Introduction. We have throughout Chapter II assumed a constant probability underlying, the frequency ratios obtained from observation. It is fairly obvious that frequency ratios are often found from material in which the underlying probability is not constant. Then the statistician should make use of all available knowledge of the material for appropriate classification into subsets for analysis and comparison. It thus becomes important to consider a set of observations which may be broken into subsets for examination and comparison as to whether the underlying probability seems to be constant from subset to subset. In the separation of a large number of relative frequencies into n subsets according to some appropriate principle of classification, it is useful to make the classification so that the theory of Lexis may be applied. In the theory of Lexis we consider three types of series or distributions characterized by the following properties:

1. The underlying probability p may remain a constant throughout the whole field of observation. Such a series is called a Bernoulli series, and has been considered in Chapter II.

2. Suppose next that the probability of an event varies from trial to trial within a set of s trials, but that the several probabilities for one set of s trials are identical to those of every other of n sets of s trials. Then the series is called a Poisson series.