Law of Total Variance

The Law of Total Variance (also called the Eve's law or conditional variance formula) decomposes the total variance of $X$ into two pieces: $$\text{Var}(X) = E[\text{Var}(X \mid Y)] + \text{Var}(E[X \mid Y])$$ Intuition: Think of grouping data by some variable $Y$. The first term captures the average spread *within* each group. The second term captures how much the group means *differ from each other*. Total variance = average within-group variance + between-group variance. Concrete example: You roll a fair die to get $N$, then flip $N$ fair coins. Let $X$ = number of heads. Given $N$, $X \sim \text{Bin}(N,$$\frac{1}{2}$), so $E[X \mid N] = N/2$ and $\text{Var}(X \mid N) = N/4$. $$E[\text{Var}(X \mid N)] = E[N/4] = E[N]/4 = 3.$$$\frac{5}{4}$ = 0.875 $$\text{Var}(E[X \mid N]) = \text{Var}(N/2) = \text{Var}(N)/4 = ($$$\frac{35}{12}$)/4 \approx 0.729 $$\text{Var}(X) = 0.875 + 0.729 = 1.604$$ When to use: Problems with hierarchical randomness — first a parameter is drawn randomly, then observations are generated conditional on that parameter. Common in compound distributions, Bayesian models, and random sample size problems. The formula tells you whether most variability comes from randomness within groups or differences between groups.