# What is the standard deviation of 1, 2, 3, 4, and 5?

Jan 24, 2017

$\frac{\sqrt{10}}{2}$

#### Explanation:

The formula for standard deviation is

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

In this case $n = 5$

and $\overline{x} = \frac{1}{n} {\sum}_{k = 1}^{n} {x}_{k} = \frac{1}{5} {\sum}_{k = 1}^{5} k = \frac{1 + 2 + 3 + 4 + 5}{5} = \frac{15}{5} = 3$

then the sum of squares ${\sum}_{k = 1}^{n} {\left(x - \overline{x}\right)}_{k}^{2}$ is

${\sum}_{k = 1}^{5} {\left(x - 3\right)}_{k}^{2} = {\left(- 2\right)}^{2} + {\left(- 1\right)}^{2} + {0}^{2} + {1}^{2} + {2}^{2}$

$= 4 + 1 + 0 + 1 + 4 = 10$

Then we plug in

$s = \sqrt{\frac{\sum {\left(x - \overline{x}\right)}^{2}}{n - 1}} = \sqrt{\frac{10}{4}} = \frac{\sqrt{10}}{2}$