The Law of Total Variance states that
This relationship has come up on several questions on recent exams.
This relationship is also sometimes called Eve's Law (from the expression on the right-hand side: ).
Intuitively, the Law of Total Variance says that the ("total") variance can be decomposed into a sum of "within-group" variance and "between-group" variance. For example, if the random variable measures the age of a driver and measures the number of claims made in a year, the variance in the number of claims (X) has two sources of variance: (1) the number of claims varies within age groups (EV) and (2) the average number of claims for a fixed age group varies between age groups (VE).
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