# The application of the logarithmic series to the frequency of occurrence of plant species in quadrats

A - Papers appearing in refereed journals

Williams, C. B. 1950. The application of the logarithmic series to the frequency of occurrence of plant species in quadrats. Journal of Ecology. 38 (1), pp. 107-138. https://doi.org/10.2307/2256527

Authors Williams, C. B. The object of the present paper is to show the possible application of certain statistical methods, which have already been found useful in the study of the structure of animal populations, to the structure of plant populations. The main difference between the two problems is that with animals it is easy to define an individual, but often difficult to say to which particular area it belongs; whereas with the plants it is often difficult to define an individual but easy to say where it is growing. To overcome difficulties arising from this a `plant-unit' is considered which, when not an easily defined individual plant, is a quantity of a species which in distribution behaves as an individual. It takes into consideration a certain amount of aggregation. The number of units in a series of samples from the same association is considered to be proportional to the area sampled. Section II (p. 108) discusses the increase of the number of species represented as we increase the number of quadrats, and a formula is presented, based on the difference between the number of species in different sized samples (i.e. different numbers of quadrats), and the differences between these differences, from which it is possible to get values for the numbers of species found on any 1 particular quadrat only, or on any 2, or 3, etc., particular quadrats. From this it is possible to calculate the frequency distribution of species found only in 1 quadrat, in 2 quadrats or in 3 quadrats, and so on, out of any number of quadrats, if the total number of species is known for each additional quadrat added. In section III it is shown that if the plant units in the population are distributed among the species in a logarithmic series, and if the quadrats are true random samples of a population distributed by chance; and further if we know: (1) the size of the quadrat as measured either by the number of species or the number of units, and (2) the Index of Diversity of the population, we can calculate the theoretical frequency distribution of species in any number of quadrats out of any total number. The mathematical formulae are given and Table 2 shows actual results for up to 25 quadrats, for certain values of the ratio between N and α or S and α. Section IV is a discussion of the distribution of species in certain percentages of the total number of quadrats. Botanists have divided the species in an association into five groups: (1) those which occur in 1-20% of the quadrats, (2) in 21-40%, (3) on 41-60%, (4) on 61-80%, and (5) the common species which occur in 81-100% of the quadrats. Theoretical considerations, based on the assumption of distribution in a logarithmic series, indicate that the result of any such classification depends on the number of quadrats, on the size of the quadrats, and on the Index of Diversity of the population. Increase in the size of the quadrat increases the proportion of species in group V, and particularly those found in all of the quadrats. Increase in the number of quadrats increases the proportion of species found in a few only (group I) and reduces those in group V. No deductions can be made from the distribution of species in these groups unless all three factors are taken into consideration. In section V there is a discussion of the sources of error in the field that might be expected to make observed values differ from the theoretical. The most important of these is lack of uniformity in the environment, which will reduce the numbers of species found on a very high proportion of quadrats. In section VI observations by Jaccard on the distribution of 92 species of plants in 52 quadrats on an Alpine Valley is analysed, and the results are shown to fit moderately well to the calculated figures. The differences found between the two are in the direction that would be expected from the known sources of error. The values obtained for the Index of Diversity are quite consistent. In section VII observations in New South Wales by Pidgeon & Ashby, which include the `number of individuals' of each species, are s own to fit moderately well to the distribution calculated from the logarithmic series. A point of great interest is that Pidgeon & Ashby had developed empirically a formula relating the number of quadrats and the number of species, and this formula is identical with that deduced from the logarithmic series provided that the number of plant units is large, i.e. the quadrats are large in size or in numbers or both. Section VIII is a discussion of various problems arising, particularly pointing out how it is impossible to avoid the conception of some Index of Diversity as a measure of the richness of the Flora, and how many of the difficulties of botanists in the past in the analysis of quadrat results have been due to neglecting this property of a population. 1950 Journal of Ecology 38 (1), pp. 107-138 https://doi.org/10.2307/2256527 Published as green open access 2256527.pdf 00220477 Wiley