Barnard Statistics Concepts Practice Test

The variance of a fair six-sided die measures how spread out the outcomes are around the average. For values 1 through 6 with equal probability, start with the mean: E[X] = (1+6)/2 = 3.5. Next, compute E[X^2] = (1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2)/6 = 91/6. The variance is Var(X) = E[X^2] − (E[X])^2 = 91/6 − (3.5)^2 = 91/6 − 12.25 = 35/12. So the variance is 35/12. This matches the general result for a discrete uniform on 1 to n, (n^2 − 1)/12, with n = 6.