Sample Variance Computation

When computing the sample variance numerically, the mean must be computed before can be determined. This requires storing the set of sample values. However, it is possible to calculate using a recursion relationship involving only the last sample as follows. This means itself need not be precomputed, and only a running set of values need be stored at each step.

In the following, use the somewhat less than optimal notation to denote calculated from the first samples (i.e., not the th moment)

and let denotes the value for the bias-corrected sample variance calculated from the first samples. The first few values calculated for the mean are

Therefore, for , 3 it is true that

Therefore, by induction,

By the definition of the sample variance,

for . Defining , can then be computed using the recurrence equation

Working on the first term,

Use (◇) to write

so

Now work on the second term in (◇),

Considering the third term in (◇),

But

so

Finally, plugging (◇), (◇), and (◇) into (◇),

gives the desired expression for in terms of , , and ,