Ixxviii Tablf* for Statixticiniix an*l /ii>nti<fn'ri,iits 1. 1 1 



The mean will be mn,/N and tin- standard deviation A/ m f 1 "'} . 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,jN the value m,/m, where m, is the number 

 found in the th cell of the sample. If n,/N or m,/m be very small and i 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 / of cases there will be 

 a deviation in defect of 6 or more and in 9'526 / of cases a deviation in excess of 

 6 or more. Hence in 17'993 % say 18 "/ 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 6 or more from the true value. 



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

 deviation is 



Here TO is supposed large compared with 17, so that the S. D. = Vl7 = 4 - 123 

 nearly. But suppose HI = 800, we should have 



S. D. = Vl7 (1 - <)21-25) = Vl7x-9787r ) = 4-079. 



Now we want deviations in excess of 5, i.e. we must take 5 > 5/4'079 = T348. 

 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 "/ of cases. Thus the actual per- 

 centages are very close, but the Poissou series tells us that 8 P 47 / of cases will 

 be in defect and 9'53 / in 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 % f 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. 



