# StandardDeviation

StandardDeviation[list]

gives the sample standard deviation of the elements in list.

StandardDeviation[dist]

gives the standard deviation of the distribution dist.

# Examples

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## Basic Examples(3)

Standard deviation of a list of numbers:

Standard deviation of elements in each column:

Standard deviation of a parametric distribution:

## Scope(14)

### Data(11)

Exact input yields exact output:

Approximate input yields approximate output:

StandardDeviation for a matrix gives columnwise standard deviations:

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

Works with large arrays:

SparseArray data can be used just like dense arrays:

Find the standard deviation of WeightedData:

Find the standard deviation of EventData:

Find the standard deviation of TemporalData:

Find the standard deviation of a TimeSeries:

The standard deviation depends only on the values:

Find a three-element moving standard deviation:

Find the standard deviation of data involving quantities:

### Distributions and Processes(3)

Find the standard deviation for univariate distributions:

Multivariate distributions:

Standard deviation for derived distributions:

Data distribution:

Standard deviation function for a random process:

## Applications(7)

StandardDeviation is a measure of dispersion:

Transform data to have mean 0 and unit variance:

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

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

Plot the data:

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

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

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

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:

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

The heights within one standard deviation from the mean:

## Properties & Relations(9)

The square of StandardDeviation is Variance:

StandardDeviation is a scaled Norm of deviations from the Mean:

StandardDeviation is the square root of a scaled CentralMoment:

StandardDeviation is a scaled RootMeanSquare of the deviations:

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

StandardDeviation as a scaled EuclideanDistance from the Mean:

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

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

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

## Neat Examples(1)

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

Wolfram Research (2003), StandardDeviation, Wolfram Language function, https://reference.wolfram.com/language/ref/StandardDeviation.html (updated 2007).

#### Text

Wolfram Research (2003), StandardDeviation, Wolfram Language function, https://reference.wolfram.com/language/ref/StandardDeviation.html (updated 2007).

#### CMS

Wolfram Language. 2003. "StandardDeviation." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2007. 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

#### BibTeX

@misc{reference.wolfram_2022_standarddeviation, author="Wolfram Research", title="{StandardDeviation}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/StandardDeviation.html}", note=[Accessed: 08-June-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2022_standarddeviation, organization={Wolfram Research}, title={StandardDeviation}, year={2007}, url={https://reference.wolfram.com/language/ref/StandardDeviation.html}, note=[Accessed: 08-June-2023 ]}