Kurtosis

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`Kurtosis`

Updated in 14.1

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Kurtosis[data]

gives the coefficient of kurtosis for the elements in data.

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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.
Kurtosis[…] is equivalent to CentralMoment[…,4]/CentralMoment[…,2]².
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[{{x₁,y₁,…},{x₂,y₂,…},…}] gives {Kurtosis[{x₁,x₂,…}],Kurtosis[{y₁,y₂,…}],…}.
Kurtosis handles both numerical and symbolic data.
The data can have the following additional forms and interpretations:

	Association	the values (the keys are ignored) »
	SparseArray	as an array, equivalent to Normal[data] »
	QuantityArray	quantities as an array »
	WeightedData	weighted mean, based on the underlying EmpiricalDistribution »
	EventData	based on the underlying SurvivalDistribution »
	TimeSeries, TemporalData, …	vector or array of values (the time stamps ignored) »
	Image,Image3D	RGB channel's values or grayscale intensity value »
	Audio	amplitude values of all channels »
	DateObject, TimeObject	list of dates or list of times

For a random process proc, the kurtosis function can be computed for slice distribution at time t, SliceDistribution[proc,t], as β[t]=Kurtosis[SliceDistribution[proc,t]]. »

Examples

open allclose all

Basic Examples (4)Summary of the most common use cases

Kurtosis for a list of values:

In[1]:=1

✖

https://wolfram.com/xid/0mlj9yhaospg-d698z0

Out[1]=1

Kurtosis for a symbolic data:

In[1]:=1

✖

https://wolfram.com/xid/0mlj9yhaospg-n9n

Out[1]=1

Kurtosis for a list of dates:

In[1]:=1

✖

https://wolfram.com/xid/0mlj9yhaospg-bpyuss

Out[1]=1

Kurtosis for a parametric distribution:

In[1]:=1

✖

https://wolfram.com/xid/0mlj9yhaospg-cdzcrp

Out[1]=1

Scope (23)Survey of the scope of standard use cases

Basic Uses (7)

Exact input yields exact output:

In[1]:=1

✖

https://wolfram.com/xid/0mlj9yhaospg-ug7y2

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0mlj9yhaospg-bcry2t

Out[2]=2

Approximate input yields approximate output:

In[1]:=1

✖

https://wolfram.com/xid/0mlj9yhaospg-xqxn3

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0mlj9yhaospg-gnz3p9

Out[2]=2

Find the kurtosis of WeightedData:

In[1]:=1

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https://wolfram.com/xid/0mlj9yhaospg-d0wc9z

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0mlj9yhaospg-mab31

In[3]:=3

✖

https://wolfram.com/xid/0mlj9yhaospg-gc8lnk

Out[3]=3

Find the kurtosis of EventData:

In[1]:=1

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https://wolfram.com/xid/0mlj9yhaospg-kiuumt

In[2]:=2

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https://wolfram.com/xid/0mlj9yhaospg-cv4d9n

Out[2]=2

Find the kurtosis of TemporalData:

In[1]:=1

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https://wolfram.com/xid/0mlj9yhaospg-gx0rsr

In[2]:=2

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https://wolfram.com/xid/0mlj9yhaospg-f5ij1t

In[3]:=3

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https://wolfram.com/xid/0mlj9yhaospg-8e999

Out[3]=3

Find the kurtosis of TimeSeries:

In[1]:=1

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https://wolfram.com/xid/0mlj9yhaospg-hf056t

Out[1]=1

The kurtosis depends only on the values:

In[2]:=2

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https://wolfram.com/xid/0mlj9yhaospg-ikztk4

Out[2]=2

Find the kurtosis of data involving quantities:

In[1]:=1

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https://wolfram.com/xid/0mlj9yhaospg-jopin9

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0mlj9yhaospg-e8c21s

Out[2]=2

Array Data (5)

Kurtosis for a matrix gives columnwise kurtosis:

In[1]:=1

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https://wolfram.com/xid/0mlj9yhaospg-ezu2uz

