2

Example 3 Find the mean deviation about the median for the following data:

3, 9, 5, 3, 12, 10, 18, 4, 7, 19, 21.

Solution Here the number of observations is 11 which is odd. Arranging the data into

ascending order, we have 3, 3, 4, 5, 7, 9, 10, 12, 18, 19, 21

Now Median =

th

11 1

2

+

or 6

th

observation = 9

The absolute values of the respective deviations from the median, i.e.,

M

i

x −

are

6, 6, 5, 4, 2, 0, 1, 3, 9, 10, 12

Therefore

11

1

M 58

i

i

x

=

− =

∑

and

( )

11

1

1 1

M.D. M M 58 5.27

11 11

i

i

x

=

= − = × =

∑

15.4.2 Mean deviation for grouped data We know that data can be grouped into

two ways :

(a) Discrete frequency distribution,

(b) Continuous frequency distribution.

Let us discuss the method of finding mean deviation for both types of the data.

(a) Discrete frequency distribution Let the given data consist of n distinct values

x

1

, x

2

, ..., x

n

occurring with frequencies f

1

, f

2

, ..., f

n

respectively. This data can be

represented in the tabular form as given below, and is called discrete frequency

distribution:

x : x

1

x

2

x

3

... x

n

f : f

1

f

2

f

3

... f

n

(i) Mean deviation about mean

First of all we find the mean

x

of the given data by using the formula

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