lb'b* Prof. Karl Pearson on Deviations. from the 



merit as to goodness of fit will be based on the general order 

 of magnitude of the probability P, and not on slight differ- 

 ences in its value. Hence, if we reject the series as a random 

 variation from the frequency distribution determined from 

 the sample, we must also reject it as a random variation from a 

 theoretical frequency distribution differing by quantities of the 

 order of the probable errors of the constants from the sample 

 theoretical distribution. On the other hand, if we accept it 

 as a random deviation from the sample theoretical distribu- 

 tion, we may ac3ept it as a random variation from a system 

 differing by quantities of the order of the probable errors of 

 the constants from this distribution. 



Thus I think we can conclude, when we are dealing with a 

 sufficiently long series to give small probable errors to the 

 constants of the series, that : — 



(i.) If xi? be so small as to warrant us in speaking of the 

 distribution as a random variation on the frequency distribu- 

 tion determined from itself, then we may also speak of it as a 

 random sample from a general population whose theoretical 

 distribution differs only by quantities of the order of the 

 probable errors of the constants, from the distribution deduced 

 from the observed sample. 



(ii.) If p$ s be so large as to make it impossible for us to 

 regard the observed distribution as a sample from a general 

 population following the law of distribution deduced from the 

 sample itself, it will be impossible to consider it as a sample 

 from any general population following a distribution differing 

 only by quantities of the order of the probable errors of the 

 sample distribution constants from that sample distribution. 



In other words, if a curve is a good fit to a sample, to the 

 same fineness of grouping it may be used to describe other 

 samples from the same general population. If it is a bad fit, 

 then this curve cannot serve to the same fineness of grouping 

 to describe other samples from the same population. 



We thus seem in a position to determine whether a given 

 form of frequency curve will effectively describe the samples 

 drawn from a given population to a certain degree of fineness 

 of grouping. 



If it serves to this degree, it will serve for all rouoher 

 groupings, but it does not follow that it will suffice for °still 

 finer groupings. Nor again does it appear to follow that 

 if the number in the sample be largely increased the same 

 curve will still be a good fit, Roughly the % 2 's of two samples 

 appear i;o vary for the same grouping as their total contents. 

 Hence if a curve be a good fit for a large sample it will be 

 good for a small one, but the converse is not true, and a larger 



