Variance under translation

Statement

Let $X$ be an integrable real random variable and let $b \in \mathbb{R}$. Then $$\operatorname{Var}(X + b) = \operatorname{Var}(X).$$

Remarks

Proof: $E[X + b] = E[X] + b$ by linearity of expectation, so $(X + b) - E[X + b] = X - E[X]$. Together with the scaling rule, this gives $\operatorname{Var}(aX + b) = a^2 \operatorname{Var}(X)$ for all real $a$ and $b$.

Explanations

L1 human

If you add the same constant to every value, the whole distribution moves, but its spread does not change.