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Sample Central Moment

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The rth sample central moment m_r of a sample with sample size n is defined as

m_r=1/nsum_(k=1)^n(x_k-m)^r,
(1)

where m=m_1^' is the sample mean. The first few sample central moments are related to power sums S_k by

m_1=0
(2)
m_2=-(S_1^2)/(n^2)+(S_2)/n
(3)
m_3=(2S_1^3)/(n^3)-(3S_1S_2)/(n^2)+(S_3)/n
(4)
m_4=-(3S_1^4)/(n^4)+(6S_1^2S_2)/(n^3)-(4S_1S_3)/(n^2)+(S_4)/n
(5)
m_5=(4S_1^5)/(n^5)-(10S_1^3S_2)/(n^4)+(10S_1^2S_3)/(n^3)-(5S_1S_4)/(n^2)+(S_5)/n.
(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

<m_2>=((n-1)mu_2)/n
(7)
<m_3>=((n-1)(n-2)mu_3)/(n^2)
(8)
<m_4>=((n-1)[3(2n-3)mu_2^2+(n^2-3n+3)mu_4])/(n^3)
(9)
<m_5>=((n-1)(n-2)[10(n-2)mu_2mu_3+(n^2-2n+2)mu_5])/(n^4).
(10)

See also

Central Moment, Raw Moment, Sample, Sample Mean, Sample Raw Moment, Sample Variance

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References

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

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Sample Central Moment

Cite this as:

Weisstein, Eric W. "Sample Central Moment." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SampleCentralMoment.html

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