no. 3.] PSYCHOLOGICAL EXAMINING IN THE UNITED STATES ARMY. 



631 



s x 2 =m 2 — m l 2 



_5L 



V2ir 



&i 



e 2 dx 



-+1- 



e 



h« 

 2 



ft 



X2 



2 dx 



Therefore 



- v *\ 1 + 1(1 + «) Li(l+«)J"1 



, = 1+J 



J=^-l 



If we substitute this value of J in (2) we obtain 



Sx 



which is the formula derived by Prof. Pearson ' showing the effect of selection upon the correla- 

 tion between two measurements of the selected sample. It is evident, therefore, that the use 

 of formula (5) is merely a device for proceeding from the correlation within a selected popula- 

 tion to that for an unselected population, the standard deviation of the directly selected 

 variates, s being given by the data, and the corresponding standard deviation of the total 

 population, a, being approximately determinable by the use of Sheppard's tables, if the dis- 

 tribution of variates does not diverge widely from the normal. 



Besides the correlation coefficients for all pairs of variables, we need to know the mean 

 and standard deviation of each distribution. The following formulae are readily deducible 

 from the normal correlation surface. If t is the variable corresponding to the truncated dis- 

 tribution and n is the variable of the nontruncated distribution, we have: 



' n "Vf+rv 



■Jl+J 



~r ' * ' fft 



M a = m n -r(T n 



M t = m t -<r t 



Z h —Zk 



1 n/N 



Zh — Zj 



n/N 



where s and m are respectively the standard deviation and mean of the incomplete distribution, 

 and a and M are the corresponding moments of the hypothetically complete distribution. 

 The symbols z h , s k , and n/N have the same significance as in formula (4). With the aid of 

 these formulae we are enabled to obtain the desired constants from all but eighteen of our 

 correlation tables. Although many more than eighteen are obviously truncated with respect 

 to both scales, we are able to avoid the necessity of treating them as such by cutting the table 

 higher in one scale than its limitation alone would require. Moreover, the unreliability of 



1 Pearson, K., On the Influence of Natural Selection on the Variability and Correlation of Organs, Phil. Trans,, Series A, vol. 200, 1903, pp. Iff- 



