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# kurtosis

Kurtosis

## Syntax

k = kurtosis(X)
k = kurtosis(X,flag)
k = kurtosis(X,flag,dim)

## Description

k = kurtosis(X) returns the sample kurtosis of X. For vectors, kurtosis(x) is the kurtosis of the elements in the vector x. For matrices kurtosis(X) returns the sample kurtosis for each column of X. For N-dimensional arrays, kurtosis operates along the first nonsingleton dimension of X.

k = kurtosis(X,flag) specifies whether to correct for bias (flag is 0) or not (flag is 1, the default). When X represents a sample from a population, the kurtosis of X is biased, that is, it will tend to differ from the population kurtosis by a systematic amount that depends on the size of the sample. You can set flag to 0 to correct for this systematic bias.

k = kurtosis(X,flag,dim) takes the kurtosis along dimension dim of X.

kurtosis treats NaNs as missing values and removes them.

## Examples

```X = randn([5 4])
X =
1.1650  1.6961  -1.4462  -0.3600
0.6268  0.0591  -0.7012  -0.1356
0.0751  1.7971  1.2460  -1.3493
0.3516  0.2641  -0.6390  -1.2704
-0.6965  0.8717  0.5774  0.9846

k = kurtosis(X)
k =
2.1658  1.2967  1.6378  1.9589```

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### Algorithms

Kurtosis is a measure of how outlier-prone a distribution is. The kurtosis of the normal distribution is 3. Distributions that are more outlier-prone than the normal distribution have kurtosis greater than 3; distributions that are less outlier-prone have kurtosis less than 3.

The kurtosis of a distribution is defined as

$k=\frac{E{\left(x-\mu \right)}^{4}}{{\sigma }^{4}}$

where μ is the mean of x, σ is the standard deviation of x, and E(t) represents the expected value of the quantity t. kurtosis computes a sample version of this population value.

 Note   Some definitions of kurtosis subtract 3 from the computed value, so that the normal distribution has kurtosis of 0. The kurtosis function does not use this convention.

When you set flag to 1, the following equation applies:

${k}_{1}=\frac{\frac{1}{n}\sum _{i=1}^{n}{\left({x}_{i}-\overline{x}\right)}^{4}}{{\left(\frac{1}{n}\sum _{i=1}^{n}{\left({x}_{i}-\overline{x}\right)}^{2}\right)}^{2}}$

When you set flag to 0, the following equation applies:

${k}_{0}=\frac{n-1}{\left(n-2\right)\left(n-3\right)}\left(\left(n+1\right){k}_{1}-3\left(n-1\right)\right)+3$

This bias-corrected formula requires that X contain at least four elements.