  Research evaluations and comparison of the assessments of academic
institutions (scientific areas, departments, universities etc.) are among the
major issues in recent years in higher education systems. One method, followed
by some national evaluation agencies, is to assess the research quality by the
evaluation of a limited number of publications in a way that each publication
is rated among $n$ classes. This method produces, for each institution, a
distribution of the publications in the $n$ classes. In this paper we introduce
a natural geometric way to compare these assessments by introducing an ad hoc
distance from the performance of an institution to the best possible achievable
assessment. Moreover, to avoid the methodological error of comparing
non-homogeneous institutions, we introduce a {\em geometric score} based on
such a distance. The latter represents the probability that an ideal
institution, with the same configuration as the one under evaluation, performs
worst. We apply our method, based on the geometric score, to rank, in two
specific scientific areas, the Italian universities using the results of the
evaluation exercise VQR 2011-2014.
