Standard deviation

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## Syntax

`S = std(A)`

`S = std(A,w)`

`S = std(A,w,"all")`

`S = std(A,w,dim)`

`S = std(A,w,vecdim)`

`S = std(___,missingflag)`

`[S,M] = std(___)`

## Description

example

`S = std(A)`

returns the standard deviation of the elements of `A`

along the first array dimension whose size is greater than 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 is greater than 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 in`A`

.If

`A`

is a scalar, then`S`

is`0`

.If

`A`

is a`0`

-by-`0`

empty array, then`S`

is`NaN`

.If

`A`

is a table or timetable, then`std(A)`

returns a one-row table containing the standard deviation of each variable.*(since R2023a)*

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,`

returns the standard deviation over all elements of `"all"`

)`A`

when `w`

is either 0 or 1.

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.

example

`S = std(A,w,vecdim)`

returns 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])`

returns the standard deviation over all elements in `A`

because every element of a matrix is contained in the array slice defined by dimensions 1 and 2.

example

`S = std(___,missingflag)`

specifies whether to include or omit missing values in `A`

for any of the previous syntaxes. For example, `std(A,"omitmissing")`

ignores all missing values when computing the standard deviation. By default, `std`

includes missing values.

example

`[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.

## Examples

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### Standard Deviation of Matrix Columns

Open Live Script

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

### Standard Deviation of 3-D Array

Open Live Script

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 2.1213S(:,:,2) = 9.8995 4.2426S(:,:,3) = 2.8284 4.9497

### Specify Standard Deviation Weights

Open Live Script

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

### Standard Deviation Along Matrix Rows

Open Live Script

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

### Standard Deviation of Array Page

Open Live Script

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) = 2.5000S(:,:,2) = 7.7460S(:,:,3) = 4.5735

### Standard Deviation Excluding Missing Values

Open Live Script

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 standard deviation of the matrix, excluding missing values. For matrix columns that contain any `NaN`

value, `std`

computes with the non-`NaN`

elements. For columns in `A`

that contain all `NaN`

values, the standard deviation is `NaN`

.

`S = std(A,"omitmissing")`

`S = `*1×4* 0 0.2440 NaN 2.2203

*Before R2023a: Use *`"omitnan"`

* or *`"omitnat"`

* to ignore missing values.*

### Standard Deviation and Mean

Open Live Script

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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`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 `std(A)`

returns `0`

. If `A`

is a `0`

-by-`0`

empty array, then `std(A)`

returns `NaN`

.

**Data Types: **`single`

| `double`

| `datetime`

| `duration`

| `table`

| `timetable`

**Complex Number Support: **Yes

`w`

— Weight

`0`

(default) | `1`

| vector

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 correspondingto the dimension of

`A`

along which the standarddeviation 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 of size greater than 1.

Dimension `dim`

indicates the dimension whoselength 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 samesize 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 `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`

.

`missingflag`

— Missing value condition

`"includemissing"`

(default) | `"includenan"`

| `"includenat"`

| `"omitmissing"`

| `"omitnan"`

| `"omitnat"`

Missing value condition, specified as one of the values in this table.

Value | Input Data Type | Description |
---|---|---|

`"includemissing"` (since R2023a) | All supported data types | Include missing values in |

`"includenan"` | `double` , `single` , `duration` | |

`"includenat"` | `datetime` | |

`"omitmissing"` (since R2023a) | All supported data types | Ignore missing values in `A` and `w` , and compute the standard deviation over fewer points. If all elements in the operating dimension are missing, then the corresponding element in `S` is missing. |

`"omitnan"` | `double` , `single` , `duration` | |

`"omitnat"` | `datetime` |

## Output Arguments

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`S`

— Standard deviation

scalar | vector | matrix | multidimensional array | table

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

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 is not greater than 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 in`A`

.If

`A`

is a scalar, then`S`

is`0`

.If

`A`

is a`0`

-by-`0`

empty array, then`S`

is`NaN`

.If

`A`

is a table or timetable, then`S`

is a one-row table.*(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`std(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.*(since R2023a)*

If `S`

is the weighted standard deviation, then `M`

is the weighted mean.

## More About

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### Standard Deviation

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

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

where *μ* is the mean of *A*:

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

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

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

where *μ _{w}* is the weighted mean of

*A*.

### Weighted Mean

For a random variable 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:

C++ code generation supports the following syntaxes:

`S = std(A)`

`S = std(A,w)`

`S = std(A,w,"all")`

`S = std(A,w,dim)`

`S = std(A,w,vecdim)`

`S = std(__,missingflag)`

When specified, dimension must be a constant.

See Variable-Sizing Restrictions for Code Generation of Toolbox Functions (MATLAB Coder).

### GPU Code Generation

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

Usage notes and limitations:

GPU code generation supports the following syntaxes:

`S = std(A)`

`S = std(A,w)`

`S = std(A,w,"all")`

`S = std(A,w,dim)`

`S = std(A,w,vecdim)`

`S = std(__,missingflag)`

If you specify

`dim`

, then it must be a constant.

### 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™.

This function fully supports GPU arrays. 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**

expand all

### R2023a: Perform calculations directly on tables and timetables

The `std`

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 all missing values in the input arrays when computing the standard deviation by using the `"includemissing"`

or `"omitmissing"`

options. Previously, `"includenan"`

, `"omitnan"`

, `"includenat"`

, and `"omitnat"`

specified a missing value condition that was specific to the data type of the input arrays.

### R2023a: Improved performance with small group size

The `std`

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 standard deviation along the default vector dimension. The code is about 1.9x faster than in the previous release.

function timingStdA = rand(10,1);for i = 1:8e5 std(A);endend

The approximate execution times are:

**R2022b:** 1.77 s

**R2023a:** 0.93 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(@timingStd)

### R2022a: Return mean or weighted mean

The `std`

function can now return the mean of the input elements used to calculate the standard deviation by using a second output argument `M`

. If a weighting scheme is specified, then `std`

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.

## See Also

corrcoef | cov | mean | median | var

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