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)

(1)

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

			(2)
			(3)
			(4)

Therefore, for , 3 it is true that

(5)

Therefore, by induction,

			(6)
			(7)
			(8)
			(9)

By the definition of the sample variance,

(10)

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

			(11)
			(12)
			(13)
			(14)

Working on the first term,

			(15)
			(16)

Use (◇) to write

(17)

so

(18)

Now work on the second term in (◇),

(19)

Considering the third term in (◇),

			(20)
			(21)
			(22)

But

(23)

so

			(24)
			(25)

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

			(26)
			(27)
			(28)
			(29)

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