Question

Standard deviation about mean $\overline{x}$ for a given discrete frequency distribution x₁, x₂, x₃, ....x_n with frequencies f₁, f₂, f₃, .... f_n is.

Answer 1

We know variance for a discrete frequency distribution can be calculated as follows,

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

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.

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

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