Use this point-biserial correlation calculator to measure the relationship between a binary variable coded 0 or 1 and a continuous outcome. Enter paired values, upload a spreadsheet, or paste group data from Excel to instantly calculate rpb, η², Cohen's d, p-value, confidence intervals, and the mathematically equivalent independent-samples t-test result.
Best for continuous variables with a linear relationship.
Pearson r = 0.9991 based on 8 valid pairs, p = 0.0000.
Average the X values and the Y values before measuring joint movement.
Subtract the mean from every X and Y value to get centered deviations.
Multiply each pair of deviations and add them up.
Compute the denominator from the independent spread of X and Y.
The covariance term is normalized by both standard-deviation components.
Enter X as a binary variable coded 0 and 1, then enter Y as a continuous numeric variable.
Use the variable setup guide to confirm that both groups are present and that the coding is valid.
Read rpb, η², Cohen's d, the group means, confidence intervals, and the equivalent t-test result from the output cards.
Expand the step-by-step section to inspect the group split, group means, population standard deviation, and final t conversion.
Point-biserial correlation measures the relationship between one truly binary variable and one continuous variable. The binary variable must be naturally dichotomous, coded as 0 and 1, while the other variable can be any continuous numeric outcome such as test