Law of Total Variance

In probability theory, the law of total variance or variance decomposition formula, also known by the acronym EVVE (or Eve's law), states that if X and Y are random variables on the same probability space, and the variance of Y is finite, then

Some writers on probability call this the "conditional variance formula". In language perhaps better known to statisticians than to probabilists, the two terms are the "unexplained" and the "explained component of the variance" (cf. fraction of variance unexplained, explained variation).

There is a general variance decomposition formula for c ≥ 2 components (see below) . For example, with two conditioning random variables:

which follows from the law of total conditional variance:

Note that the conditional expected value E( Y | X ) is a random variable in its own right, whose value depends on the value of X. Notice that the conditional expected value of Y given the event X = x is a function of x (this is where adherence to the conventional and rigidly case-sensitive notation of probability theory becomes important!). If we write E( Y | X = x ) = g(x) then the random variable E( Y | X ) is just g(X). Similar comments apply to the conditional variance.