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STANDARD DEVIATION (S.D.)

Standard deviation (and variance) is a relative measure of the dispersion of a set of data-the larger the standard deviation, the more spread out the data.

If xl, x2, ....... , xn be a set of n observations forming a population, then its standard deviation is given by

For simple frequency distribution,

For grouped frequency distribution, X; is taken as class mark.

Note. 1. The square of S.D. is !mown as variance.

2. When x is a real number, then the following alternative formula can be used to · calculate S.D.

3. For comparing the vanability of two distributions the coefficient of variation (C.V.) is calculated as follows :

The distribution with less C.Y. is said to be more uniform or consistent or less variable or more homogeneous.

4. If the values of x and fare large then for simplicity step-deviation method can be employed in which the deviations of the given values of x from any arbitrary point A is taken.

Which shows that the variance and S.D. of a distribution is independent of change of origin.

5. In case of grouped frequency distribution, if h be the width of the class-interval then we can use

6. (Combined Variance). If .~1 and cr1 be the mean and S.D. of a group of observations of si7e n1 whereas .X2 and cr2 be the mean and S.D. of another group of observations of size n2, then the variance of the combined groups is given by :

where d1 = .X1 - x, d2 = .X2 - .X and X = Mean of the combined groups.

7. S.D. is independent of origin but not of scale. 

Example 1. A group of 100 items have a mean of 60 and a S.D. of 7. If the mean and S.D. of 60 of these items be 51 and 5.2 respectively, find tlze S.D. of the other 40 items. 

Solution. Consider the following :

From the combined mean we obtain,

From the combined variance we obtain,

 14400  =40s2+13772.4

S2  =     627.6/40 = 15.69

S   =    3.961