# 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.

• If A is a vector of observations, then the standard deviation 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 deviations 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 this dimension becomes 1 while the sizes of all other dimensions remain the same.

• By default, the standard deviation is normalized by N-1, where N is the number of observations.

example

S = std(A,w) specifies a weighting scheme for any of the previous syntaxes. When w = 0 (default), S is normalized by N-1. When w = 1, S is normalized by the number of observations, N. 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 for any of the previous syntaxes. To maintain the default normalization while specifying the dimension of operation, set w = 0 in the second argument.

example

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.

example

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.

## 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 calculate 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

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

Consider a two-dimensional input array, A.

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

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

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

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

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.

For datetime arrays, you can also use 'omitnat' or 'includenat' to omit and include NaT values, respectively.

Data Types: char

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### Standard Deviation

For a random variable 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 of N instead of N-1, which you can specify by setting w to 1.