2. Introduction
Dispersion measures the extent to which the items vary from
some central value. It may be noted that the measures of
dispersion or variation measure only the degree but not the
direction of the variation. The measures of dispersion are also
called averages of the second order because these are based on
the deviations of the different values from the mean or other
measures of central tendency which are called averages of the
first order.
Department of Economics 2
3. Objectives
After going through this unit, you will be able to:
To understand the objectives of Dispersion;
Types of Dispersion;
Define Range , Quartile deviation and Mean deviation ;
Merits and Demerits of Range , Quartile deviation and Mean deviation
Department of Economics 3
4. Dispersion refers to the variations of the items among themselves / around
an average. Greater the variation amongst different items of a series, the
more will be the dispersion.
As per Bowley, “Dispersion is a measure of the variation of the items”.
In the words of Spelgel, “The degree to which numerical data tend to
spread about an average values is called the variation or dispersion of data”
Dispersion
Definition
5. 1) To determine the reliability of an average
2) To compare the variability of two or more series
3) For facilitating the use of other statistical measures
4) To get information about the composition of the series
5) To help in controlling variability
Objectives of Measuring Dispersion
6. 1) Easy to understand
2) Simple to calculate
3) Uniquely defined
4) Based on all observations
5) Not affected by extreme observations
6) Capable of further algebraic treatment
Properties of a Good Measure of Dispersion
7. Absolute: Expressed in the
same units in which data is
expressed Ex: Rupees, Kgs, Ltr,
Km etc.
Relative: In the form of ratio or
percentage, so is independent of
units It is also called Coefficient
of Dispersion
Measures Of Dispersion
Absolute
Relative
Measures Of
Dispersion
8. Range
Interquartile Range
Quartile Deviation
Mean Deviation
Standard Deviation
Coefficient of Variation
Lorenz Curve
Methods Of Measuring Dispersion
9. It is the simplest measures of dispersion It is defined as the difference
between the largest and smallest values in the series
Formula
R = Range
R = L – S, 3 5 7 9 10 12
L = Largest Value, Min Range Max
S = Smallest Value
Coefficient of Range = 𝐿 −𝑆
𝐿+𝑆
Range (R)
Definition
Range = 12- 3 = 9
10. Example: Find the range & Coefficient of Range for the following data:
Class Frequency
1-5 2
6-10 8
11-15 15
16-20 35
21-25 20
26-30 10
In case of inclusive series, first it
should be converted into exclusive
series, in the example the lowest
limit is 0.5 and the highest limit
30.5
Range
R =L –S
= 30.5-0.5 or 30
= 30
Coefficient of Range
C. Of R. = L –S = 30.5-0.5 = 30
L+ S = 30.5+0.5 31
C Of R = 0.97
11. Merits
Simple to understand
Easy to calculate
Widely used in statistical quality control
Demerits
Can’t be calculated in open ended distributions
Not based on all the observations
Affected by sampling fluctuations
Affected by extreme values
Merits and Demerits of Range
12. Interquartile Range is the difference between the upper quartile (Q3) and the
lower quartile (Q1). It covers dispersion of middle 50% of the items of the
series
Formula
Interquartile Range = Q3 – Q1
Definition
Quartile Deviation is half of the interquartile range. It is also called Semi
Interquartile Range
Formula
Quartile Deviation = 𝑄3 −𝑄1
2
Coefficient of Quartile Deviation: It is the relative measure of quartile
deviation.
