  The paper studies the correction to the distance modulus induced by
inhomogeneities and averaged over all directions from a given observer. The
inhomogeneities are modeled as mass-compensated voids in random or regular
lattices within Swiss-cheese universes. Void radii below 300 Mpc are
considered, which are supported by current redshift surveys and limited by the
recently observed imprint such voids leave on CMB. The averaging over all
directions, performed by numerical ray tracing, is non-perturbative and
includes the supernovas inside the voids. Voids aligning along a certain
direction produce a cumulative gravitational lensing correction that increases
with their number. Such corrections are destroyed by the averaging over all
directions, even in non-randomized simple cubic void lattices. At low
redshifts, the average correction is not zero but decays with the peculiar
velocities and redshift. Its upper bound is provided by the maximal average
correction which assumes no random cancelations between different voids. It is
described well by a linear perturbation formula and, for the voids considered,
is 20% of the correction corresponding to the maximal peculiar velocity. The
average correction calculated in random and simple cubic void lattices is
severely damped below the predicted maximal one after a single void diameter.
That is traced to cancellations between the corrections from the fronts and
backs of different voids. All that implies that voids cannot imitate the effect
of dark energy unless they have radii and peculiar velocities much larger than
the currently observed. The results obtained allow one to readily predict the
redshift above which the direction-averaged fluctuation in the Hubble diagram
falls below a required precision and suggest a method to extract the background
Hubble constant from low redshift data without the need to correct for peculiar
velocities.
