Suppose we have a finite number N of observed frequencies [a(i)], and suppose further that we have a ‘reasonable belief’ that the (truncated) Poisson distribution could be used to describe those frequencies. The parameter μ (which represents both the mean and the variance of the entire unit Poisson distribution) and a suitable scaling-factor H would, in some way, be estimated from the data. In the normal course of events, the chi-squared test would probably be used to test for ‘goodness of fit’, and the number of ‘degrees of freedom’ would reflect the extent to which μ and H had been directly estimated from the data. Depending upon the degree of significance considered appropriate, the test would be used to decide whether or not a ‘reasonable fit’ had been obtained, and whether or not a particular formally-defined hypothesis should be accepted or rejected.