Understanding the Point Biserial Correlation Coefficient: A Guide for Researchers

The point biserial correlation coefficient is a valuable statistical tool used to measure the relationship between a continuous variable and a binary variable. It’s widely used in research fields like psychology, education, and social sciences. This blog will explore what the point biserial correlation coefficient measures, how to calculate it, and how to interpret the results.

What Does the Point Biserial Correlation Coefficient Measure?

The point biserial correlation coefficient (denoted as r_pb) measures the strength and direction of the relationship between a binary variable and a continuous variable. A binary variable is a categorical variable with only two categories, such as pass/fail or male/female. The continuous variable is measured on a scale, like scores on a test or temperature.

For example, if you are interested in finding the relationship between gender (binary variable: male/female) and test scores (continuous variable), the point biserial correlation can be used. It quantifies the degree to which one group tends to score higher or lower than the other group.

What Is a Good Point Biserial Correlation?

The interpretation of the point biserial correlation coefficient is similar to that of the Pearson correlation coefficient. It ranges from -1 to +1, where:

+1 indicates a perfect positive relationship (as the binary variable shifts from 0 to 1, the continuous variable consistently increases).
-1 indicates a perfect negative relationship (as the binary variable shifts from 0 to 1, the continuous variable consistently decreases).
0 indicates no relationship between the two variables.

In practice, a point biserial correlation above 0.50 is often considered a strong correlation, while a value between 0.30 and 0.50 indicates a moderate correlation. Values below 0.30 suggest a weak correlation.

What Is the Formula for the Point Biserial Correlation Coefficient?

The point biserial correlation coefficient is calculated using the following formula:

Where:

M₁ = mean of the continuous variable for the group coded as 1
M₀ = mean of the continuous variable for the group coded as 0
s = standard deviation of the continuous variable
p = proportion of cases coded as 1 (binary variable)

This formula helps assess the extent to which the difference in means between the two binary groups (coded 0 and 1) is related to the overall variability of the continuous variable.

Assumptions of the Point Biserial Correlation Coefficient

For the point biserial correlation coefficient to be valid, a few assumptions must be met:

Binary variable: One variable must be truly binary (i.e., have only two distinct categories).
Continuous variable: The second variable must be continuous, measured on an interval or ratio scale.
Normality: The continuous variable should ideally follow a normal distribution for each group of the binary variable.

How to Calculate the Point Biserial Correlation Coefficient: Step-by-Step

Collect Your Data: Ensure you have one continuous variable and one binary variable. For example, let’s say you want to measure the correlation between exercise frequency (binary: frequent/infrequent) and body weight (continuous variable: weight in kg).
Compute the Means: Calculate the mean value of the continuous variable (weight) for both groups (frequent exercisers and infrequent exercisers).
Calculate the Standard Deviation: Compute the standard deviation of the continuous variable across all data points.
Determine the Proportion of Binary Groups: Calculate the proportion (p) of participants who fall into the binary category coded as 1.
Use the Formula: Substitute these values into the point biserial correlation formula to calculate r_pb.

Example of Point Biserial Correlation Coefficient Calculation

Let’s say you have data on 20 individuals. Your binary variable is gender (coded as 0 for female, 1 for male), and the continuous variable is their height. You find that the mean height for males (M₁) is 180 cm, and for females (M₀) is 165 cm, with a standard deviation (s) of 8 cm. Half of your sample are males (p = 0.50).

Plugging these values into the formula:

This strong positive correlation indicates a substantial relationship between gender and height, with males tending to be taller than females in this sample.

How to Calculate Point Biserial Correlation in SPSS

To calculate the point biserial correlation coefficient in SPSS:

Open your dataset in SPSS.
Go to Analyze > Correlate > Bivariate.
Select the binary and continuous variables you want to correlate.
Under Correlation Coefficients, select Pearson (which also applies to point biserial).
Click OK. SPSS will calculate and display the correlation coefficient.

How to Calculate Point Biserial Correlation in Excel

In Excel, calculating the point biserial correlation coefficient requires some manual work:

Compute the means (M₁ and M₀) for the two binary groups.
Calculate the standard deviation (s) using Excel’s =STDEV.P() function.
Use the SQRT() function to calculate the square root of the proportion term.
Manually apply the formula for r_pb using the values in your Excel sheet.

How to Interpret the Point Biserial Correlation Coefficient

When interpreting r_pb, consider both the magnitude and direction of the relationship:

A positive value indicates that as the binary variable shifts from 0 to 1, the continuous variable tends to increase.
A negative value indicates that as the binary variable shifts from 0 to 1, the continuous variable tends to decrease.
The closer the value is to 1 or -1, the stronger the relationship. Values closer to 0 indicate a weak or no relationship.

Conclusion

The point biserial correlation coefficient is a robust statistical tool for analyzing the relationship between a binary and continuous variable. Understanding how to calculate and interpret this coefficient is crucial for researchers who want to assess correlations in their data. Whether you are using software like SPSS or Excel, following the correct steps will ensure accurate results that can help inform your research conclusions.

References

Field, A. (2018). Discovering Statistics Using IBM SPSS Statistics. SAGE Publications.
Hinkle, D. E., Wiersma, W., & Jurs, S. G. (2003). Applied Statistics for the Behavioral Sciences. Cengage Learning.
Gravetter, F. J., & Wallnau, L. B. (2013). Essentials of Statistics for the Behavioral Sciences. Cengage Learning.

Understanding the Point Biserial Correlation Coefficient: A Guide for Researchers