Conditional variance

Definition

Let $X$ be a real random variable on $(\Omega, \mathcal{F}, P)$ with $E[X^2] < \infty$, and let $\mathcal{G} \subseteq \mathcal{F}$ be a σ-algebra. The conditional variance of $X$ given $\mathcal{G}$ is the random variable $$\operatorname{Var}(X \mid \mathcal{G}) = E\left[\left(X - E[X \mid \mathcal{G}]\right)^2 \,\middle|\, \mathcal{G}\right],$$ where $E[\cdot \mid \mathcal{G}]$ is the conditional expectation. For a random variable $Y$, $\operatorname{Var}(X \mid Y) = \operatorname{Var}(X \mid \sigma(Y))$.

Remarks

As for the variance, $\operatorname{Var}(X \mid \mathcal{G}) = E[X^2 \mid \mathcal{G}] - (E[X \mid \mathcal{G}])^2$ almost surely. For a discrete $Y$ and $y$ with $P(Y = y) > 0$, $\operatorname{Var}(X \mid Y)$ takes the value $\operatorname{Var}(X \mid Y = y)$ on $\{Y = y\}$. This is the variance of $X$ under $P(\cdot \mid Y = y)$. $\operatorname{Var}(X \mid \mathcal{G}) = 0$ almost surely if and only if $X$ is equal almost surely to a $\mathcal{G}$-measurable random variable.

Explanations

L1 human

The conditional variance tells how uncertain $X$ still is after you learn some information. For example, the variance of the height of a person given the sex of that person is the spread of heights within each group.