# Variance

Variance[data]

gives the variance estimate of the elements in data.

Variance[dist]

gives the variance of the distribution dist.

# Details

• Variance measures dispersion of data or distributions.
• Variance[data] gives the unbiased estimate of variance.
• For VectorQ data , the variance estimate is given by for reals and for complexes and =Mean[data].
• For MatrixQ data, the variance estimate is computed for each column vector, with Variance[{{x1,y1,},{x2,y2,},}] equivalent to {Variance[{x1,x2,}],Variance[{y1,y2,}]}. »
• For ArrayQ data, variance is equivalent to ArrayReduce[Variance,data,1]. »
• For a real weighted WeightedData[{x1,x2,},{w1,w2,}], the variance is given by . »
• Variance 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 » DateObject, TimeObject list of dates or list of time »
• For a univariate distribution dist, the variance is given by σ2=Expectation[(x-μ)2,xdist] with μ=Mean[dist]. »
• For a multivariate distribution dist, the variance is given by {σx2,σy2,}=Expectation[{(x-μx)2,(y-μy)2,},{x,y,}dist]. »
• For a random process proc, the variance function can be computed for slice distribution at time t, SliceDistribution[proc,t], as σ[t]2=Variance[SliceDistribution[proc,t]]. »

# Examples

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## Basic Examples(4)

Variance of a list of numbers:

Variance of elements in each column:

Variance of a list of dates:

Variance of a parametric distribution:

## Scope(22)

### Basic Uses(7)

Exact input yields exact output:

Approximate input yields approximate output:

Find the variance of WeightedData:

Find the variance of EventData:

Find the variance of TemporalData:

Find the variance of a TimeSeries:

The variance depends only on the values:

Find the variance of data involving quantities:

### Array Data(5)

Variance for a matrix gives columnwise variances:

Variance for a tensor gives columnwise variances at the first level:

Works with large arrays:

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

SparseArray data can be used just like dense arrays:

Find the variance of a QuantityArray:

### Image and Audio Data(2)

Channelwise variance of an RGB image:

Variance of a grayscale image:

On audio objects, Variance works channelwise:

### Date and Time(5)

Compute variance of dates:

Compute the weighted variance of dates:

Compute the variance of dates given in different calendars:

Compute the variance of times:

Compute the variance of times with different time zone specifications:

### Distributions and Processes(3)

Find the variance for univariate distributions:

Multivariate distributions:

Variance for derived distributions:

Data distribution:

Variance function for a random process:

## Applications(5)

Variance is a measure of dispersion:

Compute a moving variance for samples of three random processes:

Compare data volatility by smoothing with moving variance:

Find the mean and variance for the number of great inventions and scientific discoveries in each year from 1860 to 1959:

Investigate weak stationarity of the process data by analyzing variance of slices:

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

Find the variance of the heights for the children in a class:

## Properties & Relations(11)

The square root of Variance is StandardDeviation:

Variance is a scaled squared Norm of deviations from the Mean:

Variance is a scaled CentralMoment:

The square root of Variance is a scaled RootMeanSquare of the deviations:

Variance is a scaled Mean of squared deviations from the Mean:

Variance is a scaled SquaredEuclideanDistance from the Mean:

Variance is less than MeanDeviation if all absolute deviations are less than 1:

Variance is greater than MeanDeviation if all absolute deviations are greater than 1:

Variance of a random variable as an Expectation:

Variance gives an unbiased sample estimate:

Unbiased means that the expected value of the sample variance with respect to the population distribution equals the variance of the underlying distribution:

Variance gives an unbiased weighted sample estimate:

Unbiased means that the expected value of the sample variance with respect to the population distribution equals the variance of the underlying distribution:

## Neat Examples(1)

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

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_variance, author="Wolfram Research", title="{Variance}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/Variance.html}", note=[Accessed: 06-August-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_variance, organization={Wolfram Research}, title={Variance}, year={2024}, url={https://reference.wolfram.com/language/ref/Variance.html}, note=[Accessed: 06-August-2024 ]}