Coefficient of Q.D. = 𝑄3 −𝑄1
𝑄3 +𝑄1
Interquartile Range & Quartile Deviation
Definition
13. Example: Find quartile deviation and coefficient of quartile deviation:
Central
Size
1 2 3 4 5 6 7 8 9 10
Frequency 2 9 11 14 20 24 20 16 5 2
Central Size Class Interval Frequency Cum. f
1 0.5-1.5 2 2
2 1.5-2.5 9 11
3 2.5-3.5 11 22
4 3.5-4.5 14 36
5 4.5-5.5 20 56
6 5.5-6.5 24 80
7 6.5-7.5 20 100
8 7.5-8.5 16 116
9 8.5-9.5 5 121
10 9.5-10.5 2 123
N= 123
Q₁ =Size of N th item
4
Q₁ =Size of 123 th item=30.75th item
4
Q₁ =3.5 + 1 (30.75-22) =4.125
14
Q₃ =Size of 3N th item
4
Q₃ =Size of 3x123 th item=92.25h item
4
Q ₃ =6.5 + 1 (92.25-80) =7.1125
20
C of Q.D= Q₃ -Q₁ = 0.266
Q₃ +Q₁
14. Merits
Simple to understand
Minimum effect of extreme values
Dispersion of middle part
Demerits
Formation of quartiles of two series
cannot be studied by this method
Not based on all the values of variable
Affected by sampling fluctuations
It is not suitable for further algebraic treatment
Merits and Demerits of Quartile Deviation
15. Mean Deviation is the arithmetic average of deviations of all the
values taken from a measure of central tendency (Mean, Mode or
Median) of the series. In taking deviations of values, algebraic signs +
and – are not taken into consideration. .
M.D. from Median = Σ |𝑋 −𝑀| or Σ |𝑑ₘ|
𝑁 𝑁
M.D. from Mean = Σ |𝑋 −𝑋̅| or Σ |𝑑ₓ |
𝑁 𝑁
Coefficient of M.D. ₘ = 𝑀.𝐷.ₘ
𝑀𝑒𝑑𝑖𝑎𝑛
Coefficient of M.D. ₓ = 𝑀.𝐷. ₓ
𝑀𝑒𝑎𝑛
Mean Deviation (M.D.)
Definition
16. Example : Calculate M.D. from Mean & Median & coefficient of Mean Deviation
from the following data:
Individual series
Direct Method
• Find out average
• Take deviations of given values from median( any other average) ignoring algebraic signs |d|
• There deviations are aggregated Ʃ |d|
• Apply formula: δₘ = Ʃ |d ₘ | ; δₓ = Ʃ |d ₓ |
N N
Weight Deviation from
M (50) Ʃ |d ₘ
Ignoring signs Deviations
from X (52) Ʃ |d ₓ |
45 5 7
47 3 5
47 3 5
49 1 3
50 0 2
53 3 1
58 8 6
59 9 7
60 10 8
468 42 44
Ʃ X Ʃ |d ₘ Ʃ |d ₓ |
MD from Median
Median =size of (N + 1)th item
2
= 5th item = 50
δₘ = Ʃ |d ₘ | = 42 or 4.67
N 9
C of δₘ = δₘ or 4.67 =.0934
M 50
MD from Mean
_
X = Ʃ X = 468 = 52
N 9
δₓ = Ʃ |d ₓ | =44 or 4.89
N 9
C of δₓ = δₓ or 4.89 =.0940
X 52
17. Example : Calculate M.D. from Mean & Median & coefficient of Mean Deviation
from the following data:
Discrete series
Direct Method
• Find out Mean
• Take deviations of given values from median( any other average) ignoring
algebraic signs |d|
• There deviations are aggregated Ʃ |d|, multiplying by respective frequencies
• Apply formula: δₘ = Ʃf |d ₘ | ; δₓ = Ʃ f|d ₓ |
N N
Size 4 6 8 10 12 14 16
Frequency 2 4 5 3 2 1 4
18. Department of Economics
Size frequenc
y
C.f f x X Deviation from
median
Deviation from
mean (9.71)
Median
of signs
Total
deviatio
n
Ignoring
