Summary: We compare saddlepoint approximations to the exact distributions of a Studentized mean and to its bootstrap approximation. We show that, on bounded sets, these empirical saddlepoint approximations achieve second order relative errors uniformly. We also consider the relative errors for larger deviations. It follows that the Studentized-\(t\) bootstrap \(p\)- value and the coverage of the bootstrap confidence interval have second order relative errors.