This document discusses measures of dispersion, specifically standard deviation and coefficient of variation. It begins by defining standard deviation as a measure of how spread out numbers are from the mean. It then provides the formula for calculating standard deviation and discusses its properties. Several examples are shown to demonstrate calculating standard deviation for individual data series using both the direct and shortcut methods. The document also discusses calculating standard deviation for discrete and continuous data series. It concludes by defining variance and coefficient of variation, and providing an example to calculate coefficient of variation and determine which of two company's share prices is more stable.

1. Dr Rekha Choudhary
Department of Economics
Jai Narain Vyas University, Jodhpur
Rajasthan
ECONOMICS
BASICSTATISTICS

2. Measures Of Dispersion: Standard Deviation
And Co- efficient Of Variation
Department of Economics 2

3. Department of Economics 3
Introduction
Standard deviation measures the spread of a data distribution. The more
spread out a data distribution is, the greater its standard deviation.
Interestingly, standard deviation cannot be negative. A standard deviation
close to 0 indicates that the data points tend to be close to the mean. The
further the data points are from the mean, the greater the standard
deviation. The standard deviation is a measure of the spread of scores
within a set of data. Usually, we are interested in the standard deviation of
a population

4. Department of Economics 4
Objectives
After going through this unit, you will be able
to:
To understand the concept of Standard
deviation;
 Define Standard deviation , Coefficient of
variation, Variance and Combined Standard
deviation ;
Merits and Demerits Standard deviation
Some particles problem of Standard
deviation in different series with different
methods

5. A standard deviation is the positive square root of the arithmetic mean of the squares
of the deviations of the given values from their arithmetic mean. It is denoted by a
Greek letter sigma, σ. It is also referred to as root mean square deviation
Properties
 Most important & widely used measure of dispersion
 First used by Karl Pearson in 1893
 Also called root mean square deviations
 It is defined as the square root of the arithmetic mean of the squares of the
deviation of the values taken from the mean
 Denoted by σ (sigma)
Department of Economics 5
Standard Deviation
Definition
Formula
σ =√(Ʃd²) or σ =√(Ʃ(X - X̅ ) ²
N N

6. Department of Economics 6
Calculation Of Standard Deviation

7. Department of Economics 7
Standard Deviation – Individual Series
Direct Method/ Actual Mean Method
• First of all compute arithmetic mean of the series (X̅ )
• Take deviation of different values from the value of mean d =(X - X̅ )
• Each deviation is squared up and their total is obtained (Ʃd²), than dived by N
• Square root of the mean of squared deviation is extracted. The result is standard
deviation.
σ = Standard Deviation
(Ʃd²)= Sum of Squares of deviation from mean
N = Number of items
σ =√(Ʃd²) or σ =√(Ʃ(X - X̅ ) ²
N N

8. Department of Economics 8
Example: Calculate the SD and its coefficient of the following data:
41, 44,45,49, 50, 53, 55, 55, 58,60
Weight (kg) Deviations((X̅ ) Squares of
deviation
X d= (X -X̅ ) d²= (X -X̅ )²
41 -10 100
44 -7 49
45 -2 36
49 -1 4
50 2 1
53 4 4
55 4 16
55 7 16
58 9 49
60 81
ƩX =510 Ʃd²= Ʃ (X -X̅ )²
356
Arithmetic Mean (X̅ ) = ƩX
N
= 510 = 51 kg
10
Standard Deviation (σ) = √(Ʃd²) = 356
N 10
= 5.97
Coefficient of Standard deviation
C. of σ = σ or 5.97 = 0.117
X 51.00
Direct Method

9. Department of Economics 9
Standard Deviation – Individual Series
Short cut Method/ Assumed Mean Method
• When the value of mean is not in whole number it is easy to use short- cut
method .
• Any one of the values given is assumed as mean (A)
• Take deviation of the values are taken from assumed mean (dₓ) =(X- A)
• Each deviation is squared up and their total is obtained (Ʃdₓ²), than dived by N
• Square root of the mean of squared deviation is extracted. The result is standard
deviation.
σ = Standard Deviation
(Ʃdₓ²)= Sum of Squares of deviation from mean
N = Number of items
σ =√(Ʃdₓ²) (Ʃ dₓ ) ²
N N

