MEASUREMENTS OF RELATIONSHIPS 35 



shown in 4, the variabilities of all arrays of y are usually not equal 

 make this method of no great practical service except for the few 

 cases where no better method can be used. l 



Section 4 tested the hypothesis of equal variability of all arrays 

 of' y and found it true in some cases and false for others. It is some- 

 what extraordinary that Professor Pearson should in support of his 

 coefficient of variability argue that the gross variability depends 

 on the size of the mean from which the variability is measured, be- 

 ing proportioned to it, and yet not recognize that, since the means 

 of the arrays of y in positive correlation would then increase as we 

 pass from arrays related to low values of x to arrays related to high 

 values of x, the variability of one of the latter arrays should be 

 greater than that of one of the former. 



8. The Attenuation of Measurements of Relationship 



Chance inaccuramps in flip m-lonnql measures make the relation- 

 ship obtained therefrom vary toward zero from the relationship that 

 would be found with accurate measures. C. Spearman announced 

 in the American Journal of Psychology, Vol. XV., pp. 89-91, that 

 the following formulas gave the necessary correction ; 



a) 



r q , q , 



where r p , q ,= ihe mean of the correlations between each series of 



values obtained for p with each series obtained for q ; 



yy=the average correlation between one and another of 



these several independently obtained series of values 



of p; 



r gV =the same as regards q; 



and rp<,= the required real correlation between the true objective 

 values of p and q. 





where m and n = the number of independent gradings for p and q 

 respectively ; 



1 Cases, that is, where we know the variability of a related array but lack 

 the data needed for the use of the better methods. For instance, we may find 

 the variability of 100 men eminent in engineering science in early liking for 

 arithmetic to be only 30 per cent, as great as the variability of men in general 

 and so infer the amount of relationship between early liking of arithmetic and 

 engineering ability. The actual rating of a random sampling of men in both 

 early liking for arithmetic and engineering ability would be hardly possible. 



