Law of total variance

In probability theory, the law of total variance^[1] or variance decomposition formula or conditional variance formulas or law of iterated variances also known as Eve's law,^[2] states that if ${\displaystyle X}$ and ${\displaystyle Y}$ are random variables on the same probability space, and the variance of ${\displaystyle Y}$ is finite, then

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

In language perhaps better known to statisticians than to probability theorists, the two terms are the "unexplained" and the "explained" components of the variance respectively (cf. fraction of variance unexplained, explained variation). In actuarial science, specifically credibility theory, the first component is called the expected value of the process variance (EVPV) and the second is called the variance of the hypothetical means (VHM).^[3] These two components are also the source of the term "Eve's law", from the initials EV VE for "expectation of variance" and "variance of expectation".