Variance V

The variance is a measure of how much the probability distribution is spread from the mean:

$\begin{array}{l}\displaystyle V=E\bigg[ \big(x-\bar{x}\big)^2\bigg] = E\big( x^2\big)-\bar{x}^2\end{array}$

where denotes the expected value (mean) of whatever is in the brackets, so:

$\begin{array}{l}\displaystyle V=\int_{-\infty}^\infty \big(x-\bar{x}\big)^2 .f(x) .dx\end{array}$

The variance sums up the squared distance from the mean of all possible values of x, weighted by the probability of x occurring. The variance is known as the second moment about the mean. It has units that are the square of the units of x. So, if x is cows in a random field, V has units of cows2. This limits the intuitive value of the variance.

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.

Page tree

Variance

Variance V