When we would like to calculate the correlation between two continuous variables, we typically use the .

However, when we would like to calculate the correlation between a continuous variable and a , we can use something known as point biserial correlation.

Point biserial correlation is used to calculate the correlation between a binary categorical variable (a variable that can only take on two values) and a continuous variable and has the following properties:

• Point biserial correlation can range between -1 and 1.
• For each group created by the binary variable, it is assumed that the continuous variable is normally distributed with equal variances.
• For each group created by the binary variable, it is assumed that there are no extreme outliers.

The following example shows how to calculate a point biserial correlation in practice.

Example: Calculating a Point Biserial Correlation

Suppose a college professor would like to determine if there is a correlation between gender and score on particular aptitude exam.

He collects the following data on 12 males and 12 females in his class:

Since gender is a categorical variable and score is a continuous variable, it makes sense to calculate a point-biserial correlation between the two variables.

The professor can use any statistical software (including Excel, R, Python, SPSS, Stata) to calculate the point-biserial correlation between the two variables.

The following code shows how to calculate the point-biserial correlation in R, using the value 0 to represent females and 1 to represent males for the gender variable:

#define values for gender
gender <- c(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
            1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)

#define values for score
score <- c(77, 78, 79, 79, 82, 84, 85, 88, 89, 91, 91, 94,
           84, 84, 84, 85, 85, 86, 86, 86, 89, 91, 94, 98)

#calculate point-biserial correlation
cor.test(gender, score)

	Pearson's product-moment correlation

data:  gender and score
t = 1.3739, df = 22, p-value = 0.1833
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.1379386  0.6147832
sample estimates:
      cor 
0.2810996 

From the output we can see that the point biserial correlation coefficient is 0.281 and the corresponding p-value is 0.1833.

Since the correlation coefficient is positive, it tells us that there is a positive correlation between gender and score.

Since we coded the males as 1 and females as 0, this indicates that scores tend to be higher for males (i.e. scores tend to increase as gender “increases” from 0 to 1)

However, since the p-value is not less than .05, this correlation coefficient is not statistically significant.

The following tutorials explain how to calculate point biserial correlation using different statistical software: