This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# CentralMoment

 CentralMomentgives the r central moment of the elements in list with respect to their mean. CentralMomentgives the r central moment of the symbolic distribution dist. CentralMoment[r]represents the r formal central moment.
• For the list , the central moment is given by , where is the mean of the list.
• For a symbolic distribution dist, the rcentral moment is given by Expectation[(x-Mean[dist])r, xdist].
• For a multivariate symbolic distribution dist, the central moment is given by Expectation and {1, 2, ...}==Mean[dist].
Compute central moments from data:
Use symbolic data:
Compute the second central moment of a continuous univariate distribution:
The central moment of a discrete univariate distribution:
The central moment of a multivariate distribution:
Find the relation of a formal central moment to cumulants:
Evaluate for a particular distribution:
Compute central moments from data:
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Use symbolic data:
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Compute the second central moment of a continuous univariate distribution:
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The central moment of a discrete univariate distribution:
 Out[1]=

The central moment of a multivariate distribution:
 Out[1]=

Find the relation of a formal central moment to cumulants:
 Out[1]=
Evaluate for a particular distribution:
 Out[2]=
 Out[3]=
 Scope   (10)
Compute central moment for a univariate distribution:
Compute a central moment of a specific order:
Evaluate a central moment of specific order numerically:
Compute central moments of a multivariate distribution:
Compute a central moment of a truncated distribution:
Find central moment of a formula-based distribution:
Evaluate central moment of a function of random variables:
Compute a central moment of distributions derived from data:
Compute central moment for a set of 5 independent identically distributed samples:
Compute results for a SparseArray:
Compute a multivariate central moment:
 Applications   (6)
Estimate parameters of a distribution using the method of moments:
Compare data and the estimated parametric distribution:
Find a normal approximation to GammaDistribution using the method of moments:
Show how and depend on and :
Compare an original and an approximated distribution:
Construct a sample estimator of the second central moment:
Find its sample distribution expectation, assuming sample size :
Find sample distribution variance of the estimator:
Variance of the estimator for uniformly distributed sample:
The law of large numbers states that a sample moment approaches population moment as sample size increases. Use Histogram to show the probability distribution of a second sample central moment of uniform random variates for different sample sizes:
Edgeworth expansion for near-normal data correcting for third and fourth central moments:
Function computing sample Jarque-Bera statistics []:
Accumulate statistics on samples of normal random variates:
Compare the statistics histogram with an asymptotic distribution:
The first central moment is 0:
Central moments are translation invariant:
The second central moment is a scaled Variance:
Sqrt of the second central moment is RootMeanSquare of deviations from the Mean:
Skewness is a ratio of powers of third and second central moments:
Kurtosis is a ratio of powers of fourth and second central moments:
CentralMoment is equivalent to an Expectation of a power of a random variable around its mean:
CentralMoment of order is equivalent to when both exist:
Use CentralMoment directly:
Find the central moment generating function by using GeneratingFunction:
Compare with direct evaluation of CentralMomentGeneratingFunction:
CentralMoment can be expressed in terms of Moment, Cumulant, or FactorialMoment:
CentralMoment was modified to compute a multivariate central moment:
Column-wise marginal moments can be computed as follows:
Alternatively, evaluate dimension-wise:
Moments of higher order are undefined for a heavy-tailed distribution:
Compute central moments on 5 independent samples of the distribution:
Sample central moments of higher order exhibit wild fluctuations:
Sample estimators of central moments are biased:
Find sampling population expectation assuming a sample of size :
The estimator is asymptotically unbiased:
Construct an unbiased estimator:
The expected value of the estimator is the central moment for all sample sizes: