This document discusses measures of dispersion in economics, which quantify how data values are spread around the average. It defines four main measures: range, which is the difference between highest and lowest values; mean deviation, which is the average absolute deviation from the mean or median; variance, which is the average of squared deviations from the mean; and standard deviation, which is the square root of the variance. Formulas are provided for calculating each measure from both ungrouped and grouped frequency distribution data. Examples are included to demonstrate calculating the measures.

1. ECONOMICS
TOOLS OF ECONOMIC
ANALYSIS
Short revision series

2. MEASURES OF DISPERSIONS
 A quantity that measures the variability
among the data, or how the data one
dispersed about the average, known as
Measures of dispersion, scatter, or
variations.

3. Common Measures of
Dispersion
 The main measures of dispersion
1. Range
2. Mean
3. The variance &
4. the standard deviation

4. 1. RANGE
 It is the difference between the largest and the
smallest observation in a set of data.
 Range = xm – xo
 Its relative measure known as coefficient of
dispersion.
 Coefficient of dispersion =
 It is used in daily temperature recording stick prices
rate
 It ignores all the information available in middle of
data.
 It might give a misleading picture of the spread of
data.
om
om
xx
xx



5. 1. RANGE
 Example:
1. Find the range in the following data.
31,26,15,43,19,10,12,37
Range = xm – xo 33 = 43 – 10
2. Find the range in the following F.D. (Ungrouped)
5 = 8 – 3
Range 5 = 8 – 3
3. Find the range in the following data.
Range = 60 – 10 = 50
X 3 4 5 6 7 8
f 5 8 12 10 4 2
X 10 -
20
20 -
30
30- 40 40 –
50
50 - 60
f 5 8 12 10 4

6. MEAN (OR AVERAGE) DEVIATION
 It is defined as the “Arithmetic mean of the
absolute deviation measured either from the
mean or median.
 for ungroup.
 for grouped.
n
xx
DM
 
..
N
xxf 

N
medianx 
N
medianxf 


7. MEAN (OR AVERAGE) DEVIATION
 Example:
1. Calculate mean deviation from the FD (Ungrouped Data).
MD (x) = 33.6 / 20 = 1.68
X f f.x I x – 4.9
I
f I x - 4.9 I
2 3 6 2.9 8.7
4 9 36 0.9 8.1
6 5 30 1.1 5.5
8 2 16 3.1 6.2
10 1 10 5.1 5.1
Total Σf =20 Σf.x =98 Σ f I x - 4.9 I = 33.6

8. MEAN (OR AVERAGE) DEVIATION
 Exp: Calculate mean deviation from the FD (Grouped Data).
MD (x) = 33.6 / 20 = 1.68
M.D = 23.72 / 14 = 1.69
X f Class
Mark
( x )
f.x I x –
6.57 I
f I x – 6.57 I
2 – 4 2 3 6 3.57 7.14
4 - 6 3 5 15 1.57 4.71
6 – 8 6 7 42 0.43 2.58
8 – 10 2 9 18 2.43 4.86
10 – 12 1 11 11 4.43 4.43
Total Σf =14 Σ f.x
=92
Σ f I x – 6.57 I =
23.72
ẋ=92/14=6.5
7

9. MEAN (OR AVERAGE) DEVIATION
 It is an absolute measure.
 It is relative measure is coefficient of M.D.
 Coefficient of M.D. =
 It is based on all the observed values.
median
DM
or
mean
DM ....

10. THE VARIANCE AND
STANDARD DEVIATION
 It is defined as “The mean of the squares of
deviations of all the observation from their
mean.” It’s square root is called “standard
deviation”.
 Usually it is denoted by (for population of
statistics) S2 (for sample)
 = for ungrouped
2

2

n
xx  2
)(

11. THE VARIANCE AND
STANDARD DEVIATION
 = for grouped
 It is an absolute measure;
 It is relative measure is coefficient of variation.

 Shortcut method
N
xxf  2
)(2

100. 


VC 100
..
.. 
x
DS
VC
22
2










N
x
N
x

22
2
.










N
fx
N
xf


12. VARIANCE AND STANDARD
DEVIATION
 Example:
1. Calculate Variance and SD from the FD (Ungrouped Data).
Using Short cut method
var = (564 / 20) - (98 / 20) ^ 2 = 28.2 – 24.01 = 4.09
Sd = √ σ^2 = √ 4.09 = 2.02
X f f.x X^2 f.x^2
2 3 6 4 12
4 9 36 16 144
6 5 30 36 180
8 2 16 64 128
10 1 10 100 100
Tota
l
Σf =20 Σf.x = 98 Σ f.x^2=564
22
2
.










N
fx
N
xf


13. VARIANCE AND STANDARD
DEVIATION
 Exp: Calculate Variance and Standard deviation from the FD (Grouped Data).
Using Short cut method:
var = (670 /14) - (92 / 14) ^ 2 = 47.85 – 43.18 = 4.67
Sd = √ σ^2 = √ 4.67 = 2.16
X f Class
Mark
( x )
f.x x^2 f.x^2
2 – 4 2 3 6 9 18
4 - 6 3 5 15 25 75
6 – 8 6 7 42 49 294
8 – 10 2 9 18 81 162
10 – 12 1 11 11 121 121
Total Σf =14 Σ f.x
=92
Σ f.x^2 =670
22
2
.










N
fx
N
xf
