The Covariance function computes the covariance of two data sets, or the covariance of two random variables or distributions. The CovarianceMatrix function computes the covariance matrix of multiple data sets.

The first parameter can be a data set (given as e.g. a Vector), a distribution (see Statistics[Distribution]), a random variable, or an algebraic expression involving random variables (see Statistics[RandomVariable]).

By default, all computations involving random variables are performed symbolically (see option numeric below).

All computations involving data are performed in floating-point; therefore, all data provided must have type realcons and all returned solutions are floating-point, even if the problem is specified with exact values.

For more information about computation in the Statistics package, see the Statistics[Computation] help page.

The options argument can contain one or more of the options shown below. More information for some options is available in the Statistics[DescriptiveStatistics] help page.

ignore=truefalse -- This option controls how missing data is handled by the Covariance command. Missing items are represented by undefined or Float(undefined). So, if ignore=false and A contains missing data, the Covariance command will return undefined. If ignore=true all missing items in A will be ignored. The default value is false.

weights=Vector -- Data weights. The number of elements in the weights array must be equal to the number of elements in the original data sample. By default all elements in A are assigned weight $1$.

$\mathrm{with}\left(\mathrm{Statistics}\right)\:$

$U≔⟨\mathrm{seq}\left(i\,i=57..77\right)\,\mathrm{undefined}⟩\:$

$V≔⟨\mathrm{seq}\left(\sin \left(i\right)\,i=57..77\right)\,\mathrm{undefined}⟩\:$

$W≔⟨2\,4\,14\,41\,83\,169\,394\,669\,990\,1223\,1329\,1230\,1063\,646\,392\,202\,79\,32\,16\,5\,2\,5⟩\:$

$\mathrm{Covariance}\left(U\,V\right)$

$Float\left(\mathrm{undefined}\right)$

$\mathrm{Covariance}\left(U\,V\,\mathrm{ignore}\right)$

$\mathrm{-0.226147813941922}$

$\mathrm{Covariance}\left(U\,V\,\mathrm{weights}=W\,\mathrm{ignore}=\mathrm{true}\right)$

$\mathrm{-0.167449265684222}$

$M≔\mathrm{Matrix}\left(\left[U\,V\right]\right)$

$\mathrm{CovarianceMatrix}\left(M\,\mathrm{ignore}\right)$

$\left[\begin{array}{cc}38.5000000000000& \mathrm{-0.226147813941922}\\ \mathrm{-0.226147813941922}& 0.530662127023855\end{array}\right]$

Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998. Vol. 1: Distribution Theory.