Variance of a Discrete Random Variable

For a discrete random variable \(X\) with mean \(\mu=\mathrm{E}(X)\), variance is the probability-weighted mean of the squared deviations from \(\mu\). Assume the second moment is finite.

\(\operatorname{Var}(X)=\mathrm{E}[(X-\mu)^2]=\sum\limits_{x}(x-\mu)^2P(X=x)\).

Expanding the square gives a useful calculation formula:

\(\begin{aligned}\operatorname{Var}(X)&=\mathrm{E}(X^2)-2\mu\mathrm{E}(X)+\mu^2\\&=\mathrm{E}(X^2)-\mu^2.\end{aligned}\)

Calculate \(\mathrm{E}(X^2)=\sum\limits_{x}x^2P(X=x)\), then subtract \([\mathrm{E}(X)]^2\). In each sum, include every possible value of \(X\).

Variance is non-negative. The standard deviation is \(\sigma=\sqrt{\operatorname{Var}(X)}\), in the same units as \(X\).