Compute central moments from data:

Use symbolic data:

Central moment of a list of dates:

Compute the second central moment of a univariate distribution:

The central moment of a multivariate distribution:

The first central moment is always 0:

The second central moment is a measure of dispersion:

The third central moment is a measure of skewness:

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 [link]:

Accumulate statistics on samples of normal random variates:

Compare the statistics histogram with an asymptotic distribution:

Compute a moving central moment for some data:

Use the window of length .1:

Compute central moments for slices of a collection of paths of a random process:

Choose a few slice times:

Plot central moments over these paths:

Central moments are translation invariant:

The second central moment is a scaled Variance:

The odd moments of 2×2 matrices vanish:

For a multivariate order, the total order must be odd:

The multivariate central moment of an array of depth has depth :

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:

Central 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: