Variance

✖
`Variance`

Updated in 14.1

✖

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,xdist] with μ=Mean[dist]. »

For a multivariate distribution dist, the variance is given by {σ_x²,σ_y²,…}=Expectation[{(x-μ_x)²,(y-μ_y)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

open allclose all

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

Variance of a list of numbers:

In[1]:=1

✖

https://wolfram.com/xid/0ftyv7y-gax

Out[1]=1

Variance of elements in each column:

In[1]:=1

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https://wolfram.com/xid/0ftyv7y-x2c

Out[1]=1

Variance of a list of dates:

In[1]:=1

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https://wolfram.com/xid/0ftyv7y-bpyuss

Out[1]=1

Variance of a parametric distribution:

In[1]:=1

✖

https://wolfram.com/xid/0ftyv7y-n3bzeg

Out[1]=1

Scope (22)Survey of the scope of standard use cases

Basic Uses (7)

Exact input yields exact output:

In[1]:=1

✖

https://wolfram.com/xid/0ftyv7y-ug7y2

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0ftyv7y-bcry2t

Out[2]=2

Approximate input yields approximate output:

In[1]:=1

✖

https://wolfram.com/xid/0ftyv7y-ksx55

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0ftyv7y-d02ofx

Out[2]=2

Find the variance of WeightedData:

In[1]:=1

✖

https://wolfram.com/xid/0ftyv7y-d0wc9z

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0ftyv7y-gbwndp

In[3]:=3

✖

https://wolfram.com/xid/0ftyv7y-h8f9w3

Out[3]=3

Find the variance of EventData:

In[1]:=1

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https://wolfram.com/xid/0ftyv7y-i4ryy7

In[2]:=2

✖

https://wolfram.com/xid/0ftyv7y-m07ttq

Out[2]=2

Find the variance of TemporalData:

In[1]:=1

✖

https://wolfram.com/xid/0ftyv7y-gx0rsr

In[2]:=2

✖

https://wolfram.com/xid/0ftyv7y-f5ij1t

In[3]:=3

✖

https://wolfram.com/xid/0ftyv7y-8e999

Out[3]=3

Find the variance of a TimeSeries:

In[1]:=1

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https://wolfram.com/xid/0ftyv7y-b6530u

In[2]:=2

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https://wolfram.com/xid/0ftyv7y-k0xfjy

Out[2]=2

The variance depends only on the values:

In[3]:=3

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https://wolfram.com/xid/0ftyv7y-nnfg9

Out[3]=3

Find the variance of data involving quantities:

In[1]:=1

✖

https://wolfram.com/xid/0ftyv7y-jopin9

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0ftyv7y-e8c21s

Out[2]=2

Array Data (5)

Variance for a matrix gives columnwise variances:

In[1]:=1

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https://wolfram.com/xid/0ftyv7y-ezu2uz

Out[1]=1

In[4]:=4

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https://wolfram.com/xid/0ftyv7y-h4n22m

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

In[5]:=5

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https://wolfram.com/xid/0ftyv7y-lw96ov

Out[5]=5

In[6]:=6

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https://wolfram.com/xid/0ftyv7y-dagpjp

Works with large arrays:

In[1]:=1

✖

https://wolfram.com/xid/0ftyv7y-nknun

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0ftyv7y-ma3v2m

Out[2]=2

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

In[1]:=1

✖

https://wolfram.com/xid/0ftyv7y-cs7n5q

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0ftyv7y-bv5njj

Out[2]=2

SparseArray data can be used just like dense arrays:

In[1]:=1

✖

https://wolfram.com/xid/0ftyv7y-n691tv

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0ftyv7y-drrysl

Out[2]=2

Find the variance of a QuantityArray:

In[1]:=1

✖

https://wolfram.com/xid/0ftyv7y-lgwnaj

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0ftyv7y-k03qc6

Out[2]=2

Image and Audio Data (2)

Channelwise variance of an RGB image:

In[1]:=1

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https://wolfram.com/xid/0ftyv7y-hfby9q

Out[1]=1

Variance of a grayscale image:

In[2]:=2

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https://wolfram.com/xid/0ftyv7y-ue2gq5

Out[2]=2

On audio objects, Variance works channelwise:

In[1]:=1

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https://wolfram.com/xid/0ftyv7y-nq1jnz

