Documentation

# kurtosis

## Syntax

``k = kurtosis(X)``
``k = kurtosis(X,flag)``
``k = kurtosis(X,flag,'all')``
``k = kurtosis(X,flag,dim)``
``k = kurtosis(X,flag,vecdim)``

## Description

example

````k = kurtosis(X)` returns the sample kurtosis of `X`. If `X` is a vector, then `kurtosis(X)` returns a scalar value that is the kurtosis of the elements in `X`.If `X` is a matrix, then `kurtosis(X)` returns a row vector that contains the sample kurtosis of each column in `X`.If `X` is a multidimensional array, then `kurtosis(X)` operates along the first nonsingleton dimension of `X`. ```

example

````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, meaning it tends to differ from the population kurtosis by a systematic amount based on the sample size. You can set `flag` to `0` to correct for this systematic bias.```

example

````k = kurtosis(X,flag,'all')` returns the kurtosis of all elements of `X`.```

example

````k = kurtosis(X,flag,dim)` returns the kurtosis along the operating dimension `dim` of `X`.```

example

````k = kurtosis(X,flag,vecdim)` returns the kurtosis over the dimensions specified in the vector `vecdim`. For example, if `X` is a 2-by-3-by-4 array, then `kurtosis(X,1,[1 2])` returns a 1-by-1-by-4 array. Each element of the output array is the biased kurtosis of the elements on the corresponding page of `X`.```

## Examples

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Set the random seed for reproducibility of the results.

`rng('default')`

Generate a matrix with 5 rows and 4 columns.

`X = randn(5,4)`
```X = 5×4 0.5377 -1.3077 -1.3499 -0.2050 1.8339 -0.4336 3.0349 -0.1241 -2.2588 0.3426 0.7254 1.4897 0.8622 3.5784 -0.0631 1.4090 0.3188 2.7694 0.7147 1.4172 ```

Find the sample kurtosis of `X`.

`k = kurtosis(X)`
```k = 1×4 2.7067 1.4069 2.3783 1.1759 ```

`k` is a row vector containing the sample kurtosis of each column in `X`.

For an input vector, correct for bias in the calculation of kurtosis by specifying the `flag` input argument.

Set the random seed for reproducibility of the results.

`rng('default') `

Generate a vector of length 10.

`x = randn(10,1)`
```x = 10×1 0.5377 1.8339 -2.2588 0.8622 0.3188 -1.3077 -0.4336 0.3426 3.5784 2.7694 ```

Find the biased kurtosis of `x`. By default, `kurtosis` sets the value of `flag` to `1` for computing the biased kurtosis.

`k1 = kurtosis(x) % flag is 1 by default`
```k1 = 2.3121 ```

Find the bias-corrected kurtosis of `x` by setting the value of `flag` to `0`.

`k2 = kurtosis(x,0) `
```k2 = 2.7483 ```

Find the kurtosis along different dimensions for a multidimensional array.

Set the random seed for reproducibility of the results.

`rng('default') `

Create a 4-by-3-by-2 array of random numbers.

`X = randn([4,3,2])`
```X = X(:,:,1) = 0.5377 0.3188 3.5784 1.8339 -1.3077 2.7694 -2.2588 -0.4336 -1.3499 0.8622 0.3426 3.0349 X(:,:,2) = 0.7254 -0.1241 0.6715 -0.0631 1.4897 -1.2075 0.7147 1.4090 0.7172 -0.2050 1.4172 1.6302 ```

Find the kurtosis of `X` along the default dimension.

`k1 = kurtosis(X)`
```k1 = k1(:,:,1) = 2.1350 1.7060 2.2789 k1(:,:,2) = 1.0542 2.3278 2.0996 ```

By default, `kurtosis` operates along the first dimension of `X` whose size does not equal 1. In this case, this dimension is the first dimension of `X`. Therefore, `k1` is a 1-by-3-by-2 array.

Find the biased kurtosis of `X` along the second dimension.

