Bessel's Correction - Proof of Correctness

Proof of Correctness

As a background fact, we use the identity which follows from the definition of the standard deviation and linearity of expectation.

A very helpful observation is that for any distribution, the variance equals half the expected value of when are independent samples. To prove this observation we will use that (which follows from the fact that they are independent) as well as linearity of expectation:

Now that the observation is proven, it suffices to show that the expected squared difference of two samples from the sample population equals times the expected squared difference of two samples from the original distribution. To see this, note that when we pick and via u, v being integers selected independently and uniformly from 1 to n, a fraction of the time we will have u=v and therefore the sampled squared difference is zero independent of the original distribution. The remaining of the time, the value of is the expected squared difference between two unrelated samples from the original distribution. Therefore, dividing the sample expected squared difference by, or equivalently multiplying by gives an unbiased estimate of the original expected squared difference.