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

and s^{2} 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 s ^{2} is an unbiased estimator of σ ^{2}**

Stage 1 proved that:

swapping sides

dividing by 'n=1' so that the left hand side is s^{2}