gives the r^(th) cumulant of data.

Cumulant[data,{r1,,rm} ]

gives the multivariate cumulant of order {r1,,rm} of data.


gives the r^(th) cumulant of the distribution dist.


represents the r^(th) formal cumulant.



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Basic Examples  (2)

Compute cumulants from data:

For symbolic data:

Compute the second cumulant of a continuous univariate distribution:

The cumulant for a multivariate distribution:

Scope  (22)

Basic Uses  (6)

Exact input yields exact output:

Approximate input yields approximate output:

Find cumulants of WeightedData:

Find a cumulant of EventData:

Find a cumulant of TimeSeries:

The cumulant depends only on the values:

Find a cumulant for data involving quantities:

Array Data  (5)

For a matrix, Cumulant gives columnwise cumulants:

For an array, Cumulant gives columnwise cumulants at the first level:

Works with large arrays:

When the input is an Association, Cumulant works on its values:

SparseArray data can be used just like dense arrays:

Compute the multivariate cumulant of an array in terms of its raw moments:

Image and Audio Data  (2)

Channelwise cumulant of an RGB image:

Cumulant intensity value of a grayscale image:

On audio objects, Cumulant works channelwise:

Distribution and Process Cumulants  (5)

Find the cumulants of univariate distributions:

Multivariate distributions:

Compute a cumulant for a symbolic order r:

A cumulant may only evaluate for specific orders:

A cumulant may only evaluate numerically:

Cumulants for derived distributions:

Data distribution:

Cumulant function for a random process:

Find a cumulant of TemporalData at time t=0.5:

Find the corresponding cumulant function together with all the simulations:

Formal Cumulants  (4)

TraditionalForm formatting for formal cumulants:

Convert combinations of formal moments to an expression involving Cumulant:

Evaluate an expression involving formal cumulants TemplateBox[{1}, Cumulant]+TemplateBox[{2}, Cumulant] for a distribution:

Evaluate for data:

Find a sample estimator for an expression involving Cumulant:

Evaluate the resulting estimator for data:

Applications  (5)

Estimate parameters of a distribution using the method of cumulants:

The law of large numbers states that a sample moment approaches the population moment as the sample size increases. Use Histogram to show the probability distribution of sample cumulant of standard normal random variates for different sample sizes:

Edgeworth's expansion of order :

Approximate SechDistribution:

Compute a moving cumulant for some data:

Use the window of length .1:

Compute cumulants for slices of a collection of paths of a random process:

Choose a few slice times:

Plot cumulants over these paths:

Properties & Relations  (5)

First cumulant is equivalent to the first moment :

Second cumulant is equivalent to the second central moment :

Third cumulant is equivalent to the third central moment :

Cumulant is equal to the ^(th) derivative of the cumulant-generating function at zero :

Use Cumulant directly:

Find the cumulant-generating function using GeneratingFunction:

Check using CumulantGeneratingFunction:

Formally, cumulants can be computed using the fact that CumulantGeneratingFunction[dist,t] is given by Log[MomentGeneratingFunction[dist,t]]:

Plug in the definition for the moments in terms of their generating function:

Sample estimator of Cumulant on data is biased:

Find a sampling population expectation, assuming size :

Construct an unbiased sample estimator using PowerSymmetricPolynomial:

Verify unbiasedness on a small sample size:

The sample estimator is biased:

Compare with the sampling population expectation of the sample estimator:

Possible Issues  (1)

For some distributions with long tails, cumulants of only several low orders are defined:

Neat Examples  (2)

Find an unbiased estimator for a product of cumulants:

Check the sampling population expectation:

The distribution of Cumulant estimates for 20, 100, and 300 samples:

Wolfram Research (2010), Cumulant, Wolfram Language function, (updated 2023).


Wolfram Research (2010), Cumulant, Wolfram Language function, (updated 2023).


Wolfram Language. 2010. "Cumulant." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023.


Wolfram Language. (2010). Cumulant. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_cumulant, author="Wolfram Research", title="{Cumulant}", year="2023", howpublished="\url{}", note=[Accessed: 17-July-2024 ]}


@online{reference.wolfram_2024_cumulant, organization={Wolfram Research}, title={Cumulant}, year={2023}, url={}, note=[Accessed: 17-July-2024 ]}