For two variables and ,

(1) 
where denotes Standard Deviation and
is the Covariance of these two variables. For
the general case of variables and , where , 2, ..., ,

(2) 
where are elements of the Covariance Matrix. In general, a correlation gives the strength of the
relationship between variables. The variance of any quantity is alway Nonnegative by
definition, so

(3) 
From a property of Variances, the sum can be expanded

(4) 

(5) 

(6) 
Therefore,

(7) 
Similarly,

(8) 

(9) 

(10) 

(11) 
Therefore,

(12) 
so
. For a linear combination of two variables,
Examine the cases where
,

(14) 

(15) 
The Variance will be zero if
, which requires that the argument of the
Variance is a constant. Therefore, , so . If
, is either perfectly
correlated () or perfectly anticorrelated () with .
See also Covariance, Covariance Matrix, Variance
© 19969 Eric W. Weisstein
19990525