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:

$\begin{array}{l}\displaystyle E\big(x^2 \big)=\int \frac{1}{2}x^2 dx= {\bigg[\frac{x^3}{6}\bigg] }_1^3 = \frac{27-1}{6}= \frac{13}{3}\end{array}$

$\begin{array}{l}\bar{x}=2\end{array}$

from here

and therefore

$\begin{array}{l}\displaystyle V=\frac{26}{6}-2^2=\frac{1}{3}\end{array}$

and the standard deviation s is therefore:

$\begin{array}{l}\displaystyle \sigma=\sqrt V= \frac{1}{\sqrt 3}\end{array}$

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

$\begin{array}{l}V\big(X\big)\geq 0\end{array}$

and

$\begin{array}{l}\sigma\big(X\big)\geq 0\end{array}$

$\begin{array}{l}V\big(aX \big)=a^2V\big(X\big)\end{array}$

and

$\begin{array}{l}\sigma\big(aX\big)=a\sigma\big(X\big)\end{array}$

$\begin{array}{l}\displaystyle V\bigg( \displaystyle\sum_{i=1}^{n} X_i \bigg)=\displaystyle\sum_{i=1}^{n}V(X_i)\end{array}$

providing the Xis are uncorrelated.

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Standard deviation

Standard deviation s

Example