630 



Where 

 and 



MEMOIRS NATIONAL ACADEMY OF SCIENCES. 



[Vol. XV, 



n 



X 2 = 3? — 2nxy + y 2 



n is the portion of the total population represented by the volume of the truncated correlation 

 surface. 



Simple integration by parts reduces these double integrals in such a way that upon substi- 

 tution in (1) we obtain the following: 



P— ■ 



Where 



(1+rV)* 



J = 



h 1* Je I 2 



— i=e 2 —!==e 2 



V27T V2tt 



• i r k * 



V27rJ h 



V27T«/h 



or, in the notation of Sheppard's tables, 



e 2 



■e 2 



l/2ij^ 



1 f k _i! 



27r Jh 



n/N 1 n/JV J 



(2) 



(3) 



(4) 



Since h is the lower limit of integration, it will be negative if less than half of the total population 

 lie below it. Likewise, Je will be positive unless more than half of the population is above it. 

 Therefore J, which depends entirely upon the percentages of cases cut off at the upper and 

 lower ends of the distribution of z-variates, can be readily determined by the aid of Sheppard's 

 tables, and will always be negative. 



From (2) we now get the following equation expressing r, the correlation coefficient of the 

 complete surface, in terms of s the analogous quantity derived from the incomplete surface, and 

 of the percentages of cases beyond the stumps of the distribution of z-variates (if the surface be 

 a truncated normal surface) : 



r = , p = (5) 



Vl + U-sV 



If the distribution of the x-variate is truncated at one end only, say the lower, so that Jc is 

 infinite, we have 



T _ Jiz h _ [ g h 



J ~*(i+a) ua+«) 



The complete distribution of z-variates is given by the equation 



N 



Z = 



2.T.2 



<7 x -v/27r 



If we find the standard deviation of the truncated portion formed by cutting off the lower tail 

 at the point x u so that x 1 /cr I = 7i, we have: 



nm,= * I e z x 2 dx 



N<r x > 



V2V 



_&2 



he 2 



+ 



Jf* 00 _x 

 r 



dx\ 





nm,=—!=^. 



2 xdx 



• No 



"2 



: V2^ 



