Question

Standard deviation about mean $\left(\overline{x}\right)$ for a given discrete frequency distribution $x_1,x_2,x_3,.....x_n$ with frequencies $f_1,f_2,f_3,...f_n$ is :

• A. $\sqrt{{\sum }_{i=1}^n\frac{(x_i-\overline{x}{)}^2}{n}}$
• B. $\sqrt{\frac{{\sum }_{i=1}^{n}fi(x_i-\overline{x}{)}^2}{{\sum }_{i=1}^{n}fi}}$
• C. $\frac{\sqrt{{\sum }_{i=1}^{n}fi(x_i-\overline{x}{)}^2}}{{\sum }_{i=1}^{n}fi}$
• D. $\sqrt{\frac{{\sum }_{i=1}^{n}fi(x_i-\overline{x}{)}^2}{n}}$

Answer 1

The correct option is B $\sqrt{\frac{{\sum }_{i=1}^{n}fi(x_i-\overline{x}{)}^2}{{\sum }_{i=1}^{n}fi}}$

Variance for a discrete frequency distribution can be calculated as follows,

${\sigma }^2$ = $\frac{{\sum }_{i=1}^{n}fi(x_i-\overline{x}{)}^2}{{\sum }_{i=1}^{n}fi}$

This can be understood from normal variance calculation as each value xi occurs fi time total number of observations is ${\sum }_{i=1}^{n}fi$ and each deviation should be added fi times i.e.

$\to {\sum }_{i=1}^n{f}_i(x_i-\overline{x}{)}^2$

$\therefore$ Standard deviation = $\sigma =\sqrt{\frac{{\sum }_{i=1}^{n}fi(x_i-\overline{x}{)}^2}{{\sum }_{i=1}^{n}fi}}$