Karl Pearson 
295 
It now remains to find Mean (8/<:8y). We have: 
K Mean (SKSy) = *S' {k, {Mean (8/f,Sy)}) 
= \S Mean (SySn,)} + S 
We must now find Mean (SyS^j) and Mean (8y,Sy). 
Mean(8y8uJ - ^ean (S.,8»,) 
N 
Mean (8ys8y) 
Nz 
' N J 
,71, Z, 
n,H, J) 
1 
N) nX'-' N )\ 
NZ71,Z, 
Thus 
K Mean (8K8y) = - ^ |l S (.^.y/) (»,y/) - S y,)| (xiii). 
We can now substitute in (iii) from (iv), (xii) and (xiii) and obtain a,^. We 
have 
(1 
+ 
(1 - 172)2 u^l^ ly 
(i + y2)2 
(1 + y2)2iV2 
(1 - v'r 
iV (1 + y2 
''''(%,y/)--^'S(".,y/)-2s(-^;'' 
I©' 
2 s Ys 
Is ''2s /y? 
.N 
XIV . 
NoAv we know all the quantities in the sums on the right of (xiv), because 
ys has had to be found for each array, and z^ can be at once determined from 
XyjyO^ which is known. But the labour would be very considerable and hardly 
commensurate with the result desired, i.e. an approximate measure of the accuracy 
of determining the correlation by the biserial iq method. We shall accordingly 
investigate the values of the above terms on the hypothesis of a large number of 
