A random sample is drawn from a population of animals of various species. (The theory may also be applied to studies of literary vocabulary, for example.) If a particular species is represented r times in the sample of size N, then r / N is not a good estimate of the population frequency, p, when r is small. Methods are given for estimating p, assuming virtually nothing about the underlying population. The estimates are expressed in terms of smoothed values of the numbers n, (r= 1, 2, 3, ...), where n, is the number of distinct species that are each represented r times in the sample. (n, may be described as 'the frequency of the frequency r'.) Turing is acknowledged for the most interesting formula in this part of the work. An estimate of the proportion of the population represented by the species occurring in the sample is an immediate corollary. Estimates are made of measures of heterogeneity of the population, including Yule's 'characteristic'and Shannon's 'entropy '. Methods are then discussed that do depend on assumptions about the underlying population. I t is here that most work has been done by other writers. I t is pointed out that a hypothesis can give a good fit to the numbers r~, but can give quite the wrong value for Yule's characteristic. An example of this is Fisher's fit to some data of Williams's on Macrolepidoptera.

Summary This paper deals first with the relationship between the theory of probability and the theory of rational behaviour. A method is then suggested for encouraging people to make accurate probability estimates, a connection with the theory of imformation being mentioned. Finally Wald's theory of statistical decision functions is summarised and generalised and its relation to the theory of rational behaviour is discussed.

From replicate trials of experimental gingivitis in four periodontally healthy subjects, 166 bacterial species and subspecies were detected among 3,034 randomly selected isolates from 96 samples. Of these bacteria, Actinomyces naeslundii (serotype III and phenotypically similar strains that were unreactive with available antisera), Actinomyces odontolyticus (serotype I and phenotypically similar strains that were unreactive with available antisera), Fusobacterium nucleatum, Lactobacillus species D-2, Streptococcus anginosus, Veillonella parvula, and Treponema species A appeared to be the most likely etiological agents of gingivitis. Statistical interpretations indicated that the greatest source of microbiological variation of the total flora observed was person-to-person differences in the floras. The next greatest source of variation was the inflammatory status of the sample sites. Person-to-person differences were smallest at experimental day 4. The floras became more diverse with time and as gingivitis developed and progressed. Analyses indicated that sequential colonization by certain species was repeatable and therefore probably predictable. Variation was relatively small between replicate trials, between two sites on the same teeth sampled on the same day, and between the same sites sampled at the same relative time in a replicate trial.

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