If it can be assumed that the dichotomous variable Y is normally distributed, a better descriptive index is given by the biserial coefficient:[3] where
A specific case of biserial correlation occurs where X is the sum of a number of dichotomous variables of which Y is one.
A statistic of interest (which is a discrimination index) is the correlation between responses to a given item and the corresponding total test scores.
We could calculate the coefficient in the same way as where X is continuous but it would have the same disadvantage that the range of values it can take on becomes more constrained as the distribution of Y becomes more unequal.
We can therefore use the reciprocal of this value to rescale the difference between the observed mean ranks on to the interval from plus one to minus one.
This formula, which simplifies the calculation from the counting of agreements and inversions, is due to Gene V Glass (1966).
If rrb is calculated as above then the smaller of and is distributed as Mann–Whitney U with sample sizes n1 and n0 when the null hypothesis is true.