## Variance and standard deviation for grouped data

Let `$(x_i,f_i), i=1,2, \cdots , n$`

be the observed frequency distribution.

## Formula

## Sample variance

The sample variance of $X$ is denoted by $s_x^2$ and is given by

`$s_x^2 =\dfrac{1}{N-1}\sum_{i=1}^{n}f_i(x_i -\overline{x})^2$`

OR

`$s_x^2 =\dfrac{1}{N-1}\bigg(\sum_{i=1}^{n}f_ix_i^2-\frac{\big(\sum_{i=1}^n f_ix_i\big)^2}{N}\bigg)$`

where,

`$N=\sum_{i=1}^n f_i$`

is the total number of observations,`$\overline{x}$`

is the sample mean.

## Sample standard deviation

The sample standard deviation of $X$ is defined as the positive square root of sample variance. The sample standard deviation of $X$ is given by