Out[1]=1

Work with large arrays:

In[1]:=1

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https://wolfram.com/xid/0mlj9yhaospg-nknun

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0mlj9yhaospg-ma3v2m

Out[2]=2

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

In[1]:=1

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https://wolfram.com/xid/0mlj9yhaospg-cs7n5q

In[2]:=2

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https://wolfram.com/xid/0mlj9yhaospg-ddw92p

Out[2]=2

SparseArray data can be used just like dense arrays:

In[1]:=1

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https://wolfram.com/xid/0mlj9yhaospg-n691tv

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0mlj9yhaospg-drrysl

Out[2]=2

Find the kurtosis of a QuantityArray:

In[1]:=1

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https://wolfram.com/xid/0mlj9yhaospg-lgwnaj

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0mlj9yhaospg-k03qc6

Out[2]=2

Image and Audio Data (2)

Channelwise kurtosis value of an RGB image:

In[1]:=1

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https://wolfram.com/xid/0mlj9yhaospg-ojba46

Out[1]=1

Mean intensity value of a grayscale image:

In[3]:=3

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https://wolfram.com/xid/0mlj9yhaospg-ue2gq5

Out[3]=3

On audio objects, Kurtosis works channelwise:

In[1]:=1

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https://wolfram.com/xid/0mlj9yhaospg-nq1jnz

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0mlj9yhaospg-mjmudf

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0mlj9yhaospg-bs38vd

Out[3]=3

Date and Time (5)

Compute kurtosis of dates:

In[1]:=1

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https://wolfram.com/xid/0mlj9yhaospg-b1smxx

In[2]:=2

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https://wolfram.com/xid/0mlj9yhaospg-pa4nmn

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0mlj9yhaospg-uok1il

Out[3]=3

Compute the weighted kurtosis of dates:

In[1]:=1

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https://wolfram.com/xid/0mlj9yhaospg-c98kbd

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0mlj9yhaospg-8c1had

In[3]:=3

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https://wolfram.com/xid/0mlj9yhaospg-t71b2h

Out[3]=3

Compute the kurtosis of dates given in different calendars:

In[1]:=1

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https://wolfram.com/xid/0mlj9yhaospg-wbzcuv

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0mlj9yhaospg-9ius88

Out[2]=2

Compute the kurtosis of times:

In[1]:=1

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https://wolfram.com/xid/0mlj9yhaospg-et9bla

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0mlj9yhaospg-ztsexm

Out[2]=2

Compute the kurtosis of times with different time zone specifications:

In[1]:=1

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https://wolfram.com/xid/0mlj9yhaospg-mrqghz

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0mlj9yhaospg-1d7sk5

Out[2]=2

Distributions and Processes (4)

Find the kurtosis for univariate distributions:

In[1]:=1

✖

https://wolfram.com/xid/0mlj9yhaospg-rxz55

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0mlj9yhaospg-hbq28j

Out[2]=2

Multivariate distributions:

In[2]:=2

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https://wolfram.com/xid/0mlj9yhaospg-lzwoz3

Out[2]=2

Kurtosis for derived distributions:

In[1]:=1

✖

https://wolfram.com/xid/0mlj9yhaospg-rgc72x

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0mlj9yhaospg-byqvvz

Out[2]=2

Data distribution:

In[3]:=3

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https://wolfram.com/xid/0mlj9yhaospg-215ry

In[4]:=4

✖

https://wolfram.com/xid/0mlj9yhaospg-fq5ptk

Out[4]=4

Kurtosis for distributions with quantities:

In[1]:=1

✖

https://wolfram.com/xid/0mlj9yhaospg-dqsioj

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0mlj9yhaospg-b53jwg

Out[2]=2

Kurtosis function for a random process:

In[1]:=1

✖

https://wolfram.com/xid/0mlj9yhaospg-c4ojmv

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0mlj9yhaospg-g9pmgp

Out[2]=2

Applications (6)Sample problems that can be solved with this function

Normal distributions have Kurtosis value 3:

