Measures of Dispersion
OR
Measures of Variations
T he term dispersion means the spreadness of the observations from the average. A quantity
that measures this characteristic is called a measure of dispersion, or measure of
variability. The observations may be close to the center or they may be spread away from the
center. If the observations are close to the center, we say that dispersion is small. If the
observations are spread away from the center, we say that dispersion is large. A measure of
central tendency and measure of dispersion gives an adequate description of the data.
Properties of good Measure of Dispersion:
It is easy to calculate and simple to understand.
It should be least affected by sampling fluctuations.
It should be easy for comparison purpose.
It should not be rough measure.
Types of Measure of Dispersion:
1. Absolute measure of Dispersion 2. Relative measure of dispersion
Absolute measure of Dispersion:
An absolute measure of dispersion is that which measures the variation present between the
observations in terms of the same units as the units of the variable.
Relative measure of Dispersion:
A relative measure of dispersion is that which measures the variation present between the
observations relative to their average. These measures are free of the units in which the original
data is measured. If the original data is in rupees or kilograms, we do not use these units with
relative measure of dispersion. It is used to compare the variation between two or more sets of
data.
Types of absolute measure of dispersion:-
(i) The Range
(ii) The Semi-interquartile Range OR the Quartile Deviation.
(iii) The mean Deviation or the average deviation.
(iv) The variance and the Standard Deviation
RANGE:-The Range R, is the difference between the largest and the smallest observation in a
set of data.
R = Xmax - Xmin
Where,
Xm = Maximum value in the data set. Xo = Minimum value in the data set.
For group data,
Range is the difference between upper class boundary of the highest class and lower class
boundary of the smallest class.
OR
Range is the difference between mid-value of the highest class and mid-value of the smallest
class.
R = Xmax - Xmin
Where,
Xm = Mid-value of the highest class. Xo = Mid-value of the smallest class.
Merits of Range:
It is easy to calculate and simple to understand.
It gives the results very quickly.
It is good measure for rough data.
It computation requires only the extreme values of the items of a series.
De-Merits of Range:
It is not reliable measure.
It gives no idea about the other values in a set of data except the extreme ends.
We cannot find range in case of open end frequency table.
It is affected by sampling fluctuations.
Coefficient of Range: The coefficient of range is obtained by dividing the difference between
largest and smallest values by the sum of the largest and smallest values. i.e.
X m− X o
Coefficient of range = X m + X o
/Semi-inter-Quartile range: The Quartile Deviation/Semi-inter-
The Quartile Deviation
Quartile range is the half of the difference between the upper and the lower quartiles in a set of
Q3 −Q1
Q . D=
data Semi-inter-Quartile range/ 2
Variance:- The Variance is defined as the arithmetic mean of the squared deviations of the
2
observations from their mean. The population variance is denoted by σ and its estimate sample
variance is denoted by S2 and defined as
For ungrouped data
σ
2
=
∑ ( X −μ )2
Population variance N
(∑ )
2
2 ∑ ( X− X̄ )2 1 2 (∑ X )
S= OR X −
n n n
For grouped data
∑ ( )
2
2 ∑ f ( X − X̄ )
2
1 ( ∑ fX )
S= OR ∑ fX 2 −
∑f f ∑f
Standard deviation:-
The positive square root of the variance is called Standard Deviation. The population
standard deviation is denoted byσ and its estimate sample standard deviation is denoted by S and
defined as
Coefficient of variation:
The standard deviation is an absolute measure of dispersion its relative measure of
dispersion is called co-efficient of standard deviation or coefficient of variation and is defined by
S
C .V = × 100
x
This co-efficient of variation is used to compare two sets of data which are expressed in different
?
units of measurements e.g one may be in hours and the other may be in kilograms. The C.V is
independent of units used.
Example # 19. Find Variance, S.D and Co-efficient of Variation.
X 2 3 6 8 11 30
( x−x) -4 -3 0 2 5 0
( x−x )2 16 9 0 4 25 54
∑X= x=30=6
√
n 5 ¿S = ∑ ¿ = √ 10.5 ¿C.V = S x 10 = 3.286 x 10 = 54.76 %¿¿
2
( x - { x̄ )
∑S = = ,54S.D= 10.8and C.Vn
( x -
Example # 20. Find 2Variance
{ x̄ )2 x 6
Class F n Midpoint 5 ( X − X̄ ) ( X − X̄ ) 2
f ( X− X̄ ) 2
(X)
20---24 1 22 -17 289 289
25---29 4 27 -12 144 576
30---34 8 32 -7 49 392
35---39 11 37 -2 4 44
40---44 15 42 3 9 135
45---49 9 47 8 64 576
50---54 2 52 13 169 338
TOTAL 50 2350
√
∑S = f(x- {x̄) =2350 =47 ¿S = ∑f(x- {x̄) ¿ = √47 = 6.85¿C.V = 6.85 x10 = 17.56¿
2 2
2
∑f 50 ∑f 39
Properties of Variance:-
The variance of a constant is equal to zero i.e. Var (a )=0 where 'a' is any constant.
Variance is independent of origin i.e. Var ( X −a )=Var ( X ) and Var ( X +a)=Var ( X )
Where ‘a’ is any constants
X 1
2 Var ( )= 2 Var ( X )
Variance is dependent of scale i.e. Var(bX )=b Var( X ) and b b Where
‘a & b’ are any constants.
Var( X ±Y )=Var( X )±Var(Y ) if X and Y are independent random variables.
2 n1 S21 + n2 S 22
Sc=
The combined variance for two sets of data denoted by S2c is given as. n1 +n 2
Variance is positive quantity. The variance is expressed in square of the units of the
observations.
Standardized Variable:- A variable is defined to be a standardized variable if it is
expressed in terms of deviation from its mean and divided by its standard deviation. It is
denoted by Z.
A variable that has mean “0” and Variance “1” is called standard variable
Values of standard variable is called standard scores
Values of standard variable i.e standard scores are unit-less
x- x
Z=
Construction S
The mean of standardized variable is zero and variance is equal to one.