`k2 = kurtosis(X,1,2)`
```k2 = k2(:,:,1) = 1.5000 1.5000 1.5000 1.5000 k2(:,:,2) = 1.5000 1.5000 1.5000 1.5000 ```

`k2` is a 4-by-1-by-2 array.

Find the biased kurtosis of `X` along the third dimension.

`k3 = kurtosis(X,1,3)`
```k3 = 4×3 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 ```

`k3` is a 4-by-3 matrix.

Find the kurtosis over multiple dimensions by using the `'all'` and `vecdim` input arguments.

Set the random seed for reproducibility of the results.

`rng('default')`

Create a 4-by-3-by-2 array of random numbers.

`X = randn([4 3 2])`
```X = X(:,:,1) = 0.5377 0.3188 3.5784 1.8339 -1.3077 2.7694 -2.2588 -0.4336 -1.3499 0.8622 0.3426 3.0349 X(:,:,2) = 0.7254 -0.1241 0.6715 -0.0631 1.4897 -1.2075 0.7147 1.4090 0.7172 -0.2050 1.4172 1.6302 ```

Find the biased kurtosis of `X`.

`kall = kurtosis(X,1,'all')`
```kall = 2.8029 ```

`kall` is the biased kurtosis of the entire input data set `X`.

Find the biased kurtosis of each page of `X` by specifying the first and second dimensions.

`kpage = kurtosis(X,1,[1 2])`
```kpage = kpage(:,:,1) = 1.9345 kpage(:,:,2) = 2.5877 ```

For example, `kpage(1,1,2)` is the biased kurtosis of the elements in `X(:,:,2)`.

Find the biased kurtosis of the elements in each `X(i,:,:)` slice by specifying the second and third dimensions.

`krow = kurtosis(X,1,[2 3])`
```krow = 4×1 3.8457 1.4306 1.7094 2.3378 ```

For example, `krow(3)` is the biased kurtosis of the elements in `X(3,:,:)`.

## Input Arguments

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Input data that represents a sample from a population, specified as a vector, matrix, or multidimensional array.

• If `X` is a vector, then `kurtosis(X)` returns a scalar value that is the kurtosis of the elements in `X`.

• If `X` is a matrix, then `kurtosis(X)` returns a row vector that contains the sample kurtosis of each column in `X`.

• If `X` is a multidimensional array, then `kurtosis(X)` operates along the first nonsingleton dimension of `X`.

To specify the operating dimension when `X` is a matrix or an array, use the `dim` input argument.

`kurtosis` treats `NaN` values in `X` as missing values and removes them.

Data Types: `single` | `double`

Indicator for the bias, specified as `0` or `1`.

• If `flag` is `1` (default), then the kurtosis of `X` is biased, meaning it tends to differ from the population kurtosis by a systematic amount based on the sample size.

• If `flag` is `0`, then `kurtosis` corrects for the systematic bias.

Data Types: `single` | `double` | `logical`

Dimension along which to operate, specified as a positive integer. If you do not specify a value for `dim`, then the default is the first dimension of `X` whose size does not equal 1.

Consider the kurtosis of a matrix `X`:

• If `dim` is equal to 1, then `kurtosis` returns a row vector that contains the sample kurtosis of each column in `X`.

• If `dim` is equal to 2, then `kurtosis` returns a column vector that contains the sample kurtosis of each row in `X`.

If `dim` is greater than `ndims(X)` or if `size(X,dim)` is 1, then `kurtosis` returns an array of `NaN`s the same size as `X`.

Data Types: `single` | `double`

Vector of dimensions, specified as a positive integer vector. Each element of `vecdim` represents a dimension of the input array `X`. The output `k` has length 1 in the specified operating dimensions. The other dimension lengths are the same for `X` and `k`.

For example, if `X` is a 2-by-3-by-3 array, then `kurtosis(X,1,[1 2])` returns a 1-by-1-by-3 array. Each element of the output is the biased kurtosis of the elements on the corresponding page of `X`. Data Types: `single` | `double`

## Output Arguments

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Kurtosis, returned as a scalar, vector, matrix, or multidimensional array.

## 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. 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.

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. The `kurtosis` function computes a sample version of this population value.

When you set `flag` to `1`, the kurtosis is biased, and 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`, `kurtosis` corrects for the systematic bias, and 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 equation requires that `X` contain at least four elements.