26 EMPIRICAL STUDIES OF MEASUREMENT 



relationship, suchthatthe number of relationships closer than it 

 e(fuals the number less close. It gives the amount of i/'s difference 



from its central tendency implied by such difference in x for this 

 mid-case. 



The Pearson r is not an observed relationship but a measure in- 

 ferred from certain features of the observed relationships on the 

 basis of certain presuppositions about them and the distribution 

 of the facts from which they come. It is of course real in the sense 

 of being the most probable real central tendency of the relationships 

 if these various presuppositions are true, but in fact they never are 

 except by chance more than approximately true, and in the majority 

 of the cases in which students of the mental and social sciences need 

 to measure relationships, they are far from true. 



The 'regression,' that is the relation between actual amounts of 

 y and actual amounts of x, is the reality at the basis of all measures 

 of the relationship. The Median Ratio expresses it directly. It can 

 be ascertained from the Pearson r only indirectly and on the hypoth- 

 esis that certain very questionable conditions are realized. 



Importance of the Fact Measured. 



There is no great advantage either way in this respect. Neither 

 the Pearson Coefficient nor the Median Ratio gives the entire fact 

 of the relationship. Only the total distribution of the relationship 

 that. For 'normal' correlation where the relationship is the 



same regardless of the amount of x and where all of the arrays 

 are distributed in normal surfaces of frequency the Pearson Coeffi- 

 cient and the Median Ratio both give the central tendency of the rela- 

 tionship. In other cases than this the Median Ratio is a trifle more 

 important because less misleading and because it is nearer the modal 

 relationship if the distribution of the relationship is skewed. 



It is also worthy of note that our thinking about relationships 

 should for practical reasons usually be in terms of the actual y/x 

 or x/y ratios, that is the 'regressions,' since what we usually need 

 to know is the implication of some actual deviation of one concern- 

 ing the related deviation of the other. It seems better then to 

 calculate the y/x or x/y ratio directly and when necessary to infer 

 the r (that is the ratio when both traits are reduced to an equivalence 

 in variability and the correlation table is one of double entry) rather 

 than to calculate the r and infer the y/x or x/y ratio. 



Comparability. 



To compare the relationship between A and B with that between 

 C and D adequately, we must compare the total distribution of the 

 relationship A B with the total distribution of the relationship 

 C D. The Pearson Coefficients of A B and C D are per- 