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0ftyv7y-mjmudf

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0ftyv7y-bs38vd

Out[3]=3

Date and Time (5)

Compute variance of dates:

In[1]:=1

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https://wolfram.com/xid/0ftyv7y-b1smxx

In[2]:=2

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https://wolfram.com/xid/0ftyv7y-pa4nmn

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0ftyv7y-uok1il

Out[3]=3

In[4]:=4

✖

https://wolfram.com/xid/0ftyv7y-jbrj2r

Out[4]=4

Compute the weighted variance of dates:

In[1]:=1

✖

https://wolfram.com/xid/0ftyv7y-c98kbd

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0ftyv7y-8c1had

In[3]:=3

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https://wolfram.com/xid/0ftyv7y-t71b2h

Out[3]=3

Compute the variance of dates given in different calendars:

In[1]:=1

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https://wolfram.com/xid/0ftyv7y-wbzcuv

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0ftyv7y-9ius88

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0ftyv7y-o3oukb

Out[3]=3

Compute the variance of times:

In[1]:=1

✖

https://wolfram.com/xid/0ftyv7y-et9bla

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0ftyv7y-ztsexm

Out[2]=2

Compute the variance of times with different time zone specifications:

In[1]:=1

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https://wolfram.com/xid/0ftyv7y-mrqghz

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0ftyv7y-1d7sk5

Out[2]=2

Distributions and Processes (3)

Find the variance for univariate distributions:

In[1]:=1

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https://wolfram.com/xid/0ftyv7y-rxz55

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0ftyv7y-hbq28j

Out[2]=2

Multivariate distributions:

In[3]:=3

✖

https://wolfram.com/xid/0ftyv7y-ek075b

Out[3]=3

In[4]:=4

✖

https://wolfram.com/xid/0ftyv7y-lzwoz3

Out[4]=4

Variance for derived distributions:

In[1]:=1

✖

https://wolfram.com/xid/0ftyv7y-rgc72x

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0ftyv7y-byqvvz

Out[2]=2

Data distribution:

In[3]:=3

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https://wolfram.com/xid/0ftyv7y-215ry

In[4]:=4

✖

https://wolfram.com/xid/0ftyv7y-fq5ptk

Out[4]=4

Variance function for a random process:

In[1]:=1

✖

https://wolfram.com/xid/0ftyv7y-fugn

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0ftyv7y-g9pmgp

Out[2]=2

Applications (5)Sample problems that can be solved with this function

Variance is a measure of dispersion:

In[1]:=1

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https://wolfram.com/xid/0ftyv7y-bhnrki

In[2]:=2

✖

https://wolfram.com/xid/0ftyv7y-t94no

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0ftyv7y-o9323q

Out[3]=3

Compute a moving variance for samples of three random processes:

In[1]:=1

✖

https://wolfram.com/xid/0ftyv7y-ebo2c

In[2]:=2

✖

https://wolfram.com/xid/0ftyv7y-e3pstb

Out[2]=2

Compare data volatility by smoothing with moving variance:

In[3]:=3

✖

https://wolfram.com/xid/0ftyv7y-ii17q9

In[4]:=4

✖

https://wolfram.com/xid/0ftyv7y-yg9t0a

Out[4]=4

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

In[1]:=1

✖

https://wolfram.com/xid/0ftyv7y-kq0wb5

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0ftyv7y-nfevvf

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0ftyv7y-b7us08

Out[3]=3

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

In[1]:=1

✖

https://wolfram.com/xid/0ftyv7y-r14k0m

In[2]:=2

✖

https://wolfram.com/xid/0ftyv7y-hq9gxo

In[3]:=3

✖

https://wolfram.com/xid/0ftyv7y-byyqpk

In[4]:=4

✖

https://wolfram.com/xid/0ftyv7y-nhaulq

Out[4]=4

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

In[5]:=5

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https://wolfram.com/xid/0ftyv7y-xwv2xr

Out[5]=5

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

In[1]:=1

✖

https://wolfram.com/xid/0ftyv7y-cevfij

In[2]:=2

✖

https://wolfram.com/xid/0ftyv7y-fllmtw

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0ftyv7y-celepo

Out[3]=3

Properties & Relations (11)Properties of the function, and connections to other functions

The square root of Variance is StandardDeviation:

