388 Prohahle Error of the Bi -serial Correlation Coefficient 
Putting y = rx + r] and integrating with respect to r] for constant x, 
, 1 
P-2 
[{rxf + il 
\''2i 
. e dx 
=/' + azr" 
.(18), 
iV= {{rxr + S(r.r){l-r^)]-=e- 
J a \l lir 
= z{% — r- + a^r^) r 
' dx 
.(19). 
When these values are put in (17), and terms collected, the mean value of the 
bi-serial correlation coefficient in samples of N is found to be 
1 r,^,4'_,',_>^/i^.A!UjH; 
.(20), 
where / = l -/= 7(7i\^. 
In the work of obtaining this approximation all powers and products of 
deviations above the second order have been neglected. The means of such 
terms in samples of N involve second and higher powers of 1/iV* and the present 
result is correct to the first approximation. 
Again squaring (13) and taking mean values and subtracting the square of 
(15) we find to the same approximation as before 
o-,.-' = mean (S?-)'- = mean Z-c (21). 
The evaluation of mean S,- being carried out precisely in the same way as 
mean So, the result is the second moment of deviations of the bi-serial corre- 
lation coefficient in samples of N , 
Z_|3Wi_>Vi-f^V...+ 
.(22). 
Writing the two results (20) and (22) 
r = r|l+i(0„+i,-)| 
^.■' = jr[x<^'-ir^r^+rq (23), 
the values of Xa'f "^a for values of ^{1 — a) [= the smaller of n/N, n"j Hi] 
from "50 to "01 are to be found in table (24). 
* See Biometrika, Vol. ix. (1913), pp. 97—99. 
t xa for ^ (1 + a) was tabled iu Biometrika, Vol. ix. p. 27, and the table is reproduced in Tables for 
Statisticians and Biometriciam, p. 35, Cambridge University Press. 
