164 Prof. Karl Pearson an Deviations from the 



investigation are pretty clear. They are : — 

 (i.) Find % from Equation (xv.) : 



(ii.) If the number of errors, n' = n+l, be odd, find the 

 improbability of the system observed from 



P = <?-§x 2 fl + ^ + -2£- -4 ^ — + ... 4- 2L . V 



V2 2. 4^2. 4. (j^2. 4. 6.. n'-3/ 



If the number of errors, n' = n -fl, be even, find the prob- 

 ability of the system observed from 



p - v/ij x Vfey * 

 + \/! e "Ki + us + nfe + • • • + ixra) 



(iii.) If n be less than 13, then the Table at the end of this 

 paper will often enable us to determine the general 

 probability or improbability of the observed system 

 without using these values for P at all. 



(5). Hitherto we have been considering cases in which 

 the theoretical probability is known a priori. But in a great 

 many cases this is not the fact ; the theoretical distribution 

 has to be judged from the sample itself. The question we 

 wish to determine is whether the sample may be reasonably 

 considered to represent a random system of deviations from 

 the theoretical frequency distribution of the general popula- 

 tion, but this distribution has to be inferred from the sample 

 itself. Let us look at this somewhat more closely. If we 

 have a fairly numerous series, and assume it to be really a 

 random sample, then the theoretical number m for the whole 

 population falling into any group and the theoretical 

 number m g as deduced from the data for the sample will only 

 differ by terms of the order of the probable errors of the 

 constants of the sample, and these probable errors will be 

 small, as the sample is supposed to be fairly large. We may 

 accordingly take : 



m = m. g + fi, 



where the ratio of fx to m s will, as a rule, be small. It is 

 only at the " tails " that fijm s may become more appreciable, 

 but here the errors or deviations will be few or small *. 



* A theoretical probability curve without limited range will never at 

 the extreme tails exactly fit observation. The difficulty is obvious where 

 the observations go by units and the theory by fractions. We ought to 

 take our final theoretical groups to cover as much of the tail area as 

 amounts to at least a unit of frequency in such cases. 



