Law of total variance

Statement

Let $X$ be a real random variable with $E[X^2] < \infty$, and let $Y$ be a random variable on the same probability space. Then $$\operatorname{Var}(X) = E\left[\operatorname{Var}(X \mid Y)\right] + \operatorname{Var}\left(E[X \mid Y]\right).$$

Remarks

Proof: by the variance formula in its conditional form, $E[X^2 \mid Y] = \operatorname{Var}(X \mid Y) + (E[X \mid Y])^2$. Take expectations with the law of total expectation, and subtract $(E[X])^2 = (E[E[X \mid Y]])^2$. The same result holds with a σ-algebra $\mathcal{G}$ in place of $Y$. The first term is the mean variance inside the groups. The second term is the variance between the group means. A corollary: $\operatorname{Var}(E[X \mid Y]) \le \operatorname{Var}(X)$.

Explanations

L1 human

The total spread of $X$ has two parts: the spread inside each group, and the spread of the group averages. This decomposition is the base of the analysis of variance in statistics.