Mr Udny Vule, Fluctuations of sampling in Mendelian Ratios. 425 
__ Fluctuations of sampling in Mendelian Ratios. By G. UDNY 
Yuts, M.A., St John’s College, University Lecturer in Statistics. 
[Read 9 March 1914. ] 
Tn any series of Mendelian experiments, the observed propor- 
tions always exhibit greater or smaller deviations from expectation. 
In the material as a whole, if it be considerable, the deviation may 
be small: but if it be broken up into small sub-groups—e.g. the 
offspring of individual matings—the fluctuations may be very large 
indeed. For example, in the series of matings of Japanese 
waltzing mice x albino hybrids inter se, cited in Table A below, the 
expectation of albinos is 25 per cent., and the average proportion 
137/555=24'68 per cent. is extremely close to this: but in the indi- 
vidual litters the percentages cover the whole range from 0 to 100 
per cent. The question whether such fluctuations have any signifi- 
cance or whether they are merely the result of “pure chance,” 
corresponding to the fluctuations that might be expected in the 
number of black balls drawn from a bag containing both black and 
white, is one that must no doubt have occurred to most workers in 
the subject. Mr Bateson and Miss Saunders, writing in the First 
Report of the Evolution Committee of the Royal Society (1902), 
were careful to emphasise that the numerical results are irregular 
and that the laws represent only an average result (pp. 10 and 
127—8). In the Second Report (1905) a long series of results 
with peas (Pisum) is tabulated and it is stated that “the possi- 
bility that departures from the expected F, ratios might not be 
purely fortuitous was a special subject of enquiry” (p. 55). The 
conclusion given is that “the ratios found in individual plants, 
62 for shape and 85 for colour, are not enough for full discussion, 
and a study of them does not, so far, suggest the presence of any 
disturbing factor” (p. 77). But this brief statement hardly seems 
to make the most of the material given in the Report. 
It is difficult in fact to understand why more use has not been 
made of well-known results in the theory of sampling in order to 
compare the fluctuations observed in Mendelian experiments with 
those to be expected if the individual families, fruits—or what- 
ever the sub-groups may be—are simply random samples of the 
whole possible material. If p be the chance of obtaining one of 
the two alternative characters (e.g. the dominant), g the chance of 
obtaining the other (e.g. the recessive), the relative frequencies of 
