Statistical Correlation
For two random variates X and Y, the correlation is defined bY
where sigma_X denotes standard deviation and cov(X,Y) is the covariance of these two variables. For the general case of variables X_i and X_j, where i,j=1, 2, ..., n,
where V_(ii) are elements of the covariance matrix. In general, a correlation gives the strength of the relationship between variables. For i=j,
The variance of any quantity is always nonnegative by definition, so
From a property of variances, the sum can be expanded
Therefore,
Similarly,
Therefore,
so -1<=cor(X,Y)<=1.
For a linear combination of two variables,
Examine the cases where cor(X,Y)=+/-1,
| var(Y-bX)=b^2sigma_X^2+sigma_Y^2∓2bsigma_Xsigma_Y=(bsigma_X∓sigma_Y)^2. |
(19)
|
The variance will be zero if b=+/-sigma_Y/sigma_X, which requires that the argument of the variance is a constant. Therefore, y-bx=a, so y=a+bx. If cor(X,Y)=+/-1, y is either perfectly correlated (b>0) or perfectly anticorrelated (b<0) with x.
See also
Covariance, Covariance Matrix, VarianceExplore with Wolfram|Alpha
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Cite this as:
Weisstein, Eric W. "Statistical Correlation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/StatisticalCorrelation.html