Ixxviii Tables for Statisticians and Biometricians [LII 



The mean will be mn s /N and the standard deviation a / m ~ ( 1 — -X\ . If 



we 



only have a single sample of m and do not know the distribution in the actual 

 population we are compelled to give n s /N the value »i s /to, where m, is the number 

 found in the sth cell of the sample. If n s jN or m s /m be very small and m large« 

 the binomial will approach Poisson's Exponential Limit, and in such cases the 

 deviations in the samples for the sth cell will be distributed very differently from 

 those following a Gaussian law, and the usual rule for deducing the probability of 

 deviations of a given size by means of the probability integral fails markedly. 

 It is not till we get something like 30 out of 1000 in a cell that we can trust the 

 Gaussian to give us at all a reasonable approach. The present table endeavours 

 to provide material in the case of cell frequencies 1 to 30, which will supply the 

 place of the probability integral. 



Illustration (i). Suppose the actual number to be expected in a cell is 17, 

 what is the probability that the observed number will deviate by more than 5 

 from this result? Looking at p. 123 we see that in 8-467 °/ Q of cases there will be 

 a deviation in defect of G or more and in 9 - 526 °/ of cases a deviation in excess of 

 6 or more. Hence in 17'993 °/ say 18 °/ o of cases we should get values less than 

 12 or greater than 22. Thus once in every 5 or 6 trials we should get values 

 which differ as widely as G or more from the true value. 



Now look at the matter from the Gaussian standpoint. The standard 

 deviation is 



y»"(>-")V"> : 



a. 



■ml 



Here m is supposed large compared with 17, so that the S. D. = V 17 = 4 - 123 

 nearly. But suppose m = 800, we should have 



S. D. = Vl7 (1 --02125) = Vl7 x -97875 = 4-079. 



Now we want deviations in excess of 5, i.e. we must take 5'5/4079 = 1348. 

 If we turn to Table II we find for this argument 



| (1 + a) = -9102 or \ (1 - a) = '0898. 



Hence we should conclude that in not more than 17"96 % of cases would deviations 

 exceed ± 5. Actually such occur in 17'99 % °f cases. Thus the actual per- 

 centages are very close, but the Poisson series tells us that 8 - 47 °/ of cases will 

 be in defect and 9'53 % m excess, while the Gaussian gives 8-98 % in both excess 

 and defect. We may further ask the percentage of times that 17 itself would 

 occur; according to the Gaussian it will occur in 9*76 % of trials, actually it will 

 occur in 9"63 °/ . With values of cell-frequency less than 17, say in the single 

 digits, far greater divergences will be encountered. 



