636 



MEMOIRS NATIONAL ACADEMY OF SCIENCES. 



[Vol. XV, 



alpha only, or beta only, or individual examination only, we might calculate the most probable 

 scores on the combined scale from the points actually earned in the examination taken. Such 

 procedure would require a great variety of regression equations to fit all of the combinations 

 of scores that are to be found, and would prove an interminable task if attempted for the 

 principal sample. 



A different procedure is therefore necessitated. The principal sample is already tabulated 

 according to total scores in the various forms of examination and combinations of them; hence, 

 the problem is one of combining distributions — i. e., one of manipulating masses instead of 

 individuals. Since the only data of the principal sample are those given by distribution of 

 total scores, in which scores in component tests are entirely submerged, it might appear that, 

 so far as the principal sample is concerned, we have gained nothing by our analysis of the 

 experimental group. But we have seen how the limitations of range of the component tests 

 of alpha and beta effect the distribution of total scores, and that individuals in the lowest 

 class intervals do not score in general in all the tests, but in different combinations of them, 

 depending upon their ability. Now if it is possible to analyze total scores typical of each 

 class interval into their typical components we shall still be able to make use of our linear 

 system. 



For normal correlation * the frequency "surface" for n variables is: 



N 



Z = 



where R is the determinant 

 1 



R I a D 3 tTatTa ' 



(2tt)~(t 1 <t 2 <t 3 



1 



I so 



<W#o 



and R pq is the minor with its proper sign of the element in the pth row and qth column of this 

 determinant. We wish to determine the typical values of the variables x„ x 2 , x 3 , x t ■ • • x n 

 for a given value of their sum. This typical set of values will be that which gives the maximum 

 frequency for the given sum, or, what amounts to the same thing, makes 



R\ Op 2 a p u q I 



a minimum, after substituting for one of the variables, say, x u its value when the sum of all 

 the quantities x is fixed, and equal to d: 



Differentiation with respect to each variable in turn, after making this substitution, gives 

 7i-l simultaneous linear equations, of which the following is typical : 



R u i?,i R l2 , -B 2 i]_ , \R U i?,i R13 . R 3 i 



(T^i ffjff, CTjCT! 



O'l '! "^1 



+ 



+ 



i? n R ti R in t i? ni | 



<7j cTjO - ] g-,<r n <r n d\\ \a 



a 3 Oi 



R u 



x 3 + 





d. 



1 We are not assuming a normal system of variates. We could readily determine by precise tests whether our data represent a normal system. 

 As a matter of fact, they do not. But we do make the assumption that our system approximates to the normal sufficiently to permit us to treat 

 it as such without introduction of serious error. 



