.■tPPLIC.IIlOX or STATISTICAI. METIIOnS 69 



I'ciiiatioii of Pearson breaks down to the normal law and the second 

 a(iproxiniation. These, of course, can be explaineti as previously. 

 The fundamental equation, however, serves to cover the condition 

 where the causes are airrelated. Thus, because of the lack of a 

 clear conception of the physical significance of the observed varia- 

 tions in the type of cur\'es indicatcxl in Table II, it was not possible 

 easily to set up experiments to fiiul the causes of these variations. 

 For this reast>n preference has been given to the use of frequency 

 distributions derivetl upon a less empirical basis following the original 

 lines laid ilown by I.a Place, Edgeworih, Kapteyn, and others previ- 

 ously referred to. Another \ery practical reason for choosing the 

 latter type of cur\e is that it involves for the most part the use of 

 only the first three moments of the distribution instead of the first 

 four required for differentiating between the Pearson types. In 

 those cases where the interest is less of physical interpretation than 

 of graduating an observed set of data, preference may go to the more 

 generalized system of Pearson. 



How Can We Choose the Best Theoretical Frequency 

 Distribution? 



We have already briefly reviewed some of the different methods 

 for obtaining a theoretical frequency distribution from a consider- 

 ation of the moments of the observed frequencies. We have seen in 

 Table III that by using different methods we obtain different degrees 

 of approximation to the hypothetically observed distribution which 

 in this case corresponds to the terms of the binomial expansion 

 1000(.I-|-.9)'°''. Similarly from Fig. 5 it is seen that the Gram- 

 Charlier series is a much closer approximation to the observed dis- 

 tribution than that derived upon the assumption of the normal law. 

 In any given case we are naturally confronted with the question: 

 What is the best theoretical distribution? We shall consider four 

 methods for obtaining an answer. 



The oldest, simplest, and in many instances the mf)St practical, 

 is that of comparing graphically or in tabular form the theoretical 

 distribution with the one observed. This method is, however, 

 inaccurate and (jualitative. It does not furnish us with a quantitative 

 method of measuring the closeness of fit between theory and practice, 

 and in certain instances it is absolutely misleading. It is of interest 

 to see how all of these things can be truly said of one and the same 

 method. The first two characteristics, that is, oldest and simplest, 

 are perhaps readily granted. It remains to be pointed out more 



