This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.
View current documentation (Version 11.2)

CentralMoment

Updated In 8 Graphic
CentralMoment
gives the r^(th) central moment of the elements in list with respect to their mean.
CentralMoment
gives the r^(th) central moment of the symbolic distribution dist.
CentralMoment[r]
represents the r^(th) formal central moment.
  • For the list , the ^(th) central moment is given by , where is the mean of the list.
  • For a symbolic distribution dist, the r^(th)central moment is given by Expectation[(x-Mean[dist])r, xDistributeddist].
  • For a multivariate symbolic distribution dist, the ^(th) 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:
In[1]:=
Click for copyable input
In[2]:=
Click for copyable input
Out[2]=
Use symbolic data:
In[3]:=
Click for copyable input
Out[3]=
In[4]:=
Click for copyable input
Out[4]=
 
Compute the second central moment of a continuous univariate distribution:
In[1]:=
Click for copyable input
Out[1]=
 
The central moment of a discrete univariate distribution:
In[1]:=
Click for copyable input
Out[1]=
 
The central moment of a multivariate distribution:
In[1]:=
Click for copyable input
Out[1]=
 
Find the relation of a formal central moment to cumulants:
In[1]:=
Click for copyable input
Out[1]=
Evaluate for a particular distribution:
In[2]:=
Click for copyable input
Out[2]=
In[3]:=
Click for copyable input
Out[3]=
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:
TraditionalForm formatting:
Compute results for a SparseArray:
Compute a multivariate central moment:
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:
New in 6 | Last modified in 8