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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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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