Variance under scaling

Statement

Let $X$ be a real random variable with finite variance and let $a \in \mathbb{R}$. Then $$\operatorname{Var}(aX) = a^2 \operatorname{Var}(X).$$

Remarks

Proof: $E[aX] = a E[X]$ by linearity of expectation. So $(aX - E[aX])^2 = a^2 (X - E[X])^2$, and one takes expectations. For the standard deviation, the formula becomes $\sigma_{aX} = |a| \, \sigma_X$. In particular, $\operatorname{Var}(-X) = \operatorname{Var}(X)$.

Explanations

L1 human

If you measure in centimeters instead of meters, every value is multiplied by $100$. The variance is then multiplied by $100^2 = 10000$, because it is measured in squared units.