If a sample is taken from an infinite population of variance σ ² where:

and s² is to be an unbiased estimator of σ ² then:

Stage 1: prove:

As must be greater than or equal to zero this will also show that:

proof:

since

since

which completes the proof.

Stage 2: Prove that s² is an unbiased estimator of σ ²

Stage 1 proved that:

swapping sides

dividing by 'n=1' so that the left hand side is s²