APPLICATION OF STATISTICAL METHODS 67 



rosult.iiit ilistriltiition was a roiiipuuiKl of a few separate but similar 

 (list ril)iit ions about different averages. When tlie distributions of 

 the ctVuiencies of the different lots of carbon were determined separ- 

 ately they were found to be consistent with the second approximation. 



Thus, although it may be impossible to concl'ude that the a priori 

 assumptions underlying a given law of distribution are fulfilled because 

 the observations are found to be consistent therewith, nevertheless, 

 the fact that the observeil and the theoretical distributions do not 

 agree suggests the necessity of seeking for certain typical causes 

 which may l)e expected to introduce such discrepancies. This point 

 is of special importance in connection with the study of ways of 

 sampling product in order to determine whether or not the manu- 

 facturing process is subject to trends. Thus, if a product is sampled 

 at two periods, and the distributions of both groups of observations 

 are found to l)c random about ilifferent averages, it is highly probable 

 that the difTerencc indicates a trend in the manufacturing methods, 

 providing the difference between the averages is greater than 3 times 

 the standard de\iation of the average. When, however, the two 

 distributions are found to be inconsistent with a random system of 

 causes, it is quite probable that the condition of sampling has not been 

 carefully controlled. 



Ilypergeometric Series. Pearson has shown several ways in which 

 a frec]ucncy distribution may be represented by a hypergeometric 

 series. Thus the chances of getting r, r— 1, ... bad transmitters 

 from a lot containing pn bad and qn good and where r instruments 

 are drawn at a time may be represented by the terms of such a series. 

 More important, however, is Pearson's solution -' of what he calls 

 the fundamental problem of statistics. He shows, follow'ing the line 

 of reasoning similar to that originally suggested by Bayes, that if in 

 a sample of ki = (w + «) trials, an e\ent has been observed to occur m 

 times and to fail n times, in a second group of ki trials the chances 

 of the event f)ccurring r times and failing 5 times are given by the 

 successive terms of a hypergeometric series. We cannot consider 

 here the questions underlying the justification of this method of 

 solution, for, as is well-known, the application of Bayes' theorem is 

 cjuestioned by many statisticians. We can profit, however, by the 

 brrwrl experience of Prof. Pearson, for he has apparently accumulated 

 an abundance of data which are consistent with the theory. 



The answer to this problem s of special importance in connecti(jn 

 with the inspection of product which in many instances runs into 

 millions yearly. We must keep the cost of inspection at a minimum, 



"Pearson, K.— Biometrika, October, 1920— pp. 1-16. 