+ and -
Total
deviatio
n
X F C.f fxX |d ₘ | f x |d ₘ | |d ₓ | fx |d ₓ |
4 2 2 8 4 8 5.71 11.42
6 4 6 24 2 8 3.71 14.84
8 5 11 40 0 0 1.71 8.55
10 3 14 30 2 6 0.29 0.87
12 2 16 24 4 8 2.29 4.58
14 1 17 14 6 6 4.29 4.29
16 4 21 64 8 32 6.29 25.16
Total 21 204 68 69.71
N Ʃ fx Ʃf |d ₘ| Ʃ f|d ₓ|
MD from Median
Median =size of (N + 1)th item
2
= 11th item = 8
δₘ = Ʃ f|d ₘ | = 68 or 3.24
N 21
C of δₘ = δₘ or 3.24 = 0.405
M 8
MD from Mean
_
X = ƩfX = 204= 9.71
N 21
δₓ = Ʃf |d ₓ | =69.21 or 3.32
N 21
C of δₓ = δₓ or 3.32 =0.342
X 9.71
18
19. Department of Economics
Example : Calculate M.D. from Mean & Median & coefficient of Mean Deviation
from the following data:
Continous series
Direct Method
• Find the mid value
• Find out Mean
• Take deviations of given values from median( any other average) ignoring
algebraic signs |d|
• There deviations are aggregated Ʃ |d|, multiplying by respective frequencies
• Apply formula: δₘ = Ʃf |d ₘ | ; δₓ = Ʃ f|d ₓ |
N N
Marks 5-15 15-25 25-35 35-45 45-55 55-65 65-75 75-85 85-95 Total
Freque
ncy
3 8 15 20 25 10 9 6 4 100
19
20. Department of Economics
MD from Mean
_
X = ƩfX = 4760 = 47.6
N 100
δₓ = Ʃf |d ₓ | =1499.2 or 14.99
N 100
C of δₓ = δₓ or 14.99 = 0.314
X 47.6
MD from Median
Median =size of (N )th item
2
= 50th item = (45-55)
M = l + i (m –c) = 45 +10 (50-46)
f 25
= 46.6
δₘ = Ʃ f|d ₘ | = 1507.2 or 15.07
N 100
C of δₘ = δₘ or 15.07 = 0.323
M 46.6
Mid
-
poin
t
frequency C.f f x X Deviation from median
(46.6)
Deviation from mean
(47.6)
Median
of signs
Total
deviation
Ignoring
+ and -
Total
deviation
X F C.f fxX |d ₘ | f x |d ₘ | |d ₓ | fx |d ₓ |
10 3 3 30 36.6 109.8 37.6 112.8
20 8 11 160 26.6 212.8 27.6 220.8
30 15 26 450 16.6 249.0 17.6 264.0
40 20 46 800 6.6 132.0 7.6 152.0
50 25 71 1250 3.4 85.0 2.4 60.0
60 10 81 600 13.4 134.0 12.4 124.0
70 9 90 630 23.4 210.6 22.4 201.6
80 6 96 480 33.4 200.4 32.4 194.4
90 4 100 360 43.4 173.6 42.4 169.6
Total 100 4760 1507.2 1499.2
N Ʃ fx Ʃf |d ₘ| Ʃ f|d ₓ|
20
21. Merits
Simple to understand
Easy to compute
Less effected by extreme items
Useful in fields like Economics, Commerce etc.
Comparisons about formation of different series can be easily made as
deviations are taken from a central value
Demerits
Ignoring ‘±’ signs are not appropriate
Not accurate for Mode
Difficult to calculate if value of Mean or Median comes in fractions
Not capable of further algebraic treatment
Not used in statistical conclusions.
Merits and Demerits of Mean Deviation
22. Unit End Questions
1. Find out Range of the following values-
20,8,10,0,-20,10,4
2. Calculate coefficient of Quartile deviation from the following –
Class 0-10 10-20 20-30 30-40 40-50
f 4 15 28 16 7
3. Calculate coefficient of Mean deviation from the following –
Class 0-10 0-20 0-30 0-40 0-50
f 12 13 28 29 50
