Sample Central Moment

The th sample central moment  of a sample with sample size  is defined as

(1)

where  is the sample mean. The first few sample central moments are related to power sums  by

			(2)
			(3)
			(4)
			(5)
			(6)

These relations can be given by SampleCentralToPowerSum[r] in the Mathematica application package mathStatica.

In terms of the population central moments, the expectation values of the first few sample central moments are

			(7)
			(8)
			(9)
			(10)

REFERENCES:

Rose, C. and Smith, M. D. Mathematical Statistics with Mathematica. New York: Springer-Verlag, p. 251, 2002.