October 22, 1915] 



SCIENCE 



577 



Hence, in view of (5) and (6), the mean value 

 of X will be 



(7) a;o' = a;„ + 1,000 p a S. 



Similarly the mean value of y after the in- 

 crease in the average value of the e's involved 

 in X, the e's involved in y but not in x remain- 

 ing constant, will be 



(8) 3/„' = 2/„+ 900 paS. 



Therefore, since the standard deviations of x 

 and y, s^ and Sy, are equal, 

 yo — 2/0 / gp' 



Sy I i 



(9) 



= 0.9. 



It is apparent from (9) that in this instance 

 a certain increase in the average ability x 

 will be accompanied by an increase almost as 

 great in the average ability y. 



If r is to be considered in all cases a reliable 

 measure of the closeness of relationship be- 

 tween two fields of mental activity, it ought to 

 be approximately equal to the ratio in (9). 

 Let us see what its value actually is. Making 

 use of equations (4), (5) and (6), we get 



^ 900pa=' 



(10) ~ •>/(900po2+10,000po2)(10,000po='+900po2) 



= 0.08 approximately. 



We have dealt here with a special case, but 

 it is easy to see from the above discussion that 

 in many other cases we would have discrep- 

 ancies of the same sort. Hence it is apparent 

 that it is not safe to assume off-hand that r is 

 always the best measure of the relationship 

 between two fields of mental activity. It may 

 be a very poor measure of the form of rela- 

 tionship in which we are interested.^ 



The question naturally arises, under what 

 conditions will r be a good approximation to 



■f We have restricted ourselves in the foregoing 

 discussion to the case of relationship between dif- 

 ferent fields of mental activity. The mathematical 

 part of the discussion, however, will undoubtedly 

 have a bearing on many applications of the theory 

 of correlation. If for any two variables x and y, 

 the o's of equation (3) satisfy the conditions of 

 our special case, the ratio of the common factors 

 involved in the variation of x and y to all the fac- 

 tors, will, for each variable, be 0.9. Hence r, which 

 is given toy (10), will not be a good measure of the 

 closeness of relationship between the two variable 

 quantities. 



the value of the ratio in (9) ? It is the pur- 

 pose of the rest of this paper to obtain certain 

 sufiicient conditions that this will be the case. 

 It is very easy to see that if all the a's of equa- 

 tion (3) which are not zero are equal to each 

 other in absolute value, and furthermore if 

 the standard deviations of the e's are all equal 

 to each other, r will be exactly equal to the 

 ratio in (9). This leads one to suspect that 

 if these conditions are fulfilled to a sufficient 

 degree of approximation, r will not differ very 

 much from this ratio. 



In discussing the general case there are 

 really two ratios of the type (9) to be con- 

 sidered, according as the training has been in 

 the field corresponding to x or in the field 

 corresponding to y. In the special case dis- 

 cussed above these two ratios were identical, 

 so we only considered one of them. Under the 

 hypotheses we shall make in what foUows, the 

 discussion for one ratio is practically the 

 same as the discussion for the other, so here 

 too we shall only consider one of them. 



We will investigate first the case where all 

 the a's on the right-hand side of the equations 

 in (3) are positive or zero. It is apparent that 

 there is no loss of generality in supposing that 

 the a's which are zero in the first equation are 

 the a's of the first p terms and the a's which 

 are zero in the second equation are the a's of 

 the last q terms. In particular p, or q, or both 

 of them, might be zero. 



Since the standard deviations of the e's in- 

 volved in x are no longer necessarily equal to 

 each other, a uniform distribution over these 

 e's of an increase in x would result in an in- 

 crease in each e proportional to its standard 

 deviation. Let us suppose then that after 

 training in the field corresponding to x the 

 mean value of each e^ involved in x has been 

 increased by an amount s„8. Eepresenting as 

 before by a;/ the mean value of x after the in- 

 crease in the e's we have 



(11) 



■0=3)+! 



Similarly, if «/„' represents the mean value of y 

 after the increase in the e's, we have 



(12) 



v=m-q 

 ■ 1/0 = 2 a2.s„5 



