var

Syntax

``V = var(A)``
``V = var(A,w)``
``V = var(A,w,"all")``
``V = var(A,w,dim)``
``V = var(A,w,vecdim)``
``V = var(___,nanflag)``
``[V,M] = var(___)``

Description

example

````V = var(A)` returns the variance of the elements of `A` along the first array dimension whose size does not equal 1. By default, the variance is normalized by `N-1`, where `N` is the number of observationsIf `A` is a vector of observations, then `V` is a scalar.If `A` is a matrix whose columns are random variables and whose rows are observations, then `V` is a row vector containing the variance corresponding to each column.If `A` is a multidimensional array, then `var(A)` operates along the first array dimension whose size does not equal 1, treating the elements as vectors. The size of `V` in this dimension becomes `1` while the sizes of all other dimensions are the same as `A`.If `A` is a scalar, then `V` is `0`.If `A` is a `0`-by-`0` empty array, then `V` is `NaN`.```

example

````V = var(A,w)` specifies a weighting scheme. When `w = 0` (default), the variance is normalized by `N-1`, where `N` is the number of observations. When `w = 1`, the variance is normalized by the number of observations. `w` can also be a weight vector containing nonnegative elements. In this case, the length of `w` must equal the length of the dimension over which `var` is operating. ```
````V = var(A,w,"all")` computes the variance over all elements of `A` when `w` is either 0 or 1. This syntax is valid for MATLAB® versions R2018b and later.```

example

````V = var(A,w,dim)` returns the variance along the dimension `dim`. To maintain the default normalization while specifying the dimension of operation, set ```w = 0``` in the second argument.```

example

````V = var(A,w,vecdim)` computes the variance over the dimensions specified in the vector `vecdim` when `w` is 0 or 1. For example, if `A` is a matrix, then `var(A,0,[1 2])` computes the variance over all elements in `A`, since every element of a matrix is contained in the array slice defined by dimensions 1 and 2.```

example

````V = var(___,nanflag)` specifies whether to include or omit `NaN` values from the calculation for any of the previous syntaxes. For example, `var(A,"includenan")` includes all `NaN` values in `A` while `var(A,"omitnan")` ignores them.```

example

````[V,M] = var(___)` also returns the mean of the elements of `A` used to calculate the variance. If `V` is the weighted variance, then `M` is the weighted mean. This syntax is valid for MATLAB versions R2022a and later.```

Examples

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Create a matrix and compute its variance.

```A = [4 -7 3; 1 4 -2; 10 7 9]; var(A)```
```ans = 1×3 21.0000 54.3333 30.3333 ```

Create a 3-D array and compute its variance.

```A(:,:,1) = [1 3; 8 4]; A(:,:,2) = [3 -4; 1 2]; var(A)```
```ans = ans(:,:,1) = 24.5000 0.5000 ans(:,:,2) = 2 18 ```

Create a matrix and compute its variance according to a weight vector `w`.

```A = [5 -4 6; 2 3 9; -1 1 2]; w = [0.5 0.25 0.25]; var(A,w)```
```ans = 1×3 6.1875 9.5000 6.1875 ```

Create a matrix and compute its variance along the first dimension.

```A = [4 -2 1; 9 5 7]; var(A,0,1)```
```ans = 1×3 12.5000 24.5000 18.0000 ```

Compute the variance of `A` along the second dimension.

`var(A,0,2)`
```ans = 2×1 9 4 ```

Create a 3-D array and compute the variance over each page of data (rows and columns).

```A(:,:,1) = [2 4; -2 1]; A(:,:,2) = [9 13; -5 7]; A(:,:,3) = [4 4; 8 -3]; V = var(A,0,[1 2])```
```V = V(:,:,1) = 6.2500 V(:,:,2) = 60 V(:,:,3) = 20.9167 ```

Create a vector and compute its variance, excluding `NaN` values.

```A = [1.77 -0.005 3.98 -2.95 NaN 0.34 NaN 0.19]; V = var(A,"omitnan")```
```V = 5.1970 ```

Create a matrix and compute the variance and mean of each column.

```A = [4 -7 3; 1 4 -2; 10 7 9]; [V,M] = var(A)```
```V = 1×3 21.0000 54.3333 30.3333 ```
```M = 1×3 5.0000 1.3333 3.3333 ```

Create a matrix and compute the weighted variance and weighted mean of each column according to a weight vector `w`.

