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Descriptive Statistics
Measures of Central Tendency
 Measures of Location
 Measures of Dispersion
 Measures of Symmetry
 Measures of Peakedness

Measures of Variability or Dispersion

The dispersion of a distribution reveals how the
observations are spread out or scattered on each side
of the center.

To measure the dispersion, scatter, or variation of a
distribution is as important as to locate the central
tendency.

If the dispersion is small, it indicates high uniformity of
the observations in the distribution.

Absence of dispersion in the data indicates perfect
uniformity. This situation arises when all observations in
the distribution are identical.

If this were the case, description of any single
observation would suffice.
Purpose of Measuring Dispersion

A measure of dispersion appears to serve two purposes.

First, it is one of the most important quantities used to
characterize a frequency distribution.

Second, it affords a basis of comparison between two or
more frequency distributions.

The study of dispersion bears its importance from the fact
that various distributions may have exactly the same
averages, but substantial differences in their variability.
Measures of Dispersion

◦
◦
◦
◦

◦
◦
◦
◦
Range
Percentile range
Quartile deviation
Mean deviation
Variance and standard deviation
Relative measure of dispersion
Coefficient of variation
Coefficient of mean deviation
Coefficient of range
Coefficient of quartile deviation
Range
The simplest and crudest measure of
dispersion is the range. This is defined as
the difference between the largest and
the smallest values in the distribution. If
are the values of
observations
in a sample, then range (R)
x1 , x 2 ,.........
., x n
of the variable X is given by:

R x1 , x 2 ,........, x n   max x1 , x 2 ,..........., x n  min x1 , x 2 ,............, x n 
Percentile Range
Difference between 10 to 90 percentile.
 It is established by excluding the highest
and the lowest 10 percent of the items,
and is the difference between the largest
and the smallest values of the remaining
80 percent of the items.

90
10
P
 P90  P10
Quartile Deviation


A measure similar to the special range (Q) is the interquartile range . It is the difference between the third
quartile (Q3) and the first quartile (Q1). Thus
Q  Q3 range
 Q1is frequently reduced to the
The inter-quartile
measure of semi-interquartile range, known as the
quartile deviation (QD), by dividing it by 2. Thus
Q3  Q1
QD 
2
Mean Deviation

The mean deviation is an average of absolute
deviations of individual observations from the central
value of a series. Average deviation about mean
k
MDx  
f
xi  x
i 1
k = Number of classes
 xi= Mid point of the i-th class
 fi= frequency of the i-th class

i
n
Standard Deviation

Standard deviation is the positive square root of the
mean-square deviations of the observations from
their arithmetic mean.
Population

2


x


 i
N
Sample
s
SD  variance
2


x

x
 i
N 1
Standard Deviation for Group Data


SD is :
s

f i xi  x 2
Where
N
s
 fx
N
i i
i
Simplified formula
2
fx

x
f




 fx 
N


2
Example-1: Find Standard Deviation of
Ungroup Data
Family
No.
1
2
3
4
5
6
7
8
9
10
Size (xi)
3
3
4
4
5
5
6
6
7
7
x

x
Here,
Family No.
n
i
50

5
10
1
2
3
4
5
6
7
8
9
10
Total
xi
3
3
4
4
5
5
6
6
7
7
50
xi  x
-2
-2
-1
-1
0
0
1
1
2
2
0
4
4
1
1
0
0
1
1
4
4
20
9
9
16
16
25
25
36
36
49
49
270
x i  x 
xi
2
2
s2 
2


x

x
 i
n 1

20
 2.2,
9
s  2.2  1.48
Example-2: Find Standard Deviation of
Group Data
x i  x  x i  x  2 f i x i  x 2
xi
fi
f i xi
3
2
6
18
-3
9
18
5
3
15
75
-1
1
3
7
2
14
98
1
1
2
8
2
16
128
2
4
8
9
1
9
81
3
9
9
Total
10
60
400
-
-
40
f x

x
f
i
i
i
60

6
10
s
f i xi
2


2
f x i  x 
n 1
2
i

40
 4.44
9
Relative Measures of Dispersion

To compare the extent of variation of different
distributions whether having differing or identical
units of measurements, it is necessary to consider
some other measures that reduce the absolute
deviation in some relative form.

These measures are usually expressed in the form
of coefficients and are pure numbers, independent
of the unit of measurements.
Relative Measures of Dispersion
Coefficient of variation
 Coefficient of mean deviation
 Coefficient of range
 Coefficient of quartile deviation

Coefficient of Variation

A coefficient of variation is computed as a
ratio of the standard deviation of the
distribution to the mean of the same
distribution.
SD
CV 
X 100
x
Example-3: Comments on Children in a
community
Mean
SD
CV

Height
weight
40 inch
5 inch
0.125
10 kg
2 kg
0.20
Since the coefficient of variation for weight is
greater than that of height, we would tend to
conclude that weight has more variability than
height in the population.
Coefficient of Mean Deviation

The third relative measure is the coefficient of mean
deviation. As the mean deviation can be computed from
mean, median, mode, or from any arbitrary value, a
general formula for computing coefficient of mean
deviation may be put as follows:
Coefficien t of mean deviation =
Mean deviation
 100
Mean
Coefficient of Range

The coefficient of range is a relative measure
corresponding to range and is obtained by the
following formula:
LS
Coefficien t of range 
100
LS

where, “L” and “S” are respectively the largest and
the smallest observations in the data set.
Coefficient of Quartile Deviation

The coefficient of quartile deviation is
computed from the first and the third
quartiles using the following formula:
Q3  Q1
Coefficien t of quartile deviation 
100
Q3  Q1
Exercise-1

Find the following measurement of dispersion
from the data set given in the next page:
◦ Range, Percentile range, Quartile Range
◦ Quartile deviation, Mean deviation, Standard deviation
◦ Coefficient of variation, Coefficient of mean deviation,
Coefficient of range, Coefficient of quartile deviation
Data for Assignment-1
Marks
No. of students
Cumulative
frequencies
40-50
6
6
50-60
11
17
60-70
19
36
70-80
17
53
80-90
13
66
90-100
4
70
Total
70