Standard deviation - MATLAB std (2024)

Table of Contents

Syntax Description Examples Standard Deviation of Matrix Columns Standard Deviation of 3-D Array Specify Standard Deviation Weights Standard Deviation Along Matrix Rows Standard Deviation of Array Page Standard Deviation Excluding Missing Values Standard Deviation and Mean Input Arguments A — Input array vector | matrix | multidimensional array | table | timetable w — Weight 0 (default) | 1 | vector dim — Dimension to operate along positive integer scalar vecdim — Vector of dimensions vector of positive integers missingflag — Missing value condition "includemissing" (default) | "includenan" | "includenat" | "omitmissing" | "omitnan" | "omitnat" Output Arguments S — Standard deviation scalar | vector | matrix | multidimensional array | table M — Mean scalar | vector | matrix | multidimensional array | table More About Standard Deviation Weighted Standard Deviation Weighted Mean Extended Capabilities Tall Arrays Calculate with arrays that have more rows than fit in memory. C/C++ Code GenerationGenerate C and C++ code using MATLAB® Coder™. GPU Code Generation Generate CUDA® code for NVIDIA® GPUs using GPU Coder™. Thread-Based Environment Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool. GPU Arrays Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™. Distributed ArraysPartition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™. Version History R2023a: Perform calculations directly on tables and timetables R2023a: Specify missing value condition R2023a: Improved performance with small group size R2022a: Return mean or weighted mean R2018b: Operate on multiple dimensions See Also MATLAB Command Americas Europe Asia Pacific FAQs

Standard deviation

collapse all in page

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,"all") returns the standard deviation over all elements of 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

collapse all

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.

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

collapse all

`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.

`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 `A` and `w` when computing the standard deviation. If any element in the operating dimension is missing, then the corresponding element in `S` is missing.
`"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

collapse all

`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

collapse all

Standard Deviation

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

$\begin{array}{l} S = {\sqrt{\frac{1}{N - 1} \sum_{i = 1}^{N} | A_{i} - μ |^{2}}}^{}, \end{array}$

where μ is the mean of A:

$μ = \frac{1}{N} \sum_{i = 1}^{N} A_{i} .$

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

$S_{w} = \sqrt{\frac{\sum_{i = 1}^{N} w_{i} | A_{i} - μ_{w} |^{2}}{\sum_{i = 1}^{N} w_{i}}}$

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

$μ_{w} = \frac{\sum_{i = 1}^{N} w_{i} A_{i}}{\sum_{i = 1}^{N} w_{i}}$

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.

MATLAB Command

You clicked a link that corresponds to this MATLAB command:

Run the command by entering it in the MATLAB Command Window. Web browsers do not support MATLAB commands.

Select a Web Site

Choose a web site to get translated content where available and see local events and offers. Based on your location, we recommend that you select: .

You can also select a web site from the following list:

Americas

América Latina (Español)
Canada (English)
United States (English)

Europe

Belgium (English)
Denmark (English)
Deutschland (Deutsch)
España (Español)
Finland (English)
France (Français)
Ireland (English)
Italia (Italiano)
Luxembourg (English)

Netherlands (English)
Norway (English)
Österreich (Deutsch)
Portugal (English)
Sweden (English)
Switzerland
- Deutsch
- English
- Français
United Kingdom (English)

Asia Pacific

Australia (English)
India (English)
New Zealand (English)
中国
- 简体中文
- English
日本 (日本語)
한국 (한국어)

Contact your local office

FAQs

What is the standard deviation in Matlab STD? ›

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.

How do you find the standard deviation of an image in Matlab? ›

J = stdfilt( I ) performs standard deviation filtering of image I and returns the filtered image J . The value of each output pixel is the standard deviation of the 3-by-3 neighborhood around the corresponding input pixel.

Discover More Details ›

What is the deviation from the mean in Matlab? ›

y = mad( X ) returns the mean absolute deviation of the values in X . If X is a vector, then mad returns the mean or median absolute deviation of the values in X . If X is a matrix, then mad returns a row vector containing the mean or median absolute deviation of each column of X .

See Details ›

What is the standard deviation of a population in Matlab? ›

Population Standard Deviation

σ = ∑ i = 1 n ( x i − μ ) 2 n . If X is a random sample from a population, then the mean μ is estimated by the sample mean, and σ is the biased maximum likelihood estimator of the population standard deviation. Notice that the denominator in this variance formula is n.

Tell Me More ›

Is standard deviation a SD or STD? ›

Standard deviation may be abbreviated SD, and is most commonly represented in mathematical texts and equations by the lowercase Greek letter σ (sigma), for the population standard deviation, or the Latin letter s, for the sample standard deviation.

Get More Info Here ›

What is the standard deviation of the normal distribution in Matlab? ›

The standard normal distribution has zero mean and unit standard deviation. If z is standard normal, then σz + µ is also normal with mean µ and standard deviation σ. Conversely, if x is normal with mean µ and standard deviation σ, then z = (x – µ) / σ is standard normal.

What is the standard deviation of a matrix? ›

For a vector or a matrix x , y=stdev(x) returns in the scalar y the standard deviation of all the entries of x . y=stdev(x,'r') (or, equivalently, y=stdev(x,1) ) is the rowwise standard deviation.

View Details ›

What is the difference between standard error and standard deviation? ›

Standard deviation describes variability within a single sample, while standard error describes variability across multiple samples of a population. Standard deviation is a descriptive statistic that can be calculated from sample data, while standard error is an inferential statistic that can only be estimated.

Read The Full Story ›

How to calculate mean variance and standard deviation in Matlab? ›

To calculate mean, variance, and standard deviation:

Create the visionhdl. ImageStatistics object and set its properties.
Call the object with arguments, as if it were a function.

Discover More ›

How to calculate the mean in Matlab? ›

M = mean( A , "all" ) returns the mean over all elements of A . M = mean( A , dim ) returns the mean along dimension dim . For example, if A is a matrix, then mean(A,2) returns a column vector containing the mean of each row.

What is the standard deviation of an array? ›

Standard Deviation is the square root of the variance. Approach: To get the standard deviation of an array, first we calculate the mean and then the variance, and then the deviation. To calculate the mean we use Array.

Learn More Now ›

What is the standard deviation of a time series in Matlab? ›

tsstd = std( ts ) returns the standard deviation of the data in a timeseries object. tsstd = std( ts , Name,Value ) specifies additional options when computing the standard deviation using one or more name-value pair arguments.

Read On ›

How do you find STD deviation? ›

Step 1: Find the mean.
Step 2: Subtract the mean from each score.
Step 3: Square each deviation.
Step 4: Add the squared deviations.
Step 5: Divide the sum by the number of scores.
Step 6: Take the square root of the result from Step 5.

Discover More Details ›

What is the standard normal mean and Stdev? ›

Standard normal distribution: a normal distribution represented in z scores. The standard normal distribution always has a mean of zero and a standard deviation of one.

Show Me More ›

What is the 2d standard deviation in Matlab? ›

Description. The 2-D Standard Deviation block computes the standard deviation of an input array. The input can be a 1-D vector, 2-D matrix, or an N-D-array. The block can compute standard deviation along a specified dimension of the input or the entire input.

See Details ›