```A = [5 -4 6; 2 3 9; -1 1 2]; w = [0.5 0.25 0.25]; [V,M] = var(A,w)```
```V = 1×3 6.1875 9.5000 6.1875 ```
```M = 1×3 2.7500 -1.0000 5.7500 ```

Input Arguments

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Input array, specified as a vector, matrix, or multidimensional array. If `A` is a scalar, then `var(A)` returns `0`. If `A` is a `0`-by-`0` empty array, then `var(A)` returns `NaN`.

Data Types: `single` | `double`
Complex Number Support: Yes

Weight, specified as one of:

• `0` — Normalize by `N-1`, where `N` is the number of observations. If there is only one observation, then the weight is 1.

• `1` — Normalize by `N`.

• Vector made up of nonnegative scalar weights corresponding to the dimension of `A` along which the variance is calculated.

Data Types: `single` | `double`

Dimension to operate along, specified as a positive integer scalar. If you do not specify the dimension, then the default is the first array dimension of size greater than 1.

Dimension `dim` indicates the dimension whose length reduces to `1`. The `size(V,dim)` is `1`, while the sizes of all other dimensions remain the same.

Consider an `m`-by-`n` input matrix, `A`:

• `var(A,0,1)` computes the variance of the elements in each column of `A` and returns a `1`-by-`n` row vector.

• `var(A,0,2)` computes the variance of the elements in each row of `A` and returns an `m`-by-`1` column vector.

If `dim` is greater than `ndims(A)`, then `var(A)` returns an array of zeros the same size as `A`.

Vector of dimensions, specified as a vector of positive integers. Each element represents a dimension of the input array. The lengths of the output in the specified operating dimensions are 1, while the others remain the same.

Consider a 2-by-3-by-3 input array, `A`. Then `var(A,0,[1 2])` returns a 1-by-1-by-3 array whose elements are the variances computed over each page of `A`.

`NaN` condition, specified as one of these values:

• `"includenan"` — The variance of input containing `NaN` values is also `NaN`.

• `"omitnan"` — All `NaN` values appearing in either the input array or weight vector are ignored.

Output Arguments

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

• If `A` is a vector of observations, then `V` is a scalar.

• If `A` is a matrix whose columns are random variables and whose rows are observations, then `V` is a row vector containing the variance corresponding to each column.

• If `A` is a multidimensional array, then `var(A)` operates along the first array dimension whose size does not equal 1, treating the elements as vectors. The size of `V` in this dimension becomes `1` while the sizes of all other dimensions are the same as `A`.

• If `A` is a scalar, then `V` is `0`.

• If `A` is a `0`-by-`0` empty array, then `V` is `NaN`.

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

• If `A` is a vector of observations, then `M` is a scalar.

• If `A` is a matrix whose columns are random variables and whose rows are observations, then `M` is a row vector containing the mean corresponding to each column.

• If `A` is a multidimensional array, then `var(A)` operates along the first array dimension whose size does not equal 1, treating the elements as vectors. The size of `M` in this dimension becomes `1` while the sizes of all other dimensions are the same as `A`.

• If `A` is a scalar, then `M` is equal to `A`.

• If `A` is a `0`-by-`0` empty array, then `M` is `NaN`.

If `V` is the weighted variance, then `M` is the weighted mean.

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Variance

For a random variable vector A made up of N scalar observations, the variance is defined as

`$V=\frac{1}{N-1}\sum _{i=1}^{N}|{A}_{i}-\mu {|}^{2}$`

where μ is the mean of A,

`$\mu =\frac{1}{N}\sum _{i=1}^{N}{A}_{i}.$`

Some definitions of variance use a normalization factor N instead of N – 1. You can use a normalization factor of N by specifying a weight of `1`, producing the second moment of the sample about its mean.

Regardless of the normalization factor for the variance, the mean is assumed to have the normalization factor N.

Weighted Variance

For a finite-length vector A made up of N scalar observations and weighting scheme `w`, the weighted variance is defined as

`${V}_{w}=\frac{\sum _{i=1}^{N}{w}_{i}|{A}_{i}-{\mu }_{w}{|}^{2}}{\sum _{i=1}^{N}{w}_{i}}$`

where μw is the weighted mean of A.

Weighted Mean

For a finite-length vector A made up of N scalar observations and weighting scheme `w`, the weighted mean is defined as

`${\mu }_{w}=\frac{\sum _{i=1}^{N}{w}_{i}{A}_{i}}{\sum _{i=1}^{N}{w}_{i}}$`

Version History

Introduced before R2006a

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