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var

Description

example

V = var(A) returns the variance of the elements of A along the first array dimension whose size is greater than 1. By default, the variance is normalized by N-1, where N is the number of observations.

  • 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 is greater than 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 in A.

  • If A is a scalar, then V is 0.

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

  • If A is a table or timetable, then var(A) returns a one-row table containing the variance of each variable. (since R2023a)

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") returns the variance over all elements of A when w is either 0 or 1.

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

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V = var(A,w,vecdim) returns 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]) returns the variance over all elements in A because 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 in A for any of the previous syntaxes. For example, var(A,"omitnan") ignores NaN values when computing the variance. By default, var includes NaN values.

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.

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 matrix containing NaN values.

A = [1.77 -0.005 NaN -2.95; NaN 0.34 NaN 0.19]
A = 2×4

    1.7700   -0.0050       NaN   -2.9500
       NaN    0.3400       NaN    0.1900

Compute the variance of the matrix, excluding NaN values. For matrix columns that contain any NaN value, var computes with non-NaN elements. For matrix columns that contain all NaN values, the variance is NaN.

V = var(A,"omitnan")
V = 1×4

         0    0.0595       NaN    4.9298

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, multidimensional array, table, or timetable. 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 | table | timetable
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,0,1) column-wise computation

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

    var(A,0,2) row-wise computation

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.

Mapping of a 2-by-3-by-3 input array to a 1-by-1-by-3 output array

Missing value condition, specified as one of these values:

  • "includemissing" or "includenan" — Include NaN values in A when computing the variance. If any element in the operating dimension is NaN, then the corresponding element in V is NaN. "includemissing" and "includenan" have the same behavior.

  • "omitmissing" or "omitnan" — Ignore NaN values in A and w, and compute the variance over fewer points. If all elements in the operating dimension are NaN, then the corresponding element in V is NaN. "omitmissing" and "omitnan" have the same behavior.

Output Arguments

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

  • 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 is greater than 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 in A.

  • If A is a scalar, then V is 0.

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

  • If A is a table or timetable, then V is a one-row table. If the variables of A have units, then the variables of V do not have those units. (since R2023a)

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

  • 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 is greater than 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 in 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 A is a table or timetable, then M is a one-row table. If the variables of A have units, then the variables of M have the same units. (since R2023a)

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

More About

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Variance

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

V=1N1i=1N|Aiμ|2

where μ is the mean of A,

μ=1Ni=1NAi.

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

Vw=i=1Nwi|Aiμw|2i=1Nwi

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

μw=i=1NwiAii=1Nwi

Extended Capabilities

Version History

Introduced before R2006a

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See Also

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