16 Fraternal and Parental Correlation Coefficients 
(c) The Product Moment, Hn^^,,'., ofU.,,, and a'-\ 
For g'-' we obtain, from the formulae (4), (3) and (12), which concern a\ by 
substituting x for y and putting q equal to 1 : 
n — \ 2 
: (^1^2) (30), 
1 „ 2(n-l) , 
and ct\; = ' s * (32). 
By multiplication of (21) and (30) we get 
n^q . a'-^ = (n -lyl {x,y,) t (.xr) + 2S {x,y,) S (x,x,) + 7, , 
for which the mean value by application of (27) is found to be 
71' n.,.y . a'- = (n^- — 1) ?'^s'''s. 
From (22) and (31) follows 
li- Thy . 0-'- = (n -ly- Tps'^s, 
so that 
^u,r,,.a'' = n^,, . a'" - U^if . o-'-' = ^^^^—-^^rps'^s (33). 
(d) The Product Moment, U^,^^.,, of a- and a"i 
For the product a'-a'- we find by multiplying (4) and (30) : 
^1^5 VV^ = [nq - 1) {n -1)2 t {y{') -2{n-l)% (x,^) 2 (y,y^ 
+ 4>S {os.os,} S (jj,y,) + y,. 
The mean value of 73 is zero, and therefore by taking the mean value and 
using (27) we get 
n'q <tV- = (n - 1) {nq - 1 + 2g?-/ - {q -l)r] sV, 
and when from this is subtracted 
n-q a" a'- — (n — 1) [nq — l—{q — l) r] s'"s^, 
we arrive at 
2 (n - 1) 
n^-2„'2 = ^ rj/s'-s- (34). 
(e) The Standard Deviation of the Parental Correlation Coefficient. 
For the logarithm of the parental correlation coefficient calculated from our 
sample we have 
lf*g Pp = log n.,y - "I log or- - -I log (r'\ 
For great values of n, which allow us to treat the deviations of a'-, a" and U^y 
from their mean values as differentials, it follows from the above equation by 
differentiation that 
2 ' — " 2 
Pp H^y a- cr'^ 
