var

Variance

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

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 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 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 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)

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.

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.

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.

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.

[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

Variance of Matrix

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

Variance of Array

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

Specify Variance Weight Vector

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

Specify Dimension for Variance

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

Variance of Array Page

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

Variance Excluding Missing Values

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

Variance and Mean

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

A — Input array

vector | matrix | multidimensional array | table | timetable

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

w — Weight

0 (default) | 1 | vector

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

dim — Dimension to operate along

positive integer scalar

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 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 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.

vecdim — Vector of dimensions

vector of positive integers

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.

nanflag — Missing value condition

"includemissing" (default) | "includenan" | "omitmissing" | "omitnan"

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

V — Variance

scalar | vector | matrix | multidimensional array | table

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 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 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)

M — Mean

scalar | vector | matrix | multidimensional array | table

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 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 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

Variance

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

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

where μ is the mean of A,

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

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

$$V_w=\frac{{\displaystyle \sum _{i=1}^N{w}_i|A_i-{\mu }_w{|}^2}}{{\displaystyle \sum _{i=1}^N{w}_i}}$$

where μ_w 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

$${\mu }_w=\frac{{\displaystyle \sum _{i=1}^N{w}_i{A}_i}}{{\displaystyle \sum _{i=1}^N{w}_i}}$$

Extended Capabilities

Tall Arrays

Calculate with arrays that have more rows than fit in memory.

This function supports tall arrays with the limitation:

• The weighting scheme cannot be a vector.

For more information, see Tall Arrays for Out-of-Memory Data.

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• Code generation does not support the output argument M.

• If used, the arguments dim, vecdim, and nanflag must be constant at code generation time.

• For input argument A:

  • If you do not specify a dimension, the code generator operates along the first dimension of the input array that is variable size or whose size does not equal 1. If this dimension is variable size at code generation time and is 1 at run time, a run-time error can occur. To avoid this error, specify the dimension.

  • If all dimensions of the input array are variable size at code generation time, this array cannot be empty at run time.

GPU Code Generation

Generate CUDA® code for NVIDIA® GPUs using GPU Coder™.

Usage notes and limitations:

• If specified, dim must be a constant.

• GPU code generation supports the following syntaxes:

  • 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)

Thread-Based Environment

Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.

This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

The var function fully supports GPU arrays. To run the function on a GPU, specify the input data as a gpuArray (Parallel Computing Toolbox). For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Distributed Arrays

Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.

This function fully supports distributed arrays. For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).

Version History

Introduced before R2006a

R2023a: Perform calculations directly on tables and timetables

The var function can calculate on all variables within a table or timetable without indexing to access those variables. All variables must have data types that support the calculation. For more information, see Direct Calculations on Tables and Timetables.

R2023a: Specify missing value condition

Include or omit missing values in the input arrays when computing the variance by using the "includemissing" or "omitmissing" options. These options have the same behavior as the "includenan" and "omitnan" options, respectively.

R2023a: Improved performance with small group size

The var function shows improved performance when computing over a real vector when the operating dimension is not specified. The function determines the default operating dimension more quickly in R2023a than in R2022b.

For example, this code computes the variance along the default vector dimension. The code is about 1.6x faster than in the previous release.

function timingVar
A = rand(10,1);
for i = 1:8e5
    var(A);
end
end

The approximate execution times are:

R2022b: 1.29 s

R2023a: 0.79 s

The code was timed on a Windows^® 10, Intel^® Xeon^® CPU E5-1650 v4 @ 3.60 GHz test system using the timeit function.

timeit(@timingVar)

R2022a: Return mean or weighted mean

The var function can now return the mean of the input elements used to calculate the variance by using a second output argument M. If a weighting scheme is specified, then var returns the weighted mean.

R2018b: Operate on multiple dimensions

Operate on multiple dimensions of the input array at a time. Specify a vector of operating dimensions, or specify the "all" option to operate on all array dimensions.