18 



EMPIRICAL STUDIES OF MEASUREMENT 



Consider now the Pearson Coefficient from another point of view. 

 Let us for the present restrict relationships to those between two 

 series of the same form of distribution, and also define perfect corre- 

 lation as a relationship such that any deviation of A from its central 

 tendency will imply a deviation of B from B's central tendency 

 which shall be the same fraction of B 7 s variability that the deviation" 

 of A is of A 's variability! That is, 



A, 



3-z, etc. 



Var. of B series Var. of A series' Var. of B Var. of A' 



j- -j j t. Var. of 5 series . ,, . 



If then all values of B are divided by TT . . - , we should in 



Var. of A series 



perfect correlation find each deviation of A accompanied by an 

 identical deviation of B. The sum of the AB products would be 

 equal to the sum of the A 2 , or to the sum of the B 2 , or to V2A 2 V5B 2 . 



In the case of two series of the same form of distribution and 

 of equal variability the Pearson Coefficient formula then measures 

 the proportion which the sum of the series A^B^ A 2 B 2 , etc., is of 

 what it would be with perfect correlation as defined. 



It can be shown that without reducing B or A to equivalence in 

 variability perfect correlation as defined would give for the sum 

 of the AB products V2A 2 V2.B 2 , provided the form of distribution 

 of A is the same as that of B. 



The Pearson Coefficient measures, then, in cases where the form 

 of distribution of the two facts to be related is the same, the propor- 



tion which foe sum of the AB products is of what it wouldJae_were 



correlation ^perfect. 



There is no ambiguity as to what is measured by the median of 

 the B/A ratios. Whatever the distributions may be or the ratios, 

 the median means always a definite thing: the ratio B/A which is 

 exceeded in magnitude by as many of the ratios as it exceeds. We 

 have only to note that the median of the B/A 's and the median of 

 the A/B's are two different things and that if we are interested in 

 representing in one number both what a given A deviation implies 

 with respect to B and what a given B deviation implies with respect 

 to A, we must use both the B/A and the A/B median. 



Certain other measures deserve mention. The directly calcu- 

 lated average of all the individual relationships B/A or A/B is a 

 perfectly comprehensible measure but rather a useless one. The 

 Modal Ratio B/A or A/B is also a perfectly clear conception and, 

 in cases where it can be easily and accurately determined, a very 

 valuable one. 



The per cent, of direct or the per cent, of inverse relationships 

 i$ equally comprehensibly ami is an important, fnnctinn nf tt 

 ness of relationship. 



