# StandardDeviation StandardDeviation[data]

gives the standard deviation estimate of the elements in data.

StandardDeviation[dist]

gives the standard deviation of the distribution dist.

# Details    • StandardDeviation is also known as volatility.
• StandardDeviation measures dispersion from the mean of data or distributions.
• For vector data with =Mean[data], the standard deviation estimate is given by for reals and for complexes.
• • For matrix data, the standard deviation estimate is computed for each column vector with StandardDeviation[{{x1,y1,},{x2,y2,},}] equivalent to {StandardDeviation[{x1,x2,}],StandardDeviation[{y1,y2,}]}. »
• • For array data, standard deviation is equivalent to ArrayReduce[StandardDeviation,data,1]. »
•                         ↓ ↓ ↓ ↓ ↓     • For a real weighted WeightedData[{x1,x2,},{w1,w2,}], the standard deviation is given by . »
• StandardDeviation handles both numerical and symbolic data.
• The data can have the following additional forms and interpretations:
•  Association the values (the keys are ignored) » SparseArray as an array, equivalent to Normal[data] » QuantityArray quantities as an array » WeightedData weighted variance, based on the underlying EmpiricalDistribution » EventData based on the underlying SurvivalDistribution » TimeSeries, TemporalData, … vector or array of values (the time stamps ignored) » Image,Image3D RGB channel's values or grayscale intensity value » Audio amplitude values of all channels »
• For a univariate distribution dist, the standard deviation is given by σ=Expectation[(x-μ)2,xdist]1/2 with μ=Mean[dist]. »
• • For multivariate distribution dist, the standard deviation is given by {σx,σy,}=Expectation[{(x-μx)2,(y-μy)2,},{x,y,}dist]1/2. »
• • For a random process proc, the standard deviation function can be computed for slice distribution at time t, SliceDistribution[proc,t], as σ[t]=StandardDeviation[SliceDistribution[proc,t]]. »
• # 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(19)

### Basic Uses(8)

Exact input yields exact output:

Approximate input yields approximate output:

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:

### Array Data(5)

StandardDeviation for a matrix gives columnwise standard deviations:

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

Works with large arrays:

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

SparseArray data can be used just like dense arrays:

Find the standard deviation of a QuantityArray:

### Image and Audio Data(2)

Channelwise standard deviation of an RGB image:

Standard deviation of a grayscale image:

On audio objects, StandardDeviation works channelwise:

### Distributions and Processes(4)

Find the standard deviation for univariate distributions:

Multivariate distributions:

Standard deviation for derived distributions:

Data distribution:

Standard deviation for distributions with quantities:

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: