Mean and variance of \(R^ 2\) in small and moderate samples (Q1822189)

We derive and use easily computable expressions for the mean and variance of \(R^ 2\) in the standard linear regression model with fixed regressors. In respect to its probability limit \(R^ 2\) is seriously biased upward in small samples; the 'adjusted' \(\bar R{}^ 2\) does much better. But at sample sizes where these distinctions matter both measures are thoroughly unreliable because of their large dispersion. \(R^ 2\) should not be quoted for samples of less than fifty observations.