Obviously, the smaller the residuals, the better the fit. One measure of the magnitude of the residuals is their standard deviation, alias the root mean square error. This standard deviation represents the amount of variation in the dependent variable, and the error represents the variation that's left over after we fit the model. So they square the standard deviation to get total variance, and they square the error to get error variance. Stats books have lots of formulae involving sums of squares, which are what we used to use to calculate statistics in the days before computers.
Sums of squares are directly related to variances.
The total sum of squares is the sum of the squares of each observed value after the mean has been subtracted from it. The residual sum of squares is exactly what it says. Subtract the residual SS from the total SS, divide by the total SS, and you have another formula for R 2 . What programs currently won't do is give you confidence limits for the R 2 . Maybe we don't really want it anyway.