General FAQ #2: Relationship between F and R-square

Question:

How can I express R-square in terms of F?

Answer:

R^2 = df1*F / (df1*F + df2) where F is distributed as F(df1,df2).

To see this, let SST be the total (corrected) sum of squares, let SSR be the sum of squares from the regression model (which must contain df1 predictors in addition to the mean), and let the error sum of squares be SSE = SST - SSR. Then R^2 = SSR / SST and F = (SSR/df1) / (SSE/df2), and the stated relationship can be obtained with a little algebra.

Similarly, F = (df2/df1) * R^2 / (1-R^2).

If you have more questions, send e-mail to stats@ssc.utexas.edu.