10. Department of Economics 10
Example: Calculate the SD and its coefficient of the following data:
41, 44,45,49, 50, 53, 55, 55, 58,60
Standard Deviation (σ)= √(Ʃdₓ²) (Ʃ dₓ )²
N N
=√(366²) (10 )²
10 10
= 5.97
Coefficient of Standard deviation
C. of σ = σ or 5.97 = 0.117
X 51.00
Weight (kg) Deviations(
A) =50
Squares of
deviation
Squares of Values
X dₓ= (X- A) dₓ² X²
41 -9 81 1681
44 -6 36 1936
45 -5 25 2025
49 -1 1 2401
50 0 0 2500
53 3 9 2809
55 5 25 3025
55 5 25 3025
58 8 64 3364
60 10 100 3600
ƩX =510 10 Ʃdₓ² = 366 ƩX² =26366
Short Cut Method

11. Department of Economics 11
Standard Deviation – Discrete Series
Direct Method/ Actual Mean Method
• First of all compute arithmetic mean of the series (X̅ )
• Take deviation of different values from the value of mean d =(X - X̅ )
• Each deviation is squared up and their total is obtained (Ʃd²), than multiplied by
their corresponding frequencies (Ʃfd²),
• Square root of the mean of squared deviation is extracted. The result is standard
deviation. σ =√(Ʃfd²) or σ =√(Ʃf(X - X̅ ) ²
N N
σ = Standard Deviation
(Ʃfd²)= Sum of Squares of deviation from mean with multiplied
by their corresponding frequencies
N or Ʃf = Total frequencies

12. Department of Economics 12
Example: Calculate the SD and its coefficient of the following data:
Arithmetic
Mean (X̅ ) = ƩfX
N
= 1650 = 16.5
100
Standard
Deviation (σ) = √(Ʃfd²) = √1059
N 100
= 3.25
Coefficient of Standard deviation
C. of σ = σ or 3.25 = 0.197
X 16.50
Size Frequenc
y
Deviatio
n from X̅
=16.5
Squared
Deviatio
n
Product
of
Squares
deviation
With f
Size X
frequenc
y
X f d d² f x d² f x X
10 5 -6.5 42.25 211.25 50
12 8 -4.5 20.25 162.00 96
14 21 -2.5 6.25 131.25 294
16 24 -0.5 0.25 6.00 384
18 18 1.5 2.25 40.50 324
20 15 3.5 12.25 183.75 300
22 7 5.5 30.25 211.75 154
24 2 7.5 56.25 112.50 48
Total 100 Ʃfd²
=1059
Ʃ fx
=1650
Size 10 12 14 16 18 20 22 24
Frequency 5 8 21 24 18 15 7 2
Direct Method

13. Department of Economics 13
Standard Deviation – Discrete Series
Short cut Method/ Assumed Mean Method
• When the value of mean is not in whole number it is easy to use short- cut
method .
• Any one of the values given is assumed as mean (A)
• Take deviation of the values are taken from assumed mean (dₓ) =(X- A)
• Each deviation is squared up and their total is obtained (Ʃdₓ²), than multiplied by
f
• Square root of the mean of squared deviation is extracted. The result is standard
deviation. σ =√(Ʃfdₓ²) (Ʃ fdₓ ) ²
N N
σ = Standard Deviation
(Ʃfdₓ²)= Sum of Squares of deviation from mean with their
frequencies
N and f = Frequency

14. Department of Economics 14
Example: Calculate the SD and its coefficient of the following data:
Standard Deviation (σ)= √(Ʃfdₓ²) (Ʃ fdₓ ) ²
N N
=√(1084²) (50 ) ²
100 100
= 3.25
Coefficient of Standard deviation
C. of σ = σ or 3.25 = 0.197
X 16.5
Short Cut Method
Size 10 12 14 16 18 20 22 24
Frequency 5 8 21 24 18 15 7 2
Size Frequency Deviation
from A=16
Product of
f and dₓ
Product of
fdₓ and dₓ
X f dₓ f dₓ f x dₓ²
10 5 -6 -30 180
12 8 -4 -32 128
14 21 -2 -42 84
16 24 0 0 0
18 18 2 36 72
20 15 4 60 240
22 7 6 42 252
24 2 8 16 128
Total 100 Ʃfdₓ =50 Ʃfdₓ²=1084

15. Department of Economics 15
Standard Deviation – Continuous Series
Short –Cut Method
(σ)= √(Ʃfdₓ²) (Ʃ fdₓ ) ²
N N
Summation Method
(σ)= i x √2F₂ -F₁ - F²₁
Step deviation Method
(σ)= √(Ʃfd́ₓ²) (Ʃ fd́ₓ ) ² x i
N N
Direct Method
(σ)= √(Ʃfdₓ²)
N

