38(5 Probable Error of the Bi-serial Correlation Coefficient 
Hence to second order terms in powers and products of deviations 
V27r 
= ^{l-«Sa-i(l-a=)(ga)-} 
1 
\/27r 
J a 
0 ^'Ztt 
It follows that 
z + 82 = z + aBf-^JSfy- 
-I 
r&a 
= , {l-a?-i(l-«=)P+...](^^ 
J 0 
= z {Ba — ^a{Bay]. 
= z+ aSpJ -^^(Bpn'y (10). 
At the same time that this value is put in (7) we may simplify the expression 
and subsequent algebra by supposing the graded character y to be measured 
from its universal mean value as origin and in terms of its standard deviation as 
unit of measurement in which case 
Pi = 0, p2 = o-y- = l, and by (5) pi=zr (11), 
and since po = /(7) becomes, 
j-. , g.. zr + Bp,' - fBp.-Spo'Sp, 
V{1 + Bp., - {Bp,y} X 1^ + aBp:- ^(« j 
Expanding to second order of deviations we find, 
r + Br = r + B, + B, (13), 
where Bi, B., are the first order and second order expressions 
Si = ^ {Bpi —fBpi — hzrBp., — ar Bp„'], 
^2 = ^ '^^urBp.,Bp^' - ^ Bpi'Bpu +•—- Bp^Bpo - ^Bp/Bp., + ^/Bp^Bp.^ 
- Bp: Bp, + \zv {Bp,y + I zr (Bp,r + (1 + 2a^) £ (%') j (14). 
Taking mean values 
r = r + mean So (1'5)> 
mean B, being zero since by (3), (4) 
mean 8p„ = S (mean S/is?//)/iV = 0, 
mean Bp„' = S (mean Bn,'ys'')IN — 0. 
