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.
Coefficient of variation 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.
(a) For firm A : total wages = 586 × 52.5 = 30,765. For firm B : Total wages = 648 × 47.5 = 30,780. i.e. Firm B pays largest amount.
(b) For firm A : σ2 = 100 ∴ σ = 10
∴ Firm B has greater variability, as its coefficient of variation is greater than that of Firm A
Example 57 : In an examination a candidate scores the following percentage of marks :
English 2nd language mathematics Science Economics
62 74 58 61 44
Find the candidates weighted mean percentage weighted of 3, 4, 4, 5 and 2 respectively are allotted of the subject. Find also the coefficient of variation.
Example 58 : The A.M. of the following frequency distribution is 1.46. Find f1 and f2 .
No. of accidents : 0 1 2 3 4 5 total
No. of days : 46 f1 f2 25 10 5 200
Also find coefficient of variation.
Putting these values of f1 and f2 we find the following distribution:
Advantages of Standard Deviation :
1. Standard deviation is based on all the observations and is rigidly defined.
2. It is amenable to algebraic treatment and possesses many mathematical properties.
3. It is less affected by fluctuations of sampling than most other measures of dispersion.
4. For comparing variability of two or more series, coefficient of variation is considered as most appropriate and this is based on standard deviation and mean.
Disadvantages of Standard Deviation :
1. It is not easy to understand and calculate.
2. It gives more weight to the extremes and less to the items nearer to the mean, since the squares of the deviations of bigger sizes would be proportionately greater than that which are comparatively small. The deviations 2 and 6 are in the ratio of 1 : 3 but their squares 4 and 36 would be in the ratio of 1 : 9.
Uses of Standard Deviation : It is best measure of dispersion, and should be used wherever possible.