Law of total variance

The law of total variance is a fundamental result in probability theory that expresses the variance of a random variable Y in terms of its conditional variances and conditional means given another random variable X. Informally, it states that the overall variability of Y can be split into an “unexplained” component (the average of within-group variances) and an “explained” component (the variance of group means).

Formally, if X and Y are random variables on the same probability space, and Y has finite variance, then:

$${\displaystyle \operatorname {Var} (Y)\;=\;\operatorname {E} {\bigl [}\operatorname {Var} (Y\mid X){\bigr ]}\;+\;\operatorname {Var} \!{\bigl (}\operatorname {E} [Y\mid X]{\bigr )}.\!}$$

This identity is also known as the variance decomposition formula, the conditional variance formula, the law of iterated variances, or colloquially as Eve’s law, in parallel to the “Adam’s law” naming for the law of total expectation.

In actuarial science (particularly in credibility theory), the two terms ${\displaystyle \operatorname {E} [\operatorname {Var} (Y\mid X)]}$ and ${\displaystyle \operatorname {Var} (\operatorname {E} [Y\mid X])}$ are called the expected value of the process variance (EVPV) and the variance of the hypothetical means (VHM) respectively.