The Logarithmic Series and Its Application to Biological Problems
C. B. Williams · Journal of Ecology · 1947
In the course of biological investigations of a numerical character the observer frequently obtains values which can be arranged in a discontinuous series of the type known as a 'frequency series'. For example, there may be a random collection of a number of insects which have been classified into species. In this case the number of species with one individual, with two, with three individuals, and so on, would form a frequency series. Alternatively, a collection might be made of a number of rats and on each the number of fleas counted. Then the number of rats with one flea, with two fleas, with three and so on, would again form a frequency series. Many other examples could be given, but most can be put under the general type of units classified into groups, or groups divided into units, which form a series of the numbers of groups with one, two or three, etc., units. It is with one of the possible mathematical interpretations of such data-the logarithmic series-that we are concerned here. It was first suggested for biological problems by R. A. Fisher in 1943 (1), and a certain amount of information has already been published during the war; but owing to paper restrictions only a few reprints were obtained and the supply of them is already exhausted. Dr R. A. Fisher has generously allowed me to quote freely from his contribution, so that all the relevant information can be collected together here. Before discussing the mathematical properties of the series it must be pointed out that the data under consideration must be a randomized sample with no selection that would affect the size of the groups, or the number of groups of any one size. For example, a museum collection of butterflies in which every effort had been made to obtain many specimens of the 'rare' species, would not be suitable for consideration. It is also important to understand that the original randomization of the sample may occur in two different ways, (1) by units and (2) by groups, as shown by the two examples given above.