TITLE: Conditional and Total Variance
QUESTION [10 upvotes]: Why does $ \text{Var}(Y) = E(\text{Var}(Y|X))+ \text{Var}(E(Y|X))$? What is the intuitive explanation for this? In laymen's terms it seems to say that the variance of $Y$ equals the expected value of the conditional variance plus the variance of the conditional expectation.

REPLY [3 votes]: Geometrically it's just the Pythagorean theorem. We may measure the "length" of random variables by standard deviation.
We start with a random variable Y. E(Y|X) is the projection of this Y to the set of random variables wich may be expressed as a deterministic function of X. 
We have a hypotenuse Y with squared length Var(Y).
The first leg is E(Y|X) with squared length Var(E(Y|X)).
The second leg is Y-E(Y|X) with squared length Var(Y-E(Y|X))=...=E(Var(Y|X)).