General formulae for the wave aberration weighted variance are obtained for rotationally symmetric systems with a non-uniform transmission pupil expressed by polynomials. They are valid for any combination of residual aberrations and any polynomial transmission function on the pupil. Expressions to obtain the position of the diffraction focus and the Strehl ratio are given. The accuracy in the evaluation of the diffraction focus is studied for optical systems with different residual aberrations. © 1991 Taylor & Francis Ltd.