ON THE PROBABLE ERROR OF THE BI-SERIAL 
EXPRESSION FOR THE CORRELATION COEFFICIENT. 
By H. E. SOPER, M.A. Biometric Laboratory, University of London. 
In a recent paper* Professor Pearson shows that where one character is in 
multiple graded groupii)g and the other in alternative categories, greater or less 
than a given magnitude, the correlation coefficient admits of simple expression; 
the assumptions being that the unmeasured character, B. has a normal distribution 
and that the measured character, A, has linear regression upon B. Under these 
conditions the data required are the numerical ratio of the alternative groups, 
the standard deviation of the measured character and the deviation from the 
general mean of this character of the mean of one of the groups. 
This expression is subject to greater fluctuations of value in samples of N 
of the population than is the product moment form, especially where one of the 
groups is relatively small; and it is proposed to find formulae for the mean and 
second moment of the errors from this mean to a first approximation, that is to 
terms in IjN. These will appear in terms of the correlation coefficient, r, of the 
original population (which will be supposed normally correlated) and the fractional 
frequency,/, in that population of the group possessing the greater or positive 
intensity of the character put into two classes. 
Let y be the graded character and x the alternative character the intenser 
value of which is possesseil by the fraction / of the population. Let x, y be the 
general means and a^, cry the standard deviations of x and y. Then it is shown 
in the paper that if x', y' are the means of the group /, 
r = tf (1), 
on the assumption that the regression of y upon x is linear; and that if x be 
normally distributed this is equivalent to 
rJl^x-^ (2), 
(Ty z 
* Biometrika, Vol. vii. (1909), p. 96: "On a New Method of Determining Correlation between 
a Measured Character A, and a Character B, of which only the Percentage of Cases wherein B 
exceeds (or falls short of) a given intensity is recorded for each grade of A." 
