Kurtosis

Kurtosis[list]

gives the coefficient of kurtosis for the elements in list.

Kurtosis[dist]

gives the coefficient of kurtosis for the distribution dist.

Details

  • Kurtosis measures the concentration of data around the peak and in the tails versus the concentration in the flanks.
  • A normal distribution has kurtosis equal to 3. In comparing shapes with normal we have:
  • more flat than normal, platykurtic
    like normal, mesokurtic
    more peaked than normal, leptokurtic
  • Kurtosis handles both numerical and symbolic data.
  • Kurtosis[{{x1,y1,…},{x2,y2,…},…}] gives {Kurtosis[{x1,x2,…}],Kurtosis[{y1,y2,…}],…}.
  • Kurtosis[…] is equivalent to CentralMoment[…,4]/CentralMoment[…,2]2.

Examples

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Basic Examples  (2)

Kurtosis for a list of values:

Kurtosis for a parametric distribution:

Scope  (14)

Data  (10)

Exact input yields exact output:

Approximate input yields approximate output:

Kurtosis for a matrix gives column-wise kurtosis:

Work with large arrays:

SparseArray data can be used just like dense arrays:

Find the kurtosis of WeightedData:

Find the kurtosis of EventData:

Find the kurtosis of TemporalData:

Find the kurtosis of TimeSeries:

The kurtosis depends only on the values:

Find the kurtosis of data involving quantities:

Distributions and Processes  (4)

Find the kurtosis for univariate distributions:

Multivariate distributions:

Kurtosis for derived distributions:

Data distribution:

Kurtosis for distributions with quantities:

Kurtosis function for a random process:

Applications  (6)

Normal distributions have Kurtosis value 3:

Leptokurtic distributions have kurtosis greater than 3:

Platykurtic distributions have kurtosis less than 3:

The limiting distribution for BinomialDistribution as is normal:

The limiting value of the kurtosis is 3:

By the central limit theorem, kurtosis of normalized sums of random variables will converge to 3:

Define a Pearson distribution with zero mean and unit variance, parameterized by skewness and kurtosis:

Obtain parameter inequalities for Pearson types 1, 4, and 6:

The region plot for Pearson types depending on the values of skewness and kurtosis:

Generate a random sample from a ParetoDistribution:

Determine the type of PearsonDistribution with moments matching the sample moments:

This time series contains the number of steps taken daily by a person during a period of five months:

Average number of steps:

Analyze the kurtosis as an indication of consistency of daily steps taken:

The histogram of the frequency of daily counts shows that distribution is mesokurtic:

Find the skewness for the heights of the children in a class:

Kurtosis larger than 3 would indicate a distribution highly concentrated around the mean:

Properties & Relations  (5)

Kurtosis for data can be computed from CentralMoment:

Kurtosis for a distribution can be computed from CentralMoment:

Kurtosis is bounded from below by 1, as TemplateBox[{4}, CentralMoment]⩵Expectation[(x-mu)^4]>=Expectation[(x-mu)^2]^2⩵TemplateBox[{2}, CentralMoment]^2:

Normal distributions have Kurtosis value 3:

Approximately normal distributions have Kurtosis values near 3:

Plot the PDF for the distribution:

Plot the PDF for the normal approximation:

Possible Issues  (1)

Kurtosis coefficient is sometimes confused with excess kurtosis coefficient:

The excess kurtosis vanishes for NormalDistribution:

Excess kurtosis is defined as Cumulant[dist,4]/Cumulant[dist,2]^2:

Neat Examples  (1)

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

Wolfram Research (2007), Kurtosis, Wolfram Language function, https://reference.wolfram.com/language/ref/Kurtosis.html.

Text

Wolfram Research (2007), Kurtosis, Wolfram Language function, https://reference.wolfram.com/language/ref/Kurtosis.html.

CMS

Wolfram Language. 2007. "Kurtosis." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Kurtosis.html.

APA

Wolfram Language. (2007). Kurtosis. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Kurtosis.html

BibTeX

@misc{reference.wolfram_2022_kurtosis, author="Wolfram Research", title="{Kurtosis}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/Kurtosis.html}", note=[Accessed: 28-March-2023 ]}

BibLaTeX

@online{reference.wolfram_2022_kurtosis, organization={Wolfram Research}, title={Kurtosis}, year={2007}, url={https://reference.wolfram.com/language/ref/Kurtosis.html}, note=[Accessed: 28-March-2023 ]}