## Variance and Standard deviation for ungrouped data

Use this calculator to find the variance and standard deviation for ungrouped (raw) data.

Variance and std. deviation Calculator | |
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Enter the X Values (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 ungrouped data

Use this calculator to find the variance and standard deviation for ungrouped (raw) 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
```
$$
\begin{eqnarray*}
\overline{X}& =\frac{1}{n}\sum_{i=1}^{n}x_i
\end{eqnarray*}
$$
```

Variance of $X$ is denoted by $s_{x}^2$ and is given by
```
$$
\begin{aligned}
s_x^2 & =\frac{1}{n-1}\sum_{i=1}^{n}(x_i -\overline{x})^2\\ &=\frac{1}{n-1}\bigg(n\sum_{i=1}^{n} x_i^2-\big(\sum_{i=1}^nx_i\big)^2\bigg)
\end{aligned}
$$
```

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

```
$$
\begin{eqnarray*}
s_x & =\sqrt{s_x^2}
\end{eqnarray*}
$$
```