No.3.j PSYCHOLOGICAL EXAMINING IN THE UNITED STATES AKMY. 629 



On the basis of the interpretations outlined above we have considered that in the main the 

 system of 16 variables might be treated as a linear one without doing serious violence to the 

 facts. But its treatment even as a linear system requires a knowledge of means, standard 

 deviations, and correlation coefficients that have been freed from the effect of incompleteness 

 of range of the tests, for without such a set of constants we could scarcely provide the most 

 probable negative scores to substitute for zero-scores. Given these constants we might obtain 

 a total score in alpha and beta for every individual that would be directly comparable with those 

 for all other individuals, because all would then be measurements upon a linear scale. We 

 should therefore be able to eliminate most if not all of the skewness of the alpha total score 

 distribution, likewise of the beta total score distribution, and consequently the skewness of the 

 regression relationship between the two. For in all but a small number of cases alpha total 

 scores could be built up within the eight tests. As we have seen, tests 1 and 2 are not seriously 

 limited, and therefore very few cases fail to score at least 1 or 2 points in one or the other of these 

 two tests, although failing in all others. 



The fact that the distributions of 13 of the variables (8 alpha tests, beta tests 4, 5, 6, and 7, 

 and Stanford-Binet mental age) do not diverge considerably from the normal form, provided 

 we assume redistribution of unmeasured or inadequately measured cases, permits us to adopt 

 a method of analysis based upon the normal correlation surface. The problem of calculating 

 the correlation coefficients, and standard deviations and means that we require is the problem 

 of obtaining these constants for a complete correlation surface in terms of the corresponding 

 constants of a truncated correlation surface. 



The equation of the normal frequency surface for two independent variables is: 



N 1 /z> 2hxy y*\ 



2 = ■ e 2(1-A»)W c*>, »,«/ 



Let us consider the truncated portion of the surface included between the two planes x = x lt 

 x = x 2 , and define a quantity, p for this portion of the surface thus : 



p "V»p'«)* (1) 



where p' n , p' 20 and p' 02 are incomplete moments corresponding to the moments used in the 

 ordinary definition of a coefficient of correlation: 



r _ Pu 



(?20 ?0 2 ) i 



After substituting 



i x t V 



<r x " a 



7 





in the double integrals giving the incomplete moments required by (1) the equations take the 

 following forms: 



N a^ 7 p r e -Wxydxdy-- n °* „ P P ' e~^xdxdy ~ n fj la P f ' e-^'ydxdy 



