  Context. Many databases on asteroid brightnesses (e.g. ALCDEF, WISE) are
potential sources for extensive asteroid shape and spin modelling. Individual
lightcurve inversion models require several apparitions and hundreds of data
points per target. However, we can analyse the coarse shape and spin
distributions over populations of at least thousands of targets even if there
are only a few points and one apparition per asteroid. This is done by
examining the distribution of the brightness variations observed within the
chosen population.
  Aims. Brightness variation has been proposed as a population-scale rather
than individual-target observable in two studies so far. We aim to examine this
approach rigorously to establish its theoretical validity, degree of
ill-posedness, and practical applicability.
  Methods. We model the observed brightness variation of a target population by
considering its cumulative distribution function (CDF) caused by the joint
distribution function of two fundamental shape and spin indicators. These are
the shape elongation and the spin latitude of a simple ellipsoidal model. The
main advantage of the model is that we can derive analytical basis functions
that yield the observed CDF as a function of the shape and spin distribution.
The inverse problem can be treated linearly. Even though the inaccuracy of the
model is considerable, databases of thousands of targets should yield some
information on the distribution.
  Results. We establish the theoretical soundness and the typical accuracy
limits of the approach both analytically and numerically. Using simulations, we
derive a practical estimate of the model distribution in the (shape,
spin)-plane. We show that databases such as Wide-field Infrared Survey Explorer
(WISE) yield coarse but robust estimates of this distribution, and as an
example compare various asteroid families with each other.