16. Department of Economics 16
Standard Deviation – Continuous Series
Direct Method
(σ)= √(Ʃfdₓ²)
N
Short –Cut Method
(σ)= √(Ʃfdₓ²) (Ʃ fdₓ ) ²
N N
Example: Calculate arithmetic mean and standard deviation and its coefficient
from the following series:
Marks less
than
10 20 30 40 50 60 70
No of
students
10 25 50 75 85 95 100
Step deviation Method
(σ)= √(Ʃfd́ₓ²) (Ʃ fd́ₓ ) ² x i
N N

17. Department of Economics 17
Mark
s
Mid-
point
X
Frequ
ency
f
Devia
tion
X̅ =31
dₓ
Squar
ed
Devia
tion
dₓ²
Produ
ct of f
and
dₓ²
Total
marks
fx
Devia
tion
from
A =35
Produ
ct of f
x dₓ
Produ
ct of
fdₓ
And
dₓ
0-10 5 10 -26 676 6760 50 -30 -300 9000
10-20 15 15 -16 256 3840 225 -20 -300 6000
20-30 25 25 -6 36 900 625 -10 -250 2500
30-40 35 25 4 16 400 875 0 0 0
40-50 45 10 14 196 1960 450 10 100 1000
50-60 55 10 24 576 5760 550 20 200 4000
60-70 65 5 34 1156 5780 325 30 150 4500
Total 100 2540
0
3100 -400 2700
0
N=Ʃf Ʃfdₓ² ƩfX fdₓ Ʃfdₓ²
Mean =ƩfX = 3100 = 31
N 100
Direct Method
(σ)= √(Ʃfdₓ²) =√(25400)
N 100
(σ) = 15.94 marks
Mean = A + Ʃfdₓ = 35+ (-400) = 31
N 100
Short –Cut Method
(σ)= √(Ʃfdₓ²) (Ʃ fdₓ ) ²
N N
= √(27000) - (-400) ²
100 100
(σ)= 15.94 marks
Direct and Short- Cut method in Continuous series

18. Department of Economics 18
Marks Mid-
point
X
Freque
ncy
f
Deviati
on from
A =35
Product
of f x d́ₓ
Product
of fd́ₓ
And d́ₓ
0-10 5 10 -3 -30 90
10-20 15 15 -2 -30 60
20-30 25 25 -1 -25 25
30-40 35 25 0 0 0
40-50 45 10 1 10 10
50-60 55 10 2 20 40
60-70 65 5 3 15 45
Total 100 -40 270
N=Ʃf fd́ₓ Ʃfd́ₓ²
Mean = A + Ʃfd́ₓ x i = 35+ (-40) x 10
N 100
Mean =31
Step deviation Method
(σ)= √(Ʃfd́ₓ²) (Ʃ fd́ₓ ) ² x i
N N
= √(270) - (-40) ² x10
100 100
(σ)= 15.94 marks
Step deviation method in Continuous series

19. Merits of Standard Deviation
 Squaring the deviations overcomes the drawback of ignoring signs in
mean deviations
 Suitable for further mathematical treatment
 Least affected by the fluctuation of the observations
 The standard deviation is zero if all the observations are constant
 Independent of change of origin
Demerits of Standard Deviation
 Not easy to calculate
 Difficult to understand for a layman
 Dependent on the change of scale
Department of Economics 19
Merits and Demerits of Standard Deviation

20.  It was developed by Karl Pearson.
 It is an important relative measure of dispersion.
 It is used in comparing the variability, homogeneity, stability, uniformity
& consistency of two or more series.
 Higher the CV, lesser the consistency.
Department of Economics 20
Coefficient Of Variation (C.V.)
Definition
Formula
C.V. = 𝜎 x 100
X̅

21. Variance is a measure also based on std. deviation. It is in fact the square of standard
deviation (σ²). It is also know as second moment of dispersion.
Variance = (SD)² = σ²
Department of Economics 21
Variance
Definition
Example: Prices of shares of B and C company are given below. Determine shares
of which company are more stable in prices-
B Co 55 54 52 53 56 58 52 50 51 49
C Co 108 107 105 105 106 107 104 103 104 101

