Problem: Mean absolute deviation.
a) Calculate the mean absolute deviation E(|X - µ|) for X, the number on a six-sided die.
Your answer should be slightly smaller than the standard deviation. This is a general phenomenon, which occurs because the operation of squaring the absolute deviations before averaging them tends to put more weight on large deviations than on small ones.
b) Use the fact that Vαr(|X - µ|) > 0 to show that SD(X) > E(| - µ|), with equality if and only if |X - µ| is a constant.
That is to say unless | X - µ| is a constant, the standard deviation of a random variable is always strictly larger than the absolute deviation. If X is a constant, then both measures of spread are zero.