WOLFRAM

gives the standard deviation estimate of the elements in data.

gives the standard deviation of the distribution dist.

Details

Examples

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Basic Examples  (4)Summary of the most common use cases

Standard deviation of a list of numbers:

Out[1]=1

Standard deviation of elements in each column:

Out[1]=1

Standard deviation of a list of dates:

Out[1]=1

Standard deviation of a parametric distribution:

Out[1]=1

Scope  (24)Survey of the scope of standard use cases

Basic Uses  (8)

Exact input yields exact output:

Out[1]=1
Out[2]=2

Approximate input yields approximate output:

Out[1]=1
Out[2]=2

Find the standard deviation of WeightedData:

Out[1]=1
Out[3]=3

Find the standard deviation of EventData:

Out[2]=2

Find the standard deviation of TemporalData:

Out[3]=3

Find the standard deviation of a TimeSeries:

Out[2]=2

The standard deviation depends only on the values:

Out[3]=3

Find a three-element moving standard deviation:

Out[1]=1

Find the standard deviation of data involving quantities:

Out[1]=1
Out[2]=2

Array Data  (5)

StandardDeviation for a matrix gives columnwise standard deviations:

Out[1]=1

StandardDeviation for a tensor gives columnwise standard deviations at the first level:

Out[2]=2

Works with large arrays:

Out[1]=1
Out[2]=2

When the input is an Association, StandardDeviation works on its values:

Out[1]=1
Out[2]=2

SparseArray data can be used just like dense arrays:

Out[1]=1
Out[2]=2

Find the standard deviation of a QuantityArray:

Out[1]=1
Out[2]=2

Image and Audio Data  (2)

Channelwise standard deviation of an RGB image:

Out[1]=1

Standard deviation of a grayscale image:

Out[2]=2

On audio objects, StandardDeviation works channelwise:

Out[1]=1
Out[2]=2
Out[3]=3

Date and Time  (5)

Compute standard deviation of dates:

Out[2]=2
Out[3]=3

Compute the weighted standard deviation of dates:

Out[1]=1
Out[3]=3

Compute the standard deviation of dates given in different calendars:

Out[1]=1
Out[2]=2
Out[3]=3

Compute the standard deviation of times:

Out[1]=1
Out[2]=2

Compute the standard deviation of times with different time zone specifications:

Out[1]=1
Out[2]=2

Distributions and Processes  (4)

Find the standard deviation for univariate distributions:

Out[1]=1
Out[2]=2

Multivariate distributions:

Out[3]=3
Out[4]=4

Standard deviation for derived distributions:

Out[1]=1
Out[2]=2

Data distribution:

Out[4]=4

Standard deviation for distributions with quantities:

Out[1]=1
Out[2]=2
Out[3]=3

Standard deviation function for a random process:

Out[1]=1
Out[2]=2

Applications  (7)Sample problems that can be solved with this function

StandardDeviation is a measure of dispersion:

Out[3]=3
Out[7]=7

Transform data to have mean 0 and unit variance:

Out[1]=1
Out[2]=2
Out[3]=3
Out[4]=4

Identify periods of high volatility in the S&P 500 using a five-year moving standard deviation:

Out[3]=3

Find the mean and standard deviation for the number of cycles to failure of deep-groove ball-bearings:

Out[2]=2
Out[3]=3

Plot the data:

Out[5]=5

Probability that the values lie within two standard deviations of the mean:

Out[6]=6

Investigate weak stationarity of the process data by analyzing standard deviations of slices:

Out[4]=4

Use a larger plot range to see how relatively small the variations are:

Out[5]=5

Compute standard deviation for slices of a collection of paths of a random process:

Choose a few slice times:

Compute standard deviations and means:

Create a standard deviation band around the mean:

Plot standard deviations around the mean over these paths:

Out[5]=5

Find the standard deviation of the heights for the children in a class:

Out[2]=2
Out[3]=3

The heights within one standard deviation from the mean:

Out[4]=4

Properties & Relations  (9)Properties of the function, and connections to other functions

The square of StandardDeviation is Variance:

Out[1]=1
Out[2]=2

StandardDeviation is a scaled Norm of deviations from the Mean:

Out[2]=2
Out[3]=3

StandardDeviation is the square root of a scaled CentralMoment:

Out[2]=2
Out[3]=3

StandardDeviation is a scaled RootMeanSquare of the deviations:

Out[2]=2
Out[3]=3

StandardDeviation is the square root of a scaled Mean of squared deviations:

Out[2]=2
Out[3]=3

StandardDeviation as a scaled EuclideanDistance from the Mean:

Out[2]=2
Out[3]=3
Out[4]=4
Out[5]=5

StandardDeviation squared is less than MeanDeviation if all absolute deviations are less than 1:

Out[2]=2
Out[3]=3

StandardDeviation squared is greater than MeanDeviation if all absolute deviations are greater than 1:

Out[2]=2
Out[3]=3

StandardDeviation of a random variable as the square root of Variance:

Out[2]=2
Out[3]=3
Out[4]=4

Neat Examples  (1)Surprising or curious use cases

The distribution of StandardDeviation estimates for 20, 100, and 300 samples:

Out[1]=1
Out[2]=2
Wolfram Research (2003), StandardDeviation, Wolfram Language function, https://reference.wolfram.com/language/ref/StandardDeviation.html (updated 2024).
Wolfram Research (2003), StandardDeviation, Wolfram Language function, https://reference.wolfram.com/language/ref/StandardDeviation.html (updated 2024).

Text

Wolfram Research (2003), StandardDeviation, Wolfram Language function, https://reference.wolfram.com/language/ref/StandardDeviation.html (updated 2024).

Wolfram Research (2003), StandardDeviation, Wolfram Language function, https://reference.wolfram.com/language/ref/StandardDeviation.html (updated 2024).

CMS

Wolfram Language. 2003. "StandardDeviation." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. https://reference.wolfram.com/language/ref/StandardDeviation.html.

Wolfram Language. 2003. "StandardDeviation." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. https://reference.wolfram.com/language/ref/StandardDeviation.html.

APA

Wolfram Language. (2003). StandardDeviation. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/StandardDeviation.html

Wolfram Language. (2003). StandardDeviation. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/StandardDeviation.html

BibTeX

@misc{reference.wolfram_2025_standarddeviation, author="Wolfram Research", title="{StandardDeviation}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/StandardDeviation.html}", note=[Accessed: 09-July-2025 ]}

@misc{reference.wolfram_2025_standarddeviation, author="Wolfram Research", title="{StandardDeviation}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/StandardDeviation.html}", note=[Accessed: 09-July-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_standarddeviation, organization={Wolfram Research}, title={StandardDeviation}, year={2024}, url={https://reference.wolfram.com/language/ref/StandardDeviation.html}, note=[Accessed: 09-July-2025 ]}

@online{reference.wolfram_2025_standarddeviation, organization={Wolfram Research}, title={StandardDeviation}, year={2024}, url={https://reference.wolfram.com/language/ref/StandardDeviation.html}, note=[Accessed: 09-July-2025 ]}