The following formula is used to test whether an R² calculated in a sample is significantly different from zero. (The null hypothesis is that the population R² is zero.)

where N is the number of subjects, k is the number of predictor variables and R² is the squared multiple correlation coefficient. The F is based on k and N - k - 1 degrees of freedom.

In the example of predicting college GPA, N = 100, k = 3, and R² = .3997.

Therefore,

In addition to testing R² for significance, it is possible to test the individual regression coefficients for significance. This is done by dividing the regression coefficient by the standard error of the regression coefficient.