In[1]:=1

✖

https://wolfram.com/xid/0ftyv7y-iu5

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0ftyv7y-j7q

Out[2]=2

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

In[1]:=1

✖

https://wolfram.com/xid/0ftyv7y-lp8ujb

In[2]:=2

✖

https://wolfram.com/xid/0ftyv7y-dt87yu

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0ftyv7y-k1qqbw

Out[3]=3

Variance is a scaled CentralMoment:

In[1]:=1

✖

https://wolfram.com/xid/0ftyv7y-bps8zf

In[2]:=2

✖

https://wolfram.com/xid/0ftyv7y-fess9s

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0ftyv7y-ct1qwp

Out[3]=3

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

In[1]:=1

✖

https://wolfram.com/xid/0ftyv7y-hqtrtd

In[2]:=2

✖

https://wolfram.com/xid/0ftyv7y-cyyyvo

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0ftyv7y-efwiiv

Out[3]=3

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

In[1]:=1

✖

https://wolfram.com/xid/0ftyv7y-brtwin

In[2]:=2

✖

https://wolfram.com/xid/0ftyv7y-hmk853

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0ftyv7y-cibwnc

Out[3]=3

Variance is a scaled SquaredEuclideanDistance from the Mean:

In[1]:=1

✖

https://wolfram.com/xid/0ftyv7y-cqnxdq

In[2]:=2

✖

https://wolfram.com/xid/0ftyv7y-kigjh4

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0ftyv7y-dnccw

Out[3]=3

In[4]:=4

✖

https://wolfram.com/xid/0ftyv7y-enaymy

Out[4]=4

In[5]:=5

✖

https://wolfram.com/xid/0ftyv7y-u9svr

Out[5]=5

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

In[1]:=1

✖

https://wolfram.com/xid/0ftyv7y-gqobe7

In[2]:=2

✖

https://wolfram.com/xid/0ftyv7y-e1pum5

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0ftyv7y-g876u6

Out[3]=3

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

In[1]:=1

✖

https://wolfram.com/xid/0ftyv7y-h25qvy

In[2]:=2

✖

https://wolfram.com/xid/0ftyv7y-b1r9du

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0ftyv7y-felohq

Out[3]=3

Variance of a random variable as an Expectation:

In[1]:=1

✖

https://wolfram.com/xid/0ftyv7y-c5emif

In[2]:=2

✖

https://wolfram.com/xid/0ftyv7y-cl5rpb

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0ftyv7y-ekl2fe

Out[3]=3

Variance gives an unbiased sample estimate:

In[1]:=1

✖

https://wolfram.com/xid/0ftyv7y-fdwqgp

Out[1]=1

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

In[2]:=2

✖

https://wolfram.com/xid/0ftyv7y-e5xxe6

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0ftyv7y-cg6nq9

Out[3]=3

Variance gives an unbiased weighted sample estimate:

In[1]:=1

✖

https://wolfram.com/xid/0ftyv7y-6kl30

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0ftyv7y-ith2bq

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

In[3]:=3

✖

https://wolfram.com/xid/0ftyv7y-dd9yts

Out[3]=3

In[4]:=4

✖

https://wolfram.com/xid/0ftyv7y-h81hib

Out[4]=4

Neat Examples (1)Surprising or curious use cases

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

In[1]:=1

✖

https://wolfram.com/xid/0ftyv7y-fh2wz

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0ftyv7y-8raem

Out[2]=2

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Variance

✖
`Variance`

Details

Examples

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

Scope (22)Survey of the scope of standard use cases

Basic Uses (7)

Array Data (5)

Image and Audio Data (2)

Date and Time (5)

Distributions and Processes (3)

Applications (5)Sample problems that can be solved with this function

Properties & Relations (11)Properties of the function, and connections to other functions

Neat Examples (1)Surprising or curious use cases

Text

CMS

APA

BibTeX

BibLaTeX

Variance ✖ Variance

Details

Examples

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

Scope (22)Survey of the scope of standard use cases

Basic Uses (7)

Array Data (5)

Image and Audio Data (2)

Date and Time (5)

Distributions and Processes (3)

Applications (5)Sample problems that can be solved with this function

Properties & Relations (11)Properties of the function, and connections to other functions

Neat Examples (1)Surprising or curious use cases

See Also

Tech Notes

Related Guides

History

Text

CMS

APA

BibTeX

BibLaTeX

Variance

✖
`Variance`