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 cows2, 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
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 Xis are uncorrelated. |