# Kurtosis Kurtosis[data]

gives the coefficient of kurtosis for the elements in data.

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 »
• 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

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## Basic Examples(2)

Kurtosis for a list of values:

Kurtosis for a symbolic data:

Kurtosis for a parametric distribution:

## Scope(18)

### Basic Uses(7)

Exact input yields exact output:

Approximate input yields approximate output:

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:

### Array Data(5)

Kurtosis for a matrix gives columnwise kurtosis:

Work with large arrays:

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

SparseArray data can be used just like dense arrays:

Find the kurtosis of a QuantityArray:

### Image and Audio Data(2)

Channelwise kurtosis value of an RGB image:

Mean intensity value of a grayscale image:

On audio objects, Kurtosis works channelwise:

### 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 :

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(2)

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:

Kurtosis may be undefined for data:

Kurtosis may be undefined for a distribution:

## Neat Examples(1)

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