H. E. SOPER 
385 
where z is the ordinate of the normal curve cutting off the area 
/=i(l-a) or 1(1 + a) 
as defined in Sheppard's Tables of the Probability Integral. 
Now if jJo, Vi' Pi' 
be the moment coefificieots of the whole population with respect to the character 
y defined by 
i>. = 'S'(n,y/)/iV (3), 
[see bi-serial table below which is here to be looked upon as representing the 
general population] and 
To, Pi, P-2, etc. 
are moment coefficients of the group n(= fN) defined by 
p,; = S{n;y;')/N (4), 
we may write f = pn , /[/ =pi', V=lh, <^y = \Kp-2 -Pi") and 
\/{Pi-Pi^ ^ 
— ^ Grades of y in Bi-serial Table. 
.(.5). 
'3 
O 
"2" 
n-l' 
Ms" 
n" 
n 
«1 
«3 
71, 
N 
.(6). 
n'IN=f 
In samples of N the frequencies n^, n/, and consequently the moment 
coefficients p and p and the ordinate z are subject to fluctuation and the values 
of the correlation coefficient calculated from this formula will have a distribution 
of errors. Let f be the mean value in such samples, Sr the deviation from this 
mean value in any sample when Zp^, Bp^, Spo, etc. 8z are the deviations in moments 
and ordinate. Then 
Pi + Bpi - (Po + Bpo) (Pi + Vi) 
Vbs + V-2 - (Pi + BpiY] X(2 + 8Z) 
To express 82 in terms of deviations of the moments we have 
z - 
V27r 
where a is the abscissa of the point of section of the normal curve, defined by 
.(7). 
•(«)- 
