# Why OLS F Statistic close to one when there is no relationship?

I might be missing something obvious here. In linear regression, F statistic is defined as (explained variance / p) / mean squared error, where p is number of independent variables. When there is no relationship between independent and dependent variables, wouldn't explained variance close to zero and thus F statistic close to zero?

More specifically, I'm referring to page 76 of "an introduction to statistical learning with applications in R".

Thank you

• Some of the explanation here and here may help Aug 2, 2015 at 7:08

I believe this question is more or less addressed here: Intuitive explanation of the F-statistic formula?. The idea is that the regression mean square is an estimate of $\sigma^2$ when the null model holds, and so both the numerator and denominator of the $F$ statistic can roughly be expected to be close to the error variance when there is no relationship between the predictors and response.
 (error variance + regression effects) / (error variance)