Proof of unbiased estimator for variance
From Mathematics
The unbiased estimator for the variance of the distribution of a random variable 
, given a random sample 
is
That 
rather than 
appears in the denominator is counterintuitive and confuses many new students. Here it is proven that this form is the unbiased estimator for variance, i.e., that its expected value is equal to the variance itself.
[edit] Proof
Let 
be
the estimator for the variance of some random variable . Then
Then
And so
The expectations on the right-hand side are not known immediately. However, from the definition of variance
where 
and 
are the variance and mean of , respectively.
The other term can be represented thus, according to the definition of variance and the Central Limit Theorem
where 
and 
are the variance and mean of 
, respectively.
These indirect forms are then substituted back into the previously unsolvable equation
Thus the expected value of the estimator is equal to the variance and the estimator is therefore unbiased.
QED
[edit] Notes
- ↑ A subtle trick was used here.
. It can then be seen easily that 
.