22. Department of Economics 22
Computation of Coefficient of variation
Shares of B Co. X Shares of C Co. Y
Share
Prices
Deviation
from X
Squares of
deviation
Share
Prices
Deviation
from Y
Squares of
deviation
X dₓ d²ₓ Y dˠ d²ˠ
55 2 4 108 3 9
54 1 1 107 2 4
52 -1 1 105 0 0
53 0 0 105 0 0
56 3 9 106 1 1
58 5 25 107 2 4
52 -1 1 104 -1 1
50 -3 3 103 -2 4
51 -2 4 104 -1 1
49 -4 16 101 -4 16
ƩX=530 Ʃd²ₓ= 70 ƩY=1050 Ʃd²ˠ=40
X̅ =ƩX= 530 = 53
N 10
σ =√(Ʃd²) = √70 = √7 =2.64
N 10
C of V = σ x 100 =2.64 x100
X̅ 53
= 4.992 %
Y̅ =ƩY= 1050= 105
N 10
σ =√(Ʃd²) = √40 = √4 =2
N 10
C of V = σ x 100 =2 x100
Y̅ 105
= 1.905 %
The share value of C
company are more
consistent

23. It is the combined standard deviation of two or more groups as in case of
combined arithmetic mean
Formula
σ = √(N₁(σ₁² + D₁²) + N₂(σ ₂ ² + D ₂ ²) + N ₃(σ ₃ ² + D ₃ ²) …..)
N₁ +N₂ +N₃
Department of Economics 23
Combined Standard Deviation
Definition
• First combined mean is ascertained
• Then, find out differences of group means and combined mean, e.g
D₁ = X̅ ₁ - X̅ , D₂ = X̅ ₂ - X, D₃ = X̅ ₃ - X and so on…
• Apply Formula - √(N₁(σ₁² + D₁²) + N₂(σ ₂ ² + D ₂ ²) + N ₃(σ ₃ ² + D ₃ ²) …..)
N₁ +N₂ +N₃

24. Example :Two samples of sizes 40 & 60 respectively have means 20 & 25and
SD 6 & 9. Find the Combined Mean & Combined Standard Deviation.
Department of Economics 24
Calculation of Combined Mean & S.D
N₁ = 40 X̅ ₁ = 20, σ₁ = 6
N₂ = 60 X̅₂ = 25, σ₂ = 9
Combined Mean X̅ = X̅₁ N₁ + X̅ ₂ N₂ = 20x 40 +25x60 = 23
N₁ +N₂ 40 +60
D₁ = X̅ ₁ - X̅ =20-23 =-3 , D²₁ =9 , σ²₁ =36
D₂ = X̅ ₂ - X̅ = 25-23 =2 , D²₂ =4 , σ²₂ =81
Combined Standard deviation σ₁.₂ =√(N₁(σ₁² + D₁²) + N₂(σ ₂ ² + D ₂ ²) .)
N₁ +N₂
= √(40(36+ 9) + 60(81 + 4))
40 +60
=√6900 = 8.3066
100
Thus, combined std. deviation of all the groups is 8.3066

25. Department of Economics 25
Unit End Questions
1. The mean weight of 150 students is 60 kg. The mean weight of boys is 70 kg
with SD of 10 kg. The mean weight of girls is 55 kg with SD of 15 kg. Find the
number of boys & girls and their combined standard deviation.
2. For a group of 100 observations, the mean & SD were found to be 60 & 5
respectively. Later on, it was discovered that a correct item 50 was wrongly
copied as 30. Find the correct mean & correct SD.
3. Calculate Standard deviation and its coefficient from the following data –
Age
Group
0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80
Freque
ncy
3 61 223 137 53 19 4 2

26. Department of Economics 26
Required Readings
Asthana H.S, and Bhushan, B.(2007) Statistics for Social Sciences (with SPSS
Applications). Prentice Hall of India
B.L.Aggrawal (2009). Basic Statistics. New Age International Publisher, Delhi.
Gupta, S.C.(1990) Fundamentals of Statistics. Himalaya Publishing House, Mumbai
Elhance, D.N: Fundamental of Statistics
Singhal, M.L: Elements of Statistics
Nagar, A.L. and Das, R.K.: Basic Statistics
Croxton Cowden: Applied General Statistics
Nagar, K.N.: Sankhyiki ke mool tatva
Gupta, BN : Sankhyiki
https://blog.udemy.com/statistics-formula/

27. Department of Economics 27