330 Method of estimating Bacterial Density 



dilutions made from a single sample, are here insisted upon, because 

 failure in either of these two points would not necessarily affect the 

 agreement between parallel platings, from the same final dilution, which 

 is studied below. 



3. The Poisson Series 



It was shown by Poisson (i) in 1837, that if a large number of indi- 

 viduals, N , are each exposed independently to a very small risk of an 

 event of which the probability of occurrence in any instance is f, then 

 the number of occurrences, x, in any trial will be distributed according 

 to a definite law, sometimes called the Law of Small Numbers. The 

 distribution of x is found to depend on a single parameter 



m = f'N , 



in such a way that the probability that the number of occurrences shall 

 be X is given by the formula 



x\ 



It should be noted that x is always a whole number, while m may be 

 fractional; the mean value of x is equal to m, and when w is large the 

 distribution, except for its essential discontinuity, resembles a normal 

 distribution, having its mean at m and the variance (the square of the 

 standard deviation) also equal to m. 



The importance of the Poisson series in modern statistics was brought 

 out by "Student "(2) in 1907^, in discussing the accuracy of counting 

 yeast cells with the haemocytometer. Since the chance of any given 

 yeast cell settling upon any given square of the haemocytometer is 

 extremely small, while the number of cells is correspondingly great, 

 " Student" arrived independently at the Poisson formula, as a theoretical 

 result under technically perfect conditions. He was able to show that, 

 in some instances, counts of 400 squares agreed with the theoretical 



1 The Poisson Series had been successfully applied by von Bortkiewicz to the annual 

 number of deaths from horse-kick in a number of Prussian Army Corps(2<?). Miss Whitaker's 

 criticism(S) of this application is entirely vitiated by her neglect of the variation of random 

 samples. 



H. Bateman (1910)(9) arrived at the formula for the Poisson Series, as the distribution 

 of the number of a particles, emitted by a film of polonium, which strike a sensitised screen 

 in successive equal intervals of time. The formula was used by Rutherford and Geiger to 

 test the independence of simultaneous emissions. The distribution of 2608 counts shows 

 a general agreement with expectation, though there are discrepancies not easily to be 

 explained by chance. The observations are certainly not adequate, as these authors suggest, 

 as "a method of testing the laws of probability." 



