# Why are there 2 formulas for standard deviation?

Mar 17, 2018

see below

#### Explanation:

The two formulas, as shown below, are equivalent. They are alternate forms and which one is used depends on which is the most efficient method with the given data.

for a set of numbers

${x}_{1} , {x}_{2} , {x}_{3} , \ldots {x}_{i} , \ldots , {x}_{n}$

and mean

$\overline{x}$

the standard deviation is

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

arrange as follows

$s {d}^{2} = \frac{1}{n} \sum {\left({x}_{i} - \overline{x}\right)}^{2}$

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

$s {d}^{2} = \frac{\sum {x}_{i}^{2}}{n} - 2 \overline{x} \frac{\sum {x}_{i}}{n} + \frac{n {\overline{x}}^{2}}{n}$

$s {d}^{2} = \frac{\sum {x}_{i}}{n} - 2 \overline{x} \overline{x} + {\overline{x}}^{2}$

$s {d}^{2} = \frac{\sum {x}_{i}}{n} - 2 {\overline{x}}^{2} + {\overline{x}}^{2}$

$s {d}^{2} = \frac{\sum {x}_{i}}{n} - {\overline{x}}^{2}$

$s d = \sqrt{\frac{\sum {x}_{i}}{n} - {\overline{x}}^{2}} - - \left(2\right)$