SUMMARY Sorne nearest neighbour test procedures, assuming a null hypothesis of a Poisson process in an infinite plane, are shown to be inapplicable when a complete map of individuals is available. Two statistics, the squared coefficient of variation of squared nearest neighbour distances, and the ratio of the geometric mean to the arithmetic mean of the squared dis- tances, are particularly appropriate to testing for local regularity in this situation. Two methods of carrying out the test, the first based on coinputer simulation and the second an approximation not requiring simulation, are presented. Additionally, indices of local regularity are suggested. Existing tests of randomness, based on distance methods, of individuals in a plane are primarily intended for use with large populations. The null hypothesis is that their observed distribution is a realization of a two-dimensional Poisson point process, infinite in extent. For convenience we shall refer to this starting point, and the results which can be derived from it, as infinite plane theory. Tests have been based on distances to nearest neighbours of randomly chosen individuals (Skellam, 1952), distances to first and second nearest neigh- bours of randomly chosen points (Holgate, 1965), and a combination of individual to nearest individual and point to nearest individual distances (Hopkins, 1954). More recently Besag & Gleaves (1973) have proposed tests based on the T-square method of sampling. In some studies, for example of territorial behaviour, a complete map of individual locations in a relatively small population within a given boundary is available. This situation is somewhat different from that outlined above: the null hypothesis of interest then is that individual positions are a realization of a process locating them independently and at random within the given boundary. This paper presents a method for testing this null hypothesis which is designed to detect regularity of spacing of individuals on a small scale, irrespective of the global pattern. Pielou (1974, p. 155) discussed a number of ecological mechanisms which might cause such local regularity and also gave a method, based on infinite plane theory, to test for it. The method presented in this paper utilizes individual to individual nearest neighbour distances, and is based on computer simulation to overcome the problems created by the presence of a boundary and by the lack of statistical independence of the nearest neighbour distances. Two statistics are suggested as being particularly appropriate to the detection of local regularity. Their moments under infinite plane theory are derived and methods of approximation to their sampling distribution discussed. An approximate method is presented which would provide a satisfactory test of randomness in itself for some