MEASUREMENTS OF RELATIONSHIPS 25 



6. The Presuppositions of Measures of Relationship 

 The Pearson Coefficient. 



Taken at its mere face value, - or , the Pearson 



V2x 2 VSt/ 2 Tioi^ 



Coefficient has of course no presuppositions, but if it means the 

 proportion that the 2(xy) is of what it would be with perfect corre- 

 lation it presupposes sameness of form of distribution in the two 

 series. If it means the proportion which the slope of a certain 

 straight line is of the slope of the line of perfect correlation, the 

 certain line being so drawn that the sum of the squares of the 

 divergences from it of the given y values (in double entry) toward 

 greater correlation equals the sum of the squares of those toward 

 less, it presupposes the 'normal' distribution in the case of both 

 series. 



The Median Ratio. 



The Median Ratio need have no presuppositions. It is simply 

 one of the obtained individual relationships. When, however, we 

 come to draw inferences from it about the entire series of relation- 

 ships, we must state certain additional facts or use certain presup- 

 positions. 



The Modal Ratio and the Percentage of Like-signed or of Un- 

 like-signed pairs are also directly drawn from the series of indi- 

 vidual relationships themselves. In calculating the general trend 

 of relationship, r, from r= cosine irV (U being the per cent, of un- 

 like-signed pairs) we presuppose (if I understand Mr. Sheppard cor- 

 rectly) that the correlation surface is transformable into a surface 

 of revolution by a slide and two stretches. 



7. The Advantages of the Different Measures 

 The two previous sections are preliminary to the main topic 

 which forms the title of this section. 



I shall first compare the conventional measure, the Pearson Coeffi- 

 cient, with the Median Ratio and later deal very briefly with some of 

 the other measures. 



The main desiderata in any measure are that it measure some 

 real fact and that this fact be important! Other desiderata in the 

 case of a measure of relationship are that the measure be comparable 

 with other measures of other relationships, that it be conveniently 

 and easily calculated and that it diverge little from the correspond- 

 ing measure of the total series from a random sampling of which 

 it is calculated. These desiderata we will consider in the above order. 



Reality. 



The Median Ratio is a clear statement of a real fact, an observed 



