Point-Biserial Correlation Coefficient (rpb) Calculator

Formula first

Overview

The point-biserial correlation coefficient ($r_{pb}$) is a measure of association used when one variable is dichotomous (binary, e.g., pass/fail, male/female) and the other is continuous (e.g., test score, height). It quantifies the strength and direction of the linear relationship between these two types of variables. Essentially, it assesses whether there's a significant difference in the mean of the continuous variable between the two groups defined by the dichotomous variable. It is mathematically equivalent to Pearson's r when one variable is dichotomous.

Symbols

Variables

$\bar{Y}$_1 = Mean of Continuous Variable for Group 1, $\bar{Y}$_0 = Mean of Continuous Variable for Group 0, $s_Y$ = Standard Deviation of Continuous Variable (Overall), $n_1$ = Sample Size for Group 1, $n_0$ = Sample Size for Group 0

Apply it well

When To Use

When to use: Apply this formula when you want to determine the correlation between a naturally dichotomous variable (e.g., correct/incorrect answer on a test item) and a continuous variable (e.g., total test score). It's commonly used in psychometrics for item analysis to see how well individual test items discriminate between high and low overall performers.

Why it matters: The point-biserial correlation is crucial in educational and psychological testing for evaluating the quality of test items. A high positive $r_{pb}$ for an item indicates that those who scored high on the overall test tended to answer that item correctly, suggesting it's a good discriminator. It helps refine tests, ensuring they effectively measure the intended construct.

Avoid these traps

Common Mistakes

• Using it for two continuous variables (use Pearson's r).
• Using it for two dichotomous variables (use Phi coefficient).
• Misinterpreting the sign of the correlation if the dichotomous variable coding is arbitrary.

One free problem

Practice Problem

In an item analysis, students who answered a question correctly (Group 1) had a mean score of 75, while those who answered incorrectly (Group 0) had a mean score of 60. The overall standard deviation of scores was 10. There were 30 students in Group 1 and 20 in Group 0, with a total of 50 students. Calculate the point-biserial correlation coefficient ($r_{pb}$).

Solve for: $r_{p}b$

Hint: Calculate the square root term first, then multiply by the difference in means divided by the standard deviation.

The full worked solution stays in the interactive walkthrough.

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Related Formulas

References

Sources

1. Psychometric Theory by Jum C. Nunnally and Ira H. Bernstein
2. Discovering Statistics Using IBM SPSS Statistics by Andy Field
3. Wikipedia: Point-biserial correlation coefficient
4. Field, A. (2018). Discovering Statistics Using IBM SPSS Statistics (5th ed.). SAGE Publications.
5. Aron, A., Aron, E. N., & Coups, E. J. (2018). Statistics for Psychology (8th ed.). Pearson.
6. Nunnally, J. C., & Bernstein, I. H. (1994). Psychometric Theory (3rd ed.). McGraw-Hill.
7. Wikipedia: Point-biserial correlation coefficient (Retrieved 2023-10-27).
8. Gravetter, F. J., & Wallnau, L. B. (2017). Statistics for the Behavioral Sciences (10th ed.). Cengage Learning. Chapter 15: Correlation.