Variance

Definition 11.9 (Variance). The variance of $X$, written $V(X)$, $\text{Var}(X)$, or $\sigma^2$, is

It measures how the data spread out from the mean — the average of the squared deviations from $\mu$. (Variability, spread and dispersion are synonyms.) The standard deviation is $\sigma=\sqrt{V(X)}$; both $V(X)$ and $\sigma$ are always $\ge0$.

Alternative computing formula. Expanding the square and using linearity of expectation (proved formally in §11.5.3) gives the far more convenient

— compute the mean and the second moment $E(X^2)$ separately, then subtract the square of the mean. This is the formula used throughout the exercises. …