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Point-biserial Correlation Coefficient




The point-biserial correlation is mathematically equivalent to the Pearson (product moment) Correlation , that is, if we have one continuously measured variable ''X'' and a dichotomous variable ''Y'', rXY = rpb. This can be shown by assigning two distinct numerical values to the dichotomous variable.

To calculate rpb, assume that the dichotomous variable ''Y'' has the two values 0 and 1. If we divide the data set into two groups, group 1 which received the value "1" on ''Y'' and group 2 which received the value "0" on ''Y'', then the point-biserial correlation coefficient is calculated as follows:

:
r_{pb} = rac{M_1 - M_0}{s_x} \sqrt{ rac{n_1 n_0}{n(n-1)}},


where M_1 is the mean value on the continuous variable ''X'' for all data points in group 1 and M_0 is the mean value on the continuous variable ''X'' for all data points in group 2. Further, n_1 is the number of data points in group 1, n_0 is the number of data points in group 2 and n is the total sample size.
This formula is a computational formula that has been derived from the formula for rXY in order to reduce steps in the calculation - it is easier to compute than rXY. It is of much less importance these days since computers are almost exclusively used for statistical data analysis.


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