32 EMPIRICAL STUDIES OF MEASUREMENT 



TABLE XIV. 

 AVERAGE DIVERGENCE OF OBTAINED FROM TRUE MEASURE OF RELATIONSHIP 1 



(Figures in parentheses give the ranks of the three methods in freedom 

 from chance error.) 



No. of No. of Pearson Median Ratio 



Trials Cases Coefficient (Double-entry) Cosine -nil 

 Series A 



10 200 .039(1) .053(2) .058(3) 



10 100 .065(2) .062(1) .101(3) 



10 50 .100 (1) .155 (3) .135 (2) 

 Series B 



5 200 .064(2) .063(1) .082(3) 



5 100 .105(3) .072(1) .075(2) 



10 50 .153(1) .192(2) .197(3) 

 Series C 



3 200 .044 (2) .072 (3) .013 (1) 



3 100 .032(1-2) .050(3) .032(1-2) 



5 50 .119 (2) .120 (3) .077 (1) 



The Advantages of Certain Other Measures. 



The Average Ratio has no advantage over the Median Ratio and 

 suffers from the disadvantage of taking an enormous amount of 

 time and being influenced so much by extreme ratios. No experi- 

 enced worker with relationships would favor its use. 



The Modal Ratio is in some respects the most important single 

 feature of the entire series of relationships, and is probably a better 

 basis of comparison between different relationships when either is 

 not normally distributed than the Pearson Coefficient or the Median 

 Ratio. The observed Modal Ratio from a small sampling diverges 

 so much from the true Modal Ratio of the total series, however, that^ 

 it can not be well used alone unless the number of ratios is 500 or 

 more : The scale should also be fine. The most probable true 

 Modal Ratio inferred from a large part of the total distribution of 



1 It is hardly worth while to compare the empirical divergences of Table 

 XIV. for the Pearson Coefficients with the divergences to be expected from the 



.7979(1 r 2 ) 

 formula A.D. true r-obtained r = - -7= , for this formula, calculated for 



' normal ' correlation, would not be expected to fit very closely any of the three 

 sets, A, B and C, or to fit C at all closely. A certain interest does attach to the 



.7979(1 r 2 ) 

 comparison from the fact that the formula A.D. , rue r - obtained r = - 



has also been proposed as the valid one. So far as my drawings go, the former 

 is surely the better. They vary from it, moreover, with a constant deviation 

 toward a larger divergence, the divergences by theory being: 



Series A Series B Series C 



.042 .053 .027 



.059 .074 .038 



.083 .105 .054 





