how variance scales and shifts

Suppose you take a random quantity and rescale or relocate it — convert temperatures from Celsius to Fahrenheit, or shift all exam scores up by 5 points. How does its spread react? The two operations behave very differently, and the difference is intuitive once you picture it.

Adding a constant b just slides the whole distribution sideways without stretching it, so the spread is untouched: Var(X + b) = Var(X). Multiplying by a constant a stretches every deviation by a factor of a, and since variance is built from squared deviations, it stretches by a squared: Var(aX) = a^2 Var(X). Combining, Var(aX + b) = a^2 Var(X). Note the b vanishes entirely and the a comes out squared (and the sign of a is lost, since a^2 is positive — flipping a distribution does not change its spread). For standard deviation, the square root brings it back to |a|: sd(aX + b) = |a| sd(X).

This rule is the backbone of standardizing data. To turn X into a z-score with mean 0 and variance 1, you compute Z = (X - mu)/sigma: subtracting mu shifts the mean to 0 (and does nothing to variance), dividing by sigma scales the variance by 1/sigma^2, turning Var(X) = sigma^2 into 1. One caution: this clean rule is only for affine (linear-plus-constant) transformations. For a nonlinear g, Var(g(X)) is not simply some factor times Var(X) — you generally need LOTUS or an approximation like the delta method.