## Variance and Standard deviation for grouped data

Use this calculator to find the variance and standard deviation for grouped data.

Variance and std. deviation Calculator
Type of Freq. Dist. DiscreteContinuous
Enter the Classes for X (Separated by comma,)
Enter the frequencies (f) (Separated by comma,)
Results
Number of Obs. (n):
Sample Mean : ($\overline{x}$)
Sample variance : ($s^2_x$)
Sample std. Deviation : ($s_x$)

## Variance and Standard deviation for grouped data

Let $x_i, i=1,2, \cdots , n$ be $n$ observations then the mean of $X$ is denoted by $\overline{X}$ and is given by $$\begin{eqnarray*} \overline{X}& =\frac{1}{n}\sum_{i=1}^{n}x_i \end{eqnarray*}$$

## Formula

Variance of $X$ is denoted by $s_{x}^2$ and is given by

### $s_x^2 =\frac{1}{n-1}\sum_{i=1}^{n}(x_i -\overline{x})^2 =\frac{1}{n-1}\bigg(n\sum_{i=1}^{n}x_i^2-\big(\sum_{i=1}^nx_i\big)^2\bigg)$

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