## Variance and standard deviation for ungrouped data

Let $x_i, i=1,2, \cdots , n$ be $n$ observations.

## Formula

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

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

where,

• $\overline{x}=\dfrac{1}{n}\sum_{i=1}^{n}x_i$ is the sample mean

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