  A class of discrete probability distributions contains distributions with
limited support. A typical example is some variant of a Likert scale, with
response mapped to either the $\{1, 2, \ldots, 5\}$ or $\{-3, -2, \ldots, 2,
3\}$ set. An interesting subclass of discrete distributions with finite support
are distributions limited to two parameters and having no more than one change
in probability monotonicity. The main contribution of this paper is to propose
a family of distributions fitting the above description, which we call the
Generalised Score Distribution (GSD) class. The proposed GSD class covers the
whole set of possible mean and variances, for any fixed and finite support.
Furthermore, the GSD class can be treated as an underdispersed continuation of
a reparametrized beta-binomial distribution. The GSD class parameters are
intuitive and can be easily estimated by the method of moments. We also offer a
Maximum Likelihood Estimation (MLE) algorithm for the GSD class and evidence
that the class properly describes response distributions coming from 24
Multimedia Quality Assessment experiments. At last, we show that the GSD class
can be represented as a sum of dichotomous zero-one random variables, which
points to an interesting interpretation of the class.
