[absherwin]

One bit of additional intuition: Since the square of a number goes up from addition by slightly more than it goes down from subtraction, perturbations increase the average of the squares. This is why the difference between the two quantities Feynman mentions is used to measure the variance of a set of numbers.

Since that’s still not precise, let’s compare the square of the mean to the mean square for two numbers a and b.

The square of the mean is ((a+b)/2)^2=(a^2+2ab+b^2)/4

The mean of the squares is (a^2+b^2)/2

Feynman’s claim is that the second is always bigger if the numbers deviate around an average (a and b aren’t equal).

So let’s subtract the first from the second. We get (a^2-2ab+b^2)/4. The numerator is equivalent to (a-b)^2. Since a square of a real can’t be negative, when a and b are unequal the mean of the squares is always larger.

[mobileexpert]

See also https://en.m.wikipedia.org/wiki/Jensen%27s_inequality