P: Total variance includes variation between groups
Conditioning can simplify a distribution, but taking only the average conditional variance loses the variation between group means. The law of total variance separates within-group variation from between-group variation. Identify both before substituting numbers.
Worked example or practice scenario
Two equally likely groups have conditional means 2 and 6, and conditional variances 1 and 9. E[Var(X|G)] = 5. The variance of the conditional mean is 4, so Var(X)=9. The overall mean is 4.
Try this next
Recover the same result from E[X²]−E[X]²: conditional second moments are 5 and 45. Explain what would happen if both groups had the same mean, and why that eliminates only the between-group component.
Reading sources
ActNet editorial guide · October 1, 2026 · Original illustrative examples.
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