# How do you find the standard deviation of a set of numbers?

Feb 24, 2017

Population standard deviation:
$\sigma = \sqrt{\frac{{\left({x}_{1} - \overline{x}\right)}^{2} + {\left({x}_{2} - \overline{x}\right)}^{2} + \left(\ldots\right) + {\left({x}_{n} - \overline{x}\right)}^{2}}{n}}$

Sample standard deviation:
$s = \sqrt{\frac{{\left({x}_{1} - \overline{x}\right)}^{2} + {\left({x}_{2} - \overline{x}\right)}^{2} + \left(\ldots\right) + {\left({x}_{n} - \overline{x}\right)}^{2}}{n - 1}}$

#### Explanation:

This is the process to finding the standard deviation for a sample :

Find the mean of the set of numbers: $\overline{x} = \frac{{x}_{1} + {x}_{2} + \ldots + {x}_{n}}{n}$ where $n =$ the number of numbers in the set.

Subtract the mean from each number in your sample, square the difference and add: ${\left({x}_{1} - \overline{x}\right)}^{2} + {\left({x}_{2} - \overline{x}\right)}^{2} + \left(\ldots\right) + {\left({x}_{n} - \overline{x}\right)}^{2}$

Divide these numbers by $n - 1$ to find the variance of you set. Dividing by $n - 1$ provides an unbiased sample variance.

Square root the variance to get the standard deviation to the mean:

$s = \sqrt{\frac{{\left({x}_{1} - \overline{x}\right)}^{2} + {\left({x}_{2} - \overline{x}\right)}^{2} + \left(\ldots\right) + {\left({x}_{n} - \overline{x}\right)}^{2}}{n - 1}}$