Variance of Discrete Probability Distribution

While the mean locates the centre of a random variable's values, the variance measures how spread out those values tend to be around that centre. When the possible values of a random variable cluster tightly near its mean, the variance is small; when they are widely scattered, the variance is large — in this sense, variance measures the average degree to which each value differs from the expected value $E(X)$.

For a discrete random variable $X$ with possible values $x_1, x_2, \dots, x_n$ occurring with probabilities $p_1, p_2, \dots, p_n$, the variance is defined as the weighted average of the squared deviations of each value from the mean.

This is more commonly written as $\mathrm{Var}(X) = E(X^2) - [E(X)]^2$, where $E(X^2) = \sum_{i=1}^n p_i x_i^2$ is the expectation of the squared random variable.

The positive square root of the variance is the standard deviation, denoted $\sigma_x$: …