The Equilibrium Size Distribution of Freely-Forming Groups

A few years ago (1953), John James published a paper reporting a number of observations of the size distribution of "freely-forming" small groups in various public situations. These data, some of which are reproduced in Table 1, are intriguing to the quantitatively oriented sociologist, for they seem to show the outcome of some "natural processes" by which groups acquire and lose members. Yet the data seem impermeable to explanation until one approaches them as statistical equilibria of a stochastic process of some sort. Even so, there is always the possibility that there are certain particular factors which complicate matters-as would be the case if, for example, the groups were observed on the sidewalk walking to a dance. There would, of course, be a predominance of couples, and any model based on general rates of acquisition and loss of group members would be doomed to failure. However, these data of James's do not appear at first glance to have been disturbed by dances and the like. Each size appears less frequently than the next smaller size. Since this is so, can we construct a model, involving some sort of "flow" of members in and out of groups, to account for the phenomena? Perhaps the simplest model which might explain why the distributions take the form they do is a model of acquisition and loss, with the following assumptions (which include no "contagious" assumption):