In Relationship between Correlation and t Test and Relationship between Correlation and Chi-square Test we introduced the point-serial correlation coefficient, which is simply the Pearson’s correlation coefficient when one of the samples is dichotomous.

The **biserial correlation coefficient** is also a correlation coefficient where one of the samples is measured as dichotomous, but where that sample is really normally distributed. In such cases, the point-serial correlation generally under-reports the true value of the association. The biserial correlation coefficient provides a better estimate in this case.

Assuming that we have two sets *X* = {*x*_{1}, …, *x*_{n}} and *Y* = {y_{1}, …, y_{n}} where the *x _{i}* are 0 or 1, then the biserial correlation coefficient, denoted

*r*, is calculated as follows:

_{b}Where *n*_{0} = number of elements in *X* which are 0, *n*_{1} = the number of elements in *X* which are 1 (and so *n* = *n*_{0}+*n*_{1}), *p*_{0} = *n*_{0}/*n*, *p*_{1} = *n*_{1}/*n*, *m*_{0} = the mean of {y_{i}:* x _{i}* = 0},

*m*

_{1}= the mean of {y

_{i}:

*x*= 1},

_{i}*s*is the standard deviation of

*Y*and

y = NORM.S.DIST(NORM.S.INV(*p*_{0}),FALSE)

**Example 1**: Calculate the biserial correlation coefficient for the data in columns A and B of Figure 1.

**Figure 1 – Biserial Correlation Coefficient**

The biserial correlation of -.06821 (cell J15) is calculated as shown in column L. Note that the value is a little more negative than the point-serial correlation (cell C4).

**Real Statistics Function**: The following function is provided in the Real Statistics Resource Pack.

**BCORREL**(R1, R2) = the biserial correlation coefficient corresponding to the data in column ranges R1 and R2, where R1 is assumed to contain only 0’s and 1’s.

For biserial correlation coefficient for Example 1 can be calculated using the BCORREL function, as shown in cell G6 of Figure 1.