390 Pruhahle Error of the Bi-serial Correlation Coe^cient 
Thus the effect of grouping and applying the bi-serial value of the correlation 
coefficient is to add 25 to the probable error in the most accordant case where 
r is zero and the division equal, whilst if r is as large as '5 and one group as small 
as 10"/^ of the whole the probable error is nearly doubled. For higher values of 
r the errors of sampling, in the case of the product moment formula, grow smaller 
and ultimately vanish when ?■= ! ; but the bi-serial values are not invariable in 
samples drawn from a perfectly correlated population but possess a variability as 
high as '2^ /\/N in the most favourable case when the grouping is equal. 
If the standard deviation be calculated from the approximate formula, 
a,. = iJff'/z-r^)/jN (26), 
which may be written* 
<^r = (Xu-r')/jN (27), 
the error of computation will not be great for values of f and r commonly met 
with as the following table compared with the last will show : 
r 
Values of %« - 
i for 4(l-a)=: 
•500 
•309 
•159 
•067 
•023 
■006 
■00 
1-25 
rsi 
1-51 
1^93 
2^76 
4-5 
•25 
1-19 
1-25 
1-45 
1^86 
2^70 
4-4 
•50 
TOO 
ro6 
1-26 
r68 
2^51 
4^2 
•75 
•69 
"75 
•95 
r36 
2-20 
3^9 
l^OO 
•25 
•31 
•51 
•93 
1-76 
3^5 
(28). 
The difference between the two expressions only reaches 5 ''/^ when the 
smaller group is less than 7 °/\ of the whole. 
It will not be necessary, excepting in small samples, to apply a correction 
to the bi-serial formula for r in virtue of the mean of samples differing from the 
population value. The correction is less than 1/iV^th part of the value calculated 
unless one of the alternative classes is as small as 4 of the whole. 
I have to thank fellow members of the Staff for assistance in calculating the 
tables. 
* For a table of xa see Tables for Statisticians and Bionietricians, p. 37. 
