8 Fraternal and Parental Correlation Coefficients 
(e) The Product Moment, H-na^, of Yl and a'-. 
By multiplication of (4) and (15) and taking mean value for a great number of 
samples we find for the mean value of the product lia- 
(q - 1) Ua"- - - (nq -l){q- 1) (2 (2/f))= - 4(ng ~q + l)(^ {y,y,)y 
+ 2 [n^ - nq^ + 2{q- 1)] 2 (y,^) t {y,y.^ + 4 (g - 1) (^ {y,y,)f, 
the mean values of the two products being zero according to (11). Introducing the 
rest of the mean values from (11), we have 
nY = {- nq-l + r [n-cf - nq (g - 4) - 2 (^ - 1 ;] + r" [nq {q - 3) -(q- I)-]}. 
From (3) and (14) is found 
nY-^-^' = s' {-nq+l + r [n-q- - nq^ + ^ (q - 'i)] + [-"?(? " 1) + (? - 1)']!- 
As Un.2=Ua'-U .a\ 
it follows from the two foregoing e(|uations that 
nn.. = |~ (- 1 + 2r [nq -(q-l)] + r^ [,uj (q-2)-(q-l)% 
or Hn..: = ^ |r [2 + {q- 2) r] - [1 + (q - 1) r] j (18). 
( f) The Standard Deviation of tlie Fraternal Correlation Coefficient. 
If the sample is great in proportion to {q — \)r the errors of 11 and cr^ can be 
treated as differentials and we have for the correlation coefficient calculated froin a 
sample 
n + sn n 1 n . „ 
P = -o + — , OIJ — - OCT-, 
(T' + Ocr- a~ o- (cr-)- 
r-— [l + (fy-l)r} 
, ■ _ iJ nq ' 
and p = 
l-~ (l + (fi-l)r}' 
■nq 1 ' i 
and therefore neglecting the term containing ~- which according to these supposi- 
tions cannot be evaluated 
^ p = r. 
Fi-om hp = ^- Isn — ^,Scr-| we find by squaring and forming mean value 
cr- 
When the values from (3), (12), (14), (17) and (18) are introduced in this 
formula and the terms containing the higher power of are neglected, we get 
9 9)-2 4r2 
< = - - [ [1 + V (7 - 2)? + (? - 1 )} + ^- \l+{q-l) r^] - ^ [2 + (r/ - 2) r], 
nqiq — l) / J ; I i ' nq 
