576 



SCIENCE 



[N. S. Vol. XLII. No. 1086 



S iji'"''?/^ 



(i+i), 



where 



{m',vn,{vi",r,i"), •••, fc'">,'!/»') 

 are n pairs of values of rii and t;/. Hence, sub- 

 stituting in (1) the values of (x — a;„) and 

 (.y — 2/0) given by (3), we obtain 



2 aUdiiSi' 



(4) 





where the s's are the standard deviations of 

 the e's. The formula (4) for r is well adapted 

 to the discussion of the connection between 

 the value of r and the relationship between 

 X and y. We 'shall use it first to show that 

 under certain conditions r will not furnish a 

 satisfactory measure of the particular form 

 of relationship in which we are interested. 



Consider, for example, the use of correlation 

 in educational investigations. A value for r is 

 computed from the performances of a group 

 of persons in two fields of mental activity, 

 such as two school subjects, and the closeness 

 of relationship between the two fields or sub- 

 jects is discussed on the basis of this value. 

 It is clear that the value of r is a good measure 

 of the tendency of the members of the groiip 

 having a given deviation from the mean ability 

 of the whole group in one field, to have an 

 average deviation of corresponding magnitude 

 from the mean ability of the whole group in the 

 other field. It is certainly useful to be able to 

 measure such a tendency, but there is some- 

 thing else which it is more useful from the 

 educational standpoint to be able to measure. 

 Suppose the average ability of the whole 

 group in one field is increased a certain 

 amount by training in that field, and this in 

 turn causes a certain increase in the average 

 ability of the whole group in the other field. 

 The ratio of this latter increase to the former, 

 when each is measured in terms of the stand- 

 ard deviation of the group in the correspond- 

 ing field, is a very important quantity in edu- 

 cational investigations; it is vital for example 

 in the discussion of such questions as disci- 

 plinary values. 



We will now proceed to show that under cer- 

 tain conditions this ratio may be much greater 

 than r. Since ability in any complicated field 

 of mental activity like a school subject may 

 be regarded as a function of a great many ele- 

 mentary abilities, the abilities x and y in two 

 subjects may be represented as in equation (2). 

 If we expand about the mean values of the e's 

 at any given time and neglect higher powers 

 than the first,^ we get equations of the type (3). 



Since ability in each of the two subjects will 

 in general depend on certain elementary abil- 

 ities not involved in the other, we shall con- 

 sider a case where certain of the a's in the first 

 equation in (3) are zero and certain of the a's 

 in the second equation are zero. Let us sup- 

 pose then that 



0,1 = ai,= ...= flip ^0, 



[ - p + 1 — — ^2j ni - p + 2 ^^^ ■ ■ • — O^m —— ^J 



(5) 



and let us suppose further that 



(6) 



Oj, J)+l = Of, p+2 = • • • = dj. m-p = O > 



(i = 1, 2), 



fll, m-p+1 = ffll. m-p+2 = • • • = aim = lOOo, 



021 = 022 = ■ • ■ = o.ip = 100a, 

 Si = S" = • • • = Sp = s, 



m = 902p. 



If by training in one subject the average 

 ability of a group of persons in that subject is 

 increased a certain amount, it is reasonable to 

 suppose that this increase has been uniformly 

 distributed in the way of corresponding in- 

 creases in each of the elementary abilities in- 

 volved in that subject. Since from (6) the 

 standard deviations of the elementary abilities 

 are all equal in the present case, a uniform 

 distribution of the increase would imply an 

 equal increase in each elementary ability. We 

 will assume then that after training in the 

 subject, the mean value of each e of which x 

 is a function is increased by a quantity 8. 

 Since the -qs occurring on the right-hand side 

 of the first equation in (3) are deviations from 

 the original means of the e's, the mean value 

 of each of them will now be 8 instead of zero. 



« In the present instance "we neglect higher pow- 

 ers on the assumption that ability in the given sub- 

 ject is approximately a weighted mean of the ele- 

 mentary abilities on which it depends. 