In[1]:=1

✖

https://wolfram.com/xid/0mlj9yhaospg-lk6n95

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0mlj9yhaospg-baih73

Out[2]=2

Leptokurtic distributions have kurtosis greater than 3:

In[3]:=3

✖

https://wolfram.com/xid/0mlj9yhaospg-fbptm3

Out[3]=3

In[4]:=4

✖

https://wolfram.com/xid/0mlj9yhaospg-c2fgfe

Out[4]=4

In[5]:=5

✖

https://wolfram.com/xid/0mlj9yhaospg-vv0n

Out[5]=5

Platykurtic distributions have kurtosis less than 3:

In[6]:=6

✖

https://wolfram.com/xid/0mlj9yhaospg-7pmt4p

Out[6]=6

In[7]:=7

✖

https://wolfram.com/xid/0mlj9yhaospg-r1uz9k

Out[7]=7

The limiting distribution for BinomialDistribution as is normal:

In[1]:=1

✖

https://wolfram.com/xid/0mlj9yhaospg-cjcacn

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0mlj9yhaospg-ikcny4

Out[2]=2

The limiting value of the kurtosis is 3:

In[3]:=3

✖

https://wolfram.com/xid/0mlj9yhaospg-9jqn7h

Out[3]=3

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

In[1]:=1

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https://wolfram.com/xid/0mlj9yhaospg-o42rw3

In[2]:=2

✖

https://wolfram.com/xid/0mlj9yhaospg-9gpzii

Out[2]=2

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

In[1]:=1

✖

https://wolfram.com/xid/0mlj9yhaospg-hshyga

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

In[2]:=2

✖

https://wolfram.com/xid/0mlj9yhaospg-46zj1

Out[2]=2

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

In[3]:=3

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https://wolfram.com/xid/0mlj9yhaospg-26mr9

Generate a random sample from a ParetoDistribution:

In[4]:=4

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https://wolfram.com/xid/0mlj9yhaospg-hp3zyw

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

In[5]:=5

✖

https://wolfram.com/xid/0mlj9yhaospg-4eqpc

Out[5]=5

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

In[1]:=1

✖

https://wolfram.com/xid/0mlj9yhaospg-l8i1hu

In[2]:=2

✖

https://wolfram.com/xid/0mlj9yhaospg-egrdpv

Out[2]=2

Average number of steps:

In[3]:=3

✖

https://wolfram.com/xid/0mlj9yhaospg-h3rzfh

Out[3]=3

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

In[4]:=4

✖

https://wolfram.com/xid/0mlj9yhaospg-vvz2te

Out[4]=4

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

In[5]:=5

✖

https://wolfram.com/xid/0mlj9yhaospg-rfj1tf

In[6]:=6

✖

https://wolfram.com/xid/0mlj9yhaospg-kqbpv6

Out[6]=6

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

In[1]:=1

✖

https://wolfram.com/xid/0mlj9yhaospg-cevfij

In[2]:=2

✖

https://wolfram.com/xid/0mlj9yhaospg-fllmtw

Out[2]=2

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

In[3]:=3

✖

https://wolfram.com/xid/0mlj9yhaospg-celepo

Out[3]=3

Properties & Relations (5)Properties of the function, and connections to other functions

Kurtosis for data can be computed from CentralMoment:

In[1]:=1

✖

https://wolfram.com/xid/0mlj9yhaospg-gpep5v

In[2]:=2

✖

https://wolfram.com/xid/0mlj9yhaospg-pqqxoo

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0mlj9yhaospg-dlb7nb

Out[3]=3

Kurtosis for a distribution can be computed from CentralMoment:

In[1]:=1

✖

https://wolfram.com/xid/0mlj9yhaospg-b2gkzn

In[2]:=2

✖

https://wolfram.com/xid/0mlj9yhaospg-j1ep1x

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0mlj9yhaospg-gepfe5

Out[3]=3

In[4]:=4

✖

https://wolfram.com/xid/0mlj9yhaospg-cjvav

Out[4]=4

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