# std

Standard deviation

## Description

example

S = std(A) returns the standard deviation of the elements of A along the first array dimension whose size does not equal 1. By default, the standard deviation is normalized by N-1, where N is the number of observations.

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

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

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

• If A is a scalar, then S is 0.

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

example

S = std(A,w) specifies a weighting scheme. When w = 0 (default), the standard deviation is normalized by N-1, where N is the number of observations. When w = 1, the standard deviation is normalized by the number of observations. w also can be a weight vector containing nonnegative elements. In this case, the length of w must equal the length of the dimension over which std is operating.

S = std(A,w,"all") computes the standard deviation over all elements of A when w is either 0 or 1. This syntax is valid for MATLAB® versions R2018b and later.

example

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

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

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[S,M] = std(___) also returns the mean of the elements of A used to calculate the standard deviation. If S is the weighted standard deviation, 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 the standard deviation of each column.

A = [4 -5 1; 2 3 5; -9 1 7];
S = std(A)
S = 1×3

7.0000    4.1633    3.0551

Create a 3-D array and compute the standard deviation along the first dimension.

A(:,:,1) = [2 4; -2 1];
A(:,:,2) = [9 13; -5 7];
A(:,:,3) = [4 4; 8 -3];
S = std(A)
S =
S(:,:,1) =

2.8284    2.1213

S(:,:,2) =

9.8995    4.2426

S(:,:,3) =

2.8284    4.9497

Create a matrix and compute the standard deviation of each column according to a weight vector w.

A = [1 5; 3 7; -9 2];
w = [1 1 0.5];
S = std(A,w)
S = 1×2

4.4900    1.8330

Create a matrix and compute the standard deviation along each row.

A = [6 4 23 -3; 9 -10 4 11; 2 8 -5 1];
S = std(A,0,2)
S = 3×1

11.0303
9.4692
5.3229

Create a 3-D array and compute the standard deviation 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];
S = std(A,0,[1 2])
S =
S(:,:,1) =

2.5000

S(:,:,2) =

7.7460

S(:,:,3) =

4.5735

Create a vector and compute its standard deviation, excluding NaN values.

A = [1.77 -0.005 3.98 -2.95 NaN 0.34 NaN 0.19];
S = std(A,"omitnan")
S = 2.2797

Create a matrix and compute the standard deviation and mean of each column.

A = [4 -5 1; 2 3 5; -9 1 7];
[S,M] = std(A)
S = 1×3

7.0000    4.1633    3.0551

M = 1×3

-1.0000   -0.3333    4.3333

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

A = [1 5; 3 7; -9 2];
w = [1 1 0.5];
[S,M] = std(A,w)
S = 1×2

4.4900    1.8330

M = 1×2

-0.2000    5.2000

## Input Arguments

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

Data Types: single | double | datetime | duration
Complex Number Support: Yes

Weight, specified as one of these values:

• 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 standard deviation 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(S,dim) is 1, while the sizes of all other dimensions remain the same.

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

• std(A,0,1) computes the standard deviation of the elements in each column of A and returns a 1-by-n row vector.

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

If dim is greater than ndims(A), then std(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 std(A,0,[1 2]) returns a 1-by-1-by-3 array whose elements are the standard deviations computed over each page of A.

NaN condition, specified as one of these values:

• "includenan" — Include NaN values when computing the standard deviation, resulting in NaN.

• "omitnan" — Ignore NaN values appearing in either the input array or weight vector.

• "includenat" — Include NaT values when computing the standard deviation for datetime arrays.

• "omitnat" — Ignore NaT values appearing in either the input array or weight vector for datetime arrays.

## Output Arguments

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Standard deviation, returned as a scalar, vector, matrix, or multidimensional array.

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

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

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

• If A is a scalar, then S is 0.

• If A is a 0-by-0 empty array, then S 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 std(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 S is the weighted standard deviation, then M is the weighted mean.

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### Standard Deviation

For a finite-length vector A made up of N scalar observations, the standard deviation is defined as

$\begin{array}{l}S={\sqrt{\frac{1}{N-1}\sum _{i=1}^{N}|{A}_{i}-\mu {|}^{2}}}^{},\hfill \\ \hfill \end{array}$

where μ is the mean of A:

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

The standard deviation is the square root of the variance.

Some definitions of standard deviation 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 square root of the second moment of the sample about its mean.

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

### Weighted Standard Deviation

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

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

where μw is the weighted mean of A.

### Weighted Mean

For a random variable 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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