No. 3.] PSYCHOLOGICAL EXAMINING IN THE UNITED STATES AEMY. 651 



In addition to the foregoing equations, which are of practical value in calculating scores in 

 the alpha examination which are interpretable as points upon a scale having nearly linear 

 relationship with "intelligence" or the ability measured, we present the following equations of 

 mainly theoretical interest. These equations connect each alpha test with all beta tests, and 

 are given in the second of the two forms in which the equations connecting various alpha tests 

 with each other are given, i. e., each variable, a,, 6,, b 2 , b 3 , etc., is a deviation from the mean 

 divided by the standard deviation. By means of the substitutions 



O'Al CBi 



where A i; #,, etc., are scores in points, and J/ A ,, M-r x , are corresponding means, scores in alpha 

 tests might be predicted from scores in beta tests (provided none of these is zero) with a 

 degree of accuracy indicated by the coefficients of maximum correlation, #, presented with 

 each equation. 



a,= 0.0069 6, + 0. 1960 6 2 + 0.1259 & 3 + 0.2624 & 4 + 0.0582 & 5 + 0.0706 & 6 + 0. 1427 b 7 (# = 0.7109) 



a 2 = 0.0231 & 1 + 0.1894 & 2 + 0.2022 & 3 + 0.2546 & 4 + 0. 1720 & 5 + 0.0626 & 6 + 0.0097 & 7 (# = 0.7628) 



Og= -0.0566 &,- 0.0316 b 2 - 0.0021 & 3 + 0.41S5 b t + 0.1159 & 5 + 0.1027 & 6 + 0.2727 b 7 (# = 0.7299) 



o,= 0.0369 &j + 0.0501 & 2 + 0.0907 & 3 + 0.3377 6 4 + 0.0283 & 5 + 0.1361 & 6 + 0.1577 5 7 (# = 0.6964) 



a B = 0.0191 &, + 0.1579 b 2 + 0.0503 b 3 + 0.2822 b t + 0.1478 & 5 + 0.13S2 b e + 0.0469 b 7 (# = 0.6985) 



a 6 = 0.0100 &, + 0.2106& 2 + 0.1286& 3 +0.1085& 4 + 0.1927& 6 + 0.0223& 6 + 0.2213& 7 (# = 0.7315) 



o,= 0.0705 6,-0.1144 b 2 + 0.1606 b 3 + 0.3357 b 4 + 0.0862 & 5 + 0.0252 \ + 0.2368 b 7 (# = 0.6889) 



Og= -0.0674 6,-0.0400 & 2 + 0.1278 & 3 + 0.3365 & 4 + 0.0699 & 5 + 0.2454 & 6 + 0.1537 & 7 (# = 0.7261) 



Comparison of the above #'s with the correlation of each alpha test with beta total score 

 (see Table 155) indicates two things: (1) That the method of calculating correlation coefficients 

 from truncated frequency surfaces made use of in this chapter has been practically a valid one, 

 since we obtain consistent series of correlations of variables with weighted and with unweighted 

 sums of other variables, and (2) that weight of scores in beta is of no practical value in obtaining 

 scores maximally comparable with alpha scores. Hence our original use of the sum of unweighted 

 scores in all tests as the definition of the combined scale for the measurement of intelligence is 

 not in error for all practical purposes in so far as the internal evidence of the system is relevant. 



In a footnote at the beginning of this section reference was made to the fact that the partial 

 coefficients of correlation could be obtained by multiplying each term in the regression equation 



Vp x f# x # x 



■gi- 1 provided the equation is left in the form — = - 1 -jr 1 — + w 5 — + • ■ • etc 



It has been shown by Prof. Pearson * that the partial correlation between any two vari- 

 ables, Xi and Xj, of a system of n correlated variables, x lt x 2 , x s , • • • x n , with all the others kept 

 constant is given by : 



— #ij 



This may be written: 



V#ii #j; 



_ _ #jj V#)j 



thus exhibiting the relation between the coefficients in a multiple regression equation and the 

 corresponding partial correlation coefficients. 



7 12.34 • ■ ■ n- 



_R J2 / #„ 



~#12 

 V#TlV#22 



Pearson, K: Phil. Trans. Koy. Soc., Series A, vol. 200, pp. 1 S. 



