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# 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)

## Description

example

V = var(A) returns the variance of the elements of A along the first array dimension whose size does not equal 1.

• If A is a vector of observations, the variance is a scalar.

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

• If A is a multidimensional array, then var(A) treats the values along the first array dimension whose size does not equal 1 as vectors. The size of this dimension becomes 1 while the sizes of all other dimensions remain the same.

• The variance is normalized by the number of observations-1 by default.

• If A is a scalar, var(A) returns 0. If A is a 0-by-0 empty array, var(A) returns NaN.

example

V = var(A,w) specifies a weighting scheme. When w = 0 (default), V is normalized by the number of observations-1. When w = 1, it 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 variation 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.

## Examples

collapse all

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

## Input Arguments

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Input array, specified as a vector, matrix, or multidimensional array.

Data Types: single | double
Complex Number Support: Yes

Weight, specified as one of:

• 0 — normalizes by the number of observations-1. If there is only one observation, the weight is 1.

• 1 — normalizes by the number of observations.

• a 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 no value is specified, then the default is the first array dimension whose size does not equal 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 a two-dimensional input array, A.

• If dim = 1, then var(A,0,1) returns a row vector containing the variance of the elements in each column.

• If dim = 2, then var(A,0,2) returns a column vector containing the variance of the elements in each row.

var returns an array of zeros the same size as A when dim is greater than ndims(A).

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

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.

Data Types: double | single | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

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.

Data Types: char

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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 of N instead of N-1, which can be specified by setting w to 1. In either case, the mean is assumed to have the usual normalization factor N.