MEASUREMENTS OF RELATIONSHIPS 31 



chance error of it and of the Pearson Coefficient. I have also tested 

 the influence of the number of cases on the per cent, of unlike- 

 signed pairs (which I have called^ U) because at least for pre- 

 liminary investigations of mental and social relationships the 

 formula^ r = cosine -n-U (where U = the per cent, of unlike-signed 

 pairs, deviations being calculated from an exact median sectioning, 

 with no zero deviations) will often possess great advantages. 



The accuracy with which the Pearson r, the Median Ratio and 

 the cosine vU calculated from a random sampling of a series of 

 individual relationships approximate the true r, the true Median 

 Ratio, and the true cosine nil of the entire series was experimentally 

 determined in the case of the series A, B and C (shown in Tables 

 IX., XII. and XIII.). 1 These reliabilities could, I suppose, be 

 calculated by theory for any given series of relationships but it 

 seemed wise to determine them also by experiments with real cases. 

 In calculating the results for each draw of 200, 100 or of 50 cases 

 the deviations were reckoned always from the true central tend- 

 encies of the total series, not from the obtained central tendencies of 

 the draw itself. This saves much time and introduces no error 

 relevant to the problem. The Median Ratio was taken simply as 

 the observed ratio of which it was a case. That is, if the distribu- 

 tion of ratios was : 



Less than 1.00 49 



1.00 12 



over 1.00 39, 



the Median Ratio would be taken as 1.00. If one took as the Median 

 Ratio the average of this observed ratio and the ratio halfway be- 

 tween the 40 and 60 percentiles, the divergences for the Median 

 Ratio would be reduced. The results are given in Table XIV. In 

 every case the Median Ratio means the median of all the ratios 

 (y/x and x/y), the two series being reduced to an equivalence in 

 variability. 



The relationships as calculated from the entire series are: 



Series A Series B [ Series C 



Pearson Coefficient .51 .27 .73 



Median Ratio .60 .33 .83 



Cosine vU .61 .30 .79 



It is clear from Table XIV. that if A 7 is as great as 100, there is 

 no great loss in precision from the use of the Median Ratio method 

 or even of the unlike-signed pairs method. 



1 Table IX. is on page 19. 



I 



