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  "body": "\n## Definition\n\nLet $X$ be a real random variable on $(\\Omega, \\mathcal{F}, P)$ with $E[X^2] \u003c \\infty$, and let $\\mathcal{G} \\subseteq \\mathcal{F}$ be a σ-algebra.\nThe conditional variance of $X$ given $\\mathcal{G}$ is the random variable\n$$\\operatorname{Var}(X \\mid \\mathcal{G}) = E\\left[\\left(X - E[X \\mid \\mathcal{G}]\\right)^2 \\,\\middle|\\, \\mathcal{G}\\right],$$\nwhere $E[\\cdot \\mid \\mathcal{G}]$ is the [[conditional expectation|0YAMJRK04E]]. For a random variable $Y$, $\\operatorname{Var}(X \\mid Y) = \\operatorname{Var}(X \\mid \\sigma(Y))$.\n\n## Remarks\n\nAs for the [[variance|007QED3E0Y]], $\\operatorname{Var}(X \\mid \\mathcal{G}) = E[X^2 \\mid \\mathcal{G}] - (E[X \\mid \\mathcal{G}])^2$ almost surely.\nFor a discrete $Y$ and $y$ with $P(Y = y) \u003e 0$, $\\operatorname{Var}(X \\mid Y)$ takes the value $\\operatorname{Var}(X \\mid Y = y)$ on $\\{Y = y\\}$. This is the variance of $X$ under $P(\\cdot \\mid Y = y)$.\n$\\operatorname{Var}(X \\mid \\mathcal{G}) = 0$ almost surely if and only if $X$ is equal almost surely to a $\\mathcal{G}$-measurable random variable.\n\n## Explanations\n\n### L1 human\n\nThe conditional variance tells how uncertain $X$ still is after you learn some information.\nFor example, the variance of the height of a person given the sex of that person is the spread of heights within each group.\n",
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