CentralMoment
CentralMoment[list,r]
gives the r central moment of the elements in list with respect to their mean.
CentralMoment[dist,r]
gives the r central moment of the distribution dist.
represents the r formal central moment.
Details

- CentralMoment is also known as a moment about the mean.
- CentralMoment handles both numerical and symbolic data.
- For the list {x1,x2,…,xn}, the
central moment is given by
, where
is the mean of the list.
- CentralMoment[{{x1,y1,…},…,{xn,yn,…}},{rx,ry,…}] gives
.
- For a distribution dist, the r
central moment is given by Expectation[(x-Mean[dist])r,xdist].
- For a multivariate distribution dist, the {r1,r2,…}
central moment is given by Expectation[(x1-μ1)r1(x2-μ2)r2⋯,{x1,x2,…}dist] and {μ1,μ2,…}==Mean[dist].
- CentralMoment[r] can be used in functions such as MomentConvert and MomentEvaluate, etc.
Examples
open allclose allBasic Examples (2)
Scope (18)
Data Moments (9)
Exact input yields exact output:
Approximate input yields approximate output:
CentralMoment for a matrix gives column-wise moments:
CentralMoment for a tensor gives column-wise moments at the first level:
SparseArray data can be used just like dense arrays:
Find central moments of WeightedData:
Find a central moment of EventData:
Find a central moment of TimeSeries:
Distribution and Process Moments (5)
Find the central moments for univariate distributions:
Compute a central moment for a symbolic order r:
A central moment may only evaluate for specific orders:
A central moment may only evaluate numerically:
Central moments for derived distributions:
Central moment function for a random process:
Find a central moment of TemporalData at some time t=0.5:
Find the corresponding moment function together with all the simulations:
Formal Moments (4)
TraditionalForm formatting for formal moments:
Convert combinations of formal moments to an expression involving CentralMoment:
Evaluate an expression involving formal moments for a distribution:
Find a sample estimator for an expression involving CentralMoment:
Applications (11)
The first central moment is 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:
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:
Compute central moments for slices of a collection of paths of a random process:
Properties & Relations (9)
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:
Possible Issues (2)
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:
Neat Examples (1)
The distribution of CentralMoment estimates for 20, 100, and 300 samples:
Text
Wolfram Research (2007), CentralMoment, Wolfram Language function, https://reference.wolfram.com/language/ref/CentralMoment.html (updated 2010).
CMS
Wolfram Language. 2007. "CentralMoment." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2010. https://reference.wolfram.com/language/ref/CentralMoment.html.
APA
Wolfram Language. (2007). CentralMoment. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CentralMoment.html