gives the covariance between the vectors v and w.


gives the cross-covariance matrix for the matrices a and b.


gives the auto-covariance matrix for observations in matrix a.


gives the auto-covariance matrix for the multivariate symbolic distribution dist.


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


  • Covariance is typically used to measure covariation, i.e. whether one variable tends to vary similarly to another.
  • Covariance[v,w] gives the unbiased estimate of the covariance between v and w.
  • For vectors and of length , the covariance estimate Covariance[v,w] is given by with mu^^_v=Mean[v].
  • For matrices and with dimensions and and columns indexed as and , respectively, Covariance[a,b] is a matrix with elements given by :
  • ,
  • where is an -vector of ones, is Mean[a] and is Mean[b].
  • For a matrix a with columns, Covariance[a] is a matrix given by Covariance[a, a].
  • Covariance works with any vector that is VectorQ or matrix that is MatrixQ.
  • 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  (11)

Data  (6)

Exact input yields exact output:

Approximate input yields approximate output:

Covariance between vectors of complexes:

Works with large arrays:

A structured array can be used (see the guide):

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 2023).


Wolfram Research (2007), Covariance, Wolfram Language function, (updated 2023).


Wolfram Language. 2007. "Covariance." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023.


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


@misc{reference.wolfram_2024_covariance, author="Wolfram Research", title="{Covariance}", year="2023", howpublished="\url{}", note=[Accessed: 15-June-2024 ]}


@online{reference.wolfram_2024_covariance, organization={Wolfram Research}, title={Covariance}, year={2023}, url={}, note=[Accessed: 15-June-2024 ]}