Variance in the Outcome: The Black Box
Regression models engage an exercise in variance accounting. How much of the outcome is explained by the inputs, individually (slope divided by standard error is t) and collectively (Average explained/Average unexplained with averaging over degrees of freedom is F). This, of course, assumes normal errors. This document provides a function for making use of the black box. Just as in common parlance, a black box is the unexplained.
Correlations and the Impact on Sums and Differences
I will use a simple R function to illustrate the effect of correlation on sums and differences of random variables. In general, the variance [and standard deviation] of a sum of random variables is the variance of the individual variables plus twice the covariance; the variance [and standard deviation] of a difference in random variables is the variance of the individual variables minus twice the (signed) covariance.