Variance

We measure each outcome's deviation \(X-E[X]\) from the center and average it. Averaging the raw deviation always gives 0, because positives cancel negatives; so it is squared before averaging, giving \(\operatorname{Var}(X)=E[(X-E[X])^2]\). Being an average of squares, it is never negative.

To compute it one almost always uses the equivalent form \(\operatorname{Var}(X)=E[X^2]-(E[X])^2\). For a fair die, \(E[X^2]=\tfrac{91}{6}\) and \(E[X]=\tfrac72\), so \(\operatorname{Var}(X)=\tfrac{35}{12}\). A typical mistake is to forget subtracting \((E[X])^2\) and stop at \(E[X^2]\).

Its properties show up under linear transformations: \(\operatorname{Var}(aX+b)=a^2\operatorname{Var}(X)\). Shifting by \(b\) leaves the spread unchanged and rescaling by \(a\) multiplies it by \(a^2\); hence \(\operatorname{Var}(-2X)=4\operatorname{Var}(X)\). It is 0 exactly when \(X\) is constant.

Its units are those of \(X\) squared, which makes it hard to interpret; the square root restores the original scale. For two variables, the variance of a sum includes a cross term measuring how they move together, an idea developed by the covariance.