# Standard deviation *s*

The standard deviation is the positive square root of the variance, i.e. s = √*V*. Thus, if the variance has units of cows^{2}, the standard deviation has units of cows, the same as the variable x. The standard deviation is therefore more popularly used to express a measure of spread.

*Example*

*Example*

The variance *V* of the Uniform(1,3) distribution is calculated as follows:

| \bar{x}=2 | from here |

and therefore

V=\frac{26}{6}-2^2=\frac{1}{3} |

and the standard deviation *s* is therefore:

\sigma=\sqrt V= \frac{1}{\sqrt 3} |

Variance and standard deviation have the following properties, where *a* is some constant and X, Xi are random variables:

V\big(X\big)\geq 0 | and | \sigma\big(X\big)\geq 0 |

V\big(aX \big)=a^2V\big(X\big) | and | \sigma\big(aX\big)=a\sigma\big(X\big) |

| providing the |