5.58 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICSMeasures of Central Tendency and Measures of Dispersion
Where d 1 =x x ;d 1 − 123 2 =x 2 −x ;d 123 3 =x 3 −x 123
Example 54 : Two samples of sizes 40 and 50 respectively have the same mean 53, but different standard
deviations 19 and 8 respectively. Find the Standard Deviations of the combined sample of size 90.
Solution:
Here, n 1 =40, x 1 =53,σ = 1 19 ; n 2 =50, x 2 =53,σ = 2 8Now, 12 1 1 1 2 2 2
x n x n x 40 53 50 53
n n 40 50
= + = × + ×
+ + =2 120 2650 4770 53
90 90+ = =Now, d 1 =x 1 −x 12 =53 53 0,− = d 2 =x 2 −x 12 =53 53 0,− =( )^2 ( )^2 ( )^2 ( )^2
1240 19 50 8 40 0 50 0
40 50 + + +
σ = +
14440 3200 (^17640) 196 14
90 90
= + = = =
VARIANCE :
The square of the Standard Deviation is known as Variance.
COEFFICIENT OF VARIATION :
It is the ratio of the Standard Deviation to the Mean expressed as percentage. This relative measure was first
suggested by Professor Kari Pearson. According to him, coefficient is the percentage variation in the Mean,
while Standard Deviation is the total variation in the Mean.
Symbolically,
Coefficient of variation (V) =xσ×^100 = Coefficient of stand. deviation × 100.
Note : The coefficient of variation is also known as coefficient at variability. It is expressed as percentage.
Example 55 : If Mean and Standard deviation of a series are respectively 40 and 10, then the coefficient of
variations would be 10 / 40 × 100 = 25%, which means the standard deviation is 25% of the mean.
Example 56: An analysis of the monthly wages paid to workers in two firms, A and B, belonging to the same
industry gives the following results :
Firm A Firm B
No. of wage-earners 586 648
Average monthly wages 52.5 47.5
Variance of distribution of wages 100 121
(a) Which firm A and B pays out the largest amount as monthly wages?
(b) Which firm A and B has greater variability in individual wages?
(c) Find the average monthly wages and the standard deviation of the wages of all the workers in two
firms A and B together.