A generalization of Poisson's binomial limit for use in ecology.

1. It is of interest to field ecologists to estimate the abundance of a given species in a commonwealth of plants. The method usually employed for this purpose is that of sampling by quadrat. A square lattice-the quadrat-is dropped at random points in the commonwealth, and the number of plants of the given species found in the quadrat is counted. It is thus possible, with repeated sampling, to~form a frequency distribution of the number of quadrats containing k plants (k = 0, 1, 2, ...), and the mean of this distribution gives an estimate of the density of the species, that is, the frequency with which, on the average, the species occurs in the commonwealth. In describing such observations mathematically, it has been customary to assume that the individual plant has no area, and further, that they are distributed randomly within the commonwealth studied. Provided the quadrat is large compared with the individual plant, this first assumption is justifiable, but the second, that of randomness, is recognized by the plant ecologist to be far removed from reality in the case of many species. Archibald (1948) has collected material and analysed it to show that for a number of species the hypothesis of randomness will not hold owing to the tendency of the plants to cluster together. We put forward here a series which will allow for this clustering, and which will also enable us to obtain an estimate of it. 2. It is a characteristic of the observational series collected from quadrat sampling that the variance is greater than the mean, a result which is attributable to the clustering of the observations. Were the plants distributed at random, it might be expected that Poisson's binomial limit would describe the distribution-as, in fact, it does for some species-so a generalization of Poisson suggests itself as appropriate for species in which the clustering affects the variance. Archibald (1948) fitted Neyman's (1939) contagious series to such observations, the parameters m1 and m2 of the distribution being taken as proportional respectively to the number of clusters in the data and the average number of plants per cluster. Another generalization arises from the following set-up. We assume an area over which a number of points is distributed at random. With each of these points a random number of other points is associated. The area is now divided into squares, and we calculate the probabilities that a square contains 0, 1, 2, ... points. Thus if x is the random variable associated with the first distribution of points, we write