Why R Squared Does Not Equal r Squared plus r Squared

 

Each of these circles represents the variance of each variable in the regression equation. Together, they each account for a portion of the total variance in the regression.

So, homework makes its own contribution to variance as do Parental Education and GPA.

We can chose to look at only how much of the total variance is accounted for by both GPA and Homework working together. This is their shared variance, it is accounted for by lowercase r squared, and is represented by the area of overlap here.

Alternately, we can chose to look at only how much of the total variance is accounted for by both GPA and Parental Education working together. This is their shared variance, it is accounted for by another lowercase r squared, and is represented by the area of overlap here.

If, however, we chose to look at how much total variance is accounted for by GPA, Homework , and Parental Education working together we start seeing some shared overlap. So, we start with GPA, Homework enters the Equation and creates the overlap we saw before, and then Parental Education joins the equation last to recreate it’s overlap.

We now see in this very dark area that there is overlap WITHIN the overlap. This happens because Homework and Parental Education are, themselves, correlated. What we are seeing in this dark area is the variance of Grades JOINTLY ACCOUNTED FOR by the variables.

Since when we consider total variance we would not want to count those original areas of overlap that are a portion of the combined overlap twice we need to account for that in the formula.

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