SECTION III. ON THE STATISTICAL TESTS OF 



HOMOGENEITY. 



One of the main objects of our present enquiry is to investi- 

 gate the "homogeneity' of our material. For this purpose it is 

 necessary to have some precise definition of <( homogeneity." I 

 fully realise the great difficulties underlying any attempt at such 

 a definition; but in order to avoid confusion of thought I have found 

 it impossible to forego at least a working definition. I shall 

 approach the problem from a purely statistical point of view. 



<( Homogeneity " implies similarity and functional equivalency 

 among the members of a group of any class of objects. When all 

 the members are identical with respect to some definite property, 

 homogeneity is perfect with reference to that particular property. 

 This is the ideal limit of thought, but in practice it always remains a 

 mere intellectual abstraction. 



Thus in actual practice diversity is always present. But if 

 the similarity attains a certain intensity we can speak of the 

 group as being homogeneous. The actual amount of similarity 

 considered necessary to attain this intensity is of course a matter 

 of practical convenience. A group which is homogeneous for one 

 purpose may be quite heterogeneous for another. 1 



" Homogeneity ' ' thus ultimately depends on our standard of 

 discrimination. 2 If the actual difference between any two mem- 

 bers of a group is less than our unit of discrimination, we can 

 never become aware of this difference and the group will appear to 

 be homogeneous. On the other hand if the actual difference is 

 greater, heterogeneity will become evident. If our unit of dis- 

 crimination is made indefinitely small and yet no heterogeneity is 

 detected, we gradually approach identity, which is the ideal limit 

 of thought. 



The concept of " homogeneity " is thus essentially relative 

 and practical. We can never have any absolute logical criterion 

 of homogeneity. We must set up separate standards of homo- 

 geneity in each case. To this extent the definition of homo- 

 geneity is necessarily arbitrary and conventional. But having 

 once set up a standard we must rigidly adhere to it. We cannot 

 give it up in the middle of a discussion on the plea of arbitrariness. 

 The discriminant may be either qualitative or quantitative, 

 in either case it should be precise and definite. 



We can now proceed to set up tests of homogeneity for our 

 special purpose. 



' Cf. K. Pearson: "Skew Variation," Biom. Vol. 4 (1906), p. 176, 192 and 

 p. 185. 



_ e.g. In statistics, the probable error is the fundamental discriminant, 

 in Experimental Psychology the least perceptible difference is the ultimate 

 unit. 



