The Covariance( obj1, obj2 ) command returns the covariance between the quantities-with-error obj1 and obj2.

Either of the quantities-with-error obj1 and obj2 can have functional dependence on other quantities-with-error.

If neither of the quantities-with-error obj1 and obj2 has functional dependence on other quantities-with-error, the correlation between obj1 and obj2 is accessed and converted to the covariance.

The relationship between the correlation  and covariance  is

where  and  are the errors in  and , respectively.

If either of the quantities-with-error obj1 and obj2 has functional dependence on other quantities-with-error, the covariance is calculated using the usual formula of error analysis involving a first-order expansion with the dependent forms and covariances between the other quantities-with-error. This process can be recursive.

The covariance  between  and , where  depends on the , and  depends on the , is

where  is the covariance between  and , and the partials are evaluated at the central values of the  and .

Covariances involving physical constants are calculated naturally and correctly in the implied system of units because central values and errors are obtained from the interface to ScientificConstants.

Unusual cases are possible involving the covariance between the same physical constant in different systems of units, but correct results are obtained. In the case of a nonderived constant, the identical identifiers obtained from the interface to ScientificConstants cause Covariance to obtain the uncertainty from both objects. In the case of a derived constant, the general double summation of the error analysis formula is evaluated as usual (over the same functional form).