gives the covariance between the vectors v1 and v2.


gives the sample covariance matrix for observations in matrix m.


gives the covariance matrix for the matrices m1 and m2.


gives the covariance matrix for the multivariate symbolic distribution dist.


gives the (i,j)^(th) covariance for the multivariate symbolic distribution dist.


  • Covariance[v1,v2] gives the unbiased estimate of the covariance between v1 and v2.
  • The lists v1 and v2 must be the same length.
  • Covariance[v1,v2] is equivalent to (v1-Mean[v1]). Conjugate[v2-Mean[v2]]/(Length[v1]-1).
  • For a matrix m with columns, Covariance[m] is a × matrix of the covariances between columns of m.
  • For an × matrix m1 and an × matrix m2, Covariance[m1,m2] is a × matrix of the covariances between columns of m1 and columns of m2.
  • Covariance works with SparseArray objects.
  • Covariance[dist,i,j] gives Expectation[(xi-μi)(xj-μj),{x1,x2,}dist], where μi is the i^(th) component of the mean of dist.
  • Covariance[dist] gives a covariance matrix with the (i,j)^(th) entry given by Covariance[dist,i,j].


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Basic Examples  (3)

Covariance between two vectors:

Covariance matrix for a matrix:

Covariance matrix for two matrices:

Scope  (12)

Data  (7)

Exact input yields exact output:

Approximate input yields approximate output:

Covariance between vectors of complexes:

Works with large arrays:

SparseArray data can be used:

Find the covariance of WeightedData:

Find the covariance for data involving quantities:

Distributions and Processes  (5)

Covariance for a continuous multivariate distribution:

Covariance for a discrete multivariate distribution:

Covariance for derived distributions:

Data distribution:

Covariance matrix for a random process at times s and t:

Covariance matrix for TemporalData at times and :

Compare to the covariance of the process slice:

Applications  (3)

Compute the covariance of two financial time series:

Covariance can be used to measure linear association:

Covariance can only detect monotonic relationships:

HoeffdingD can be used to detect a variety of dependence structures:

Properties & Relations  (9)

The covariance matrix is symmetric and positive semidefinite:

A covariance matrix scaled by standard deviations is a correlation matrix:

Covariance and AbsoluteCorrelation are the same for a distribution with zero mean:

SpearmanRho is related to Covariance applied to ranks:

CovarianceFunction for a process is the off-diagonal entry in the covariance matrix:

Covariance and Correlation are the same for standardized vectors:

The covariance of a list with itself is the variance:

The diagonal of a covariance matrix is the variance:

The covariance tends to be large only on the diagonal of a random matrix:

Neat Examples  (1)

Compute the covariance for a GCD array:

Wolfram Research (2007), Covariance, Wolfram Language function, (updated 2010).


Wolfram Research (2007), Covariance, Wolfram Language function, (updated 2010).


Wolfram Language. 2007. "Covariance." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2010.


Wolfram Language. (2007). Covariance. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2023_covariance, author="Wolfram Research", title="{Covariance}", year="2010", howpublished="\url{}", note=[Accessed: 04-December-2023 ]}


@online{reference.wolfram_2023_covariance, organization={Wolfram Research}, title={Covariance}, year={2010}, url={}, note=[Accessed: 04-December-2023 ]}