BBA 13-09-2021
Abdurrahman sabir
Abdurrahmansabir27@gmail.com
MPhil scholar
Variance
Variance of a set of observations is defined as the mean of the squares of deviations of all the
observations from their mean. When it is calculated from the entire population, the variance is
called population variance, which is denoted by ( is the Greek letter “sigma”). If instead, the
data from the sample are used to calculate the variance, it is referred to as the sample variance
and is denoted by . Symbolically the variance is
= , for the population data,
x
= , for the sample data
Note: The term variance was all also denoted by Var(x). the term variance was introduced in
1918 by R.A Fisher
Standard Deviation
The positive square root of the variance is called standard deviation.
Symbolically
=√ . for population data
√ x
for sample data
Co-efficient of standard deviation
The relative measure of standard deviation is called Co-efficient of standard deviation
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BBA 13-09-2021
To calculate the variance and standard deviation for the following formula and showing
that: =
Proof:
=
=
= ( )
=
Thus the sum of squares of the deviations from the mean is equal to the sum of the squares of all
minus a correction factor which is the of the square of the sum of all .
The variance is the mean of the square minus the square of the mean the corresponding formula
for sample variance is
x
= =
The alternative formulas for standard deviations are
=√ =√
√ x
= √
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BBA 13-09-2021
The following alternative formulas for the sample variance and standard deviation of a frequency
distribution are obtained in a similar way.
and
Example: A population of N =10 has the observations 7, 8, 10, 13, 14, 19, 20, 25, 26 and 28
Find the variance and standard deviation.
7 -10 100 49
8 -9 81 64
10 -7 49 100
13 -4 16 169
14 -3 9 196
19 -2 4 361
20 3 9 400
25 8 64 625
26 9 81 676
28 11 121 784
=170 0 = 534 = 3424
Now, = = = 17
Therefore, = = =53.4
And =√ =√ =√ =7.31
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BBA 13-09-2021
Using alternative method
= =53.4
And √
√ =7.31
Example: Calculate the variance and standard deviation from the following marks obtained 9
students.
45. 32, 37, 46, 39, 36, 41, 48, 36
Find the variance and standard deviation from the following data.
45 5 25 2025
32 -8 64 1024
37 -3 9 1369
46 6 36 2116
39 -1 1 1521
36 -4 16 1296
41 1 1 1681
48 8 64 2304
36 -4 16 1296
=360 0 = 232 = 14632
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BBA 13-09-2021
Now, x= = = 40 marks
x
Therefore, = = = 25.78 marks
√ x
And, = √ =√ = 5.08 marks
Example: calculate the variance and standard deviation from the following data
which showing the weights of apples:
Weight 65-84 85-104 105-124 125-144 145-164 165-184 184-204
(grams)
f 9 10 17 10 5 4 5
Sol:
Weight (grams)
65-84 74.5 9 670.5 5550.25 49952.25
85-104 94.5 10 945 8930.25 89302.5
105-124 114.5 17 1946.5 13110.25 222874.3
125-144 134.5 10 1345 18090.25 180902.5
145-164 154.5 5 772.5 23870.25 119351.3
165-184 174.5 4 698 30450.25 121801
184-204 194.5 5 972.5 37830.25 189151.3
Total 60 = 7350 973335
Thus we find,
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BBA 13-09-2021
=√
=√
= 34.87 grams
Co-efficient of Variation
The variability of two or more than two sets of data cannot be compared unless we have a
relative measure of dispersion. For this purpose, Karl Pearson (1857-1936) introduced a relative
measure of variation, known as the Co-efficient of variation, abbreviated C.V. which expresses
the standard deviation as a percentage of the arithmetic mean of a data set.
Symbolically, C.V = X 100
C.V = X , for population data
x
C.V = X , for sample data
x
Note: As the coefficient of variation is a pure number without units, it is therefore used to
compare the variation in two or more data sets or distributions that are measured in different
units, e.g. one may be measured in hours and the other in kilograms or rupees. A large value of
C.V; indicates that the variability is great and a small value of C.V. indicates less variability
The coefficient of variation is also used to compare the performance of two candidates or
of two players given their scores in various papers. or games, the smaller the coefficient of
variation the mean consistent is the performance of the candidates or players. Thus it is used as a
criterion for the consistence performance of the candidates or the players. It should be noted that
this co-efficient is unreliable when the arithmetic mean is very small.
Example: Using the co-efficient of variation, determine whether or not there is greater variation
among the prices of certain similar commodities given, than among the life in hours under test.
Price in Rs 8 13 18 23 30
Life in Hrs 130 150 180 250 345
Sol:
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BBA 13-09-2021
Price in Rs (X) Life in Hrs ( Y)
X Y
8 64 130 16900
13 169 150 22500
18 324 180 32400
23 529 250 62500
30 900 345 119025
= 92 = 1986 = 1055 = 253325
Price of commodities life in hours
X= = 18.4 Rs Y = = = 211 Hrs
√ √
√ √
√ √
√ √
7.66 Rs = 78.38 Hrs
Co-efficient of variation
C.V = X C.V = X
X Y
C.V = X C.V = X
C.V = 41.63% C.V = 37.15%
Results: We see that the co-efficient of variation for the prices of commodities (X) is larger than
that for the life in hours (Y). Hence the prices of certain similar commodities are showing greater
variation than that among the life in hours under test.
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BBA 13-09-2021
Example: Goals scored by two teams A and B in a football season were as follows.
No. of goals Number of Matches (frequencies
scored in a match (xi) A B
0 27 17
1 9 9
2 8 6
3 5 5
4 4 3
Using coefficient of variation, find which team may be considered more consistent?
Sol:
No. of Team A Team B
goals ( )
0 27 0 0 17 0 0
1 9 9 9 9 9 9
2 8 16 32 6 12 24
3 5 15 45 5 15 45
4 4 16 64 3 12 48
Total 53 56 150 40 48 126
Team A: Team B:
XA = = =1.06 XB = = =1.20
√ √
√ √
√ √
1.308
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BBA 13-09-2021
Co-efficient of variation
C.V = X C.V = X
XA XB
C.V = X C.V = X
C.V = 123.4% C.V = 109 %
Results: We see that the co-efficient of variation for the team B is smaller than that for the
team A. Hence team B is more consistent than team A.
