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  "body": "\n## Statement\n\nLet $X$ be a real random variable with $E[X^2] \u003c \\infty$, and let $Y$ be a random variable on the same probability space. Then\n$$\\operatorname{Var}(X) = E\\left[\\operatorname{Var}(X \\mid Y)\\right] + \\operatorname{Var}\\left(E[X \\mid Y]\\right).$$\n\n## Remarks\n\nProof: by the [[variance formula|0NMF95YK8W]] 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|04RRB6HM00]], and subtract $(E[X])^2 = (E[E[X \\mid Y]])^2$.\nThe same result holds with a σ-algebra $\\mathcal{G}$ in place of $Y$.\nThe first term is the mean variance inside the groups. The second term is the variance between the group means.\nA corollary: $\\operatorname{Var}(E[X \\mid Y]) \\le \\operatorname{Var}(X)$.\n\n## Explanations\n\n### L1 human\n\nThe total spread of $X$ has two parts: the spread inside each group, and the spread of the group averages.\nThis decomposition is the base of the analysis of variance in statistics.\n",
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