Variance
Variance[list]
gives the sample variance of the elements in list.
Variance[dist]
gives the variance of the distribution dist.
Details

- Variance measures dispersion of data or distributions.
- Variance[list] gives the unbiased estimate of variance.
- Variance[list] is equivalent to Total[(list-Mean[list])^2]/(Length[list]-1) for real-valued data.
- For complex data, Variance[list] is equivalent to (list-Mean[list]).Conjugate[list-Mean[list]]/(Length[list]-1).
- Variance handles both numerical and symbolic data.
- Variance[{{x1,y1,…},{x2,y2,…},…}] gives {Variance[{x1,x2,…}],Variance[{y1,y2,…}]}.
- Variance[dist] is equivalent to Expectation[(x-μ)2,xdist] with μ=Mean[dist].
Examples
open allclose allBasic Examples (3)
Scope (13)
Data (10)
Exact input yields exact output:
Approximate input yields approximate output:
Variance for a matrix gives columnwise variances:
Variance for a tensor gives columnwise variances at the first level:
SparseArray data can be used just like dense arrays:
Find the variance of WeightedData:
Find the variance of EventData:
Find the variance of TemporalData:
Find the variance of a TimeSeries:
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:
Text
Wolfram Research (2003), Variance, Wolfram Language function, https://reference.wolfram.com/language/ref/Variance.html (updated 2007).
CMS
Wolfram Language. 2003. "Variance." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2007. 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