Suppose you have a list of N numbers x_i, whose mean is X. The unaveraged variance, ∑_i (x_i - X)^2 is the sum of a bunch of squares, which are non-nega

Suppose you have a list of N numbers x_i, whose mean is X. The unaveraged variance, ∑_i (x_i - X)^2 is the sum of a bunch of squares, which are non-negative things, so the variance is non-negative. Now, we can expand the square 0 ≤ ∑_i (x_i - X)^2 = ∑_i x_i^2 - 2 X ∑_i x_i + X^2 ∑_i 1 ≤ N * (mean of the squares) - 2 N X^2 + X^2 N ≤ N * (mean of the squares) - N * (square of the mean) and find square of the mean ≤ mean of the squares equality being saturated if and only if the list is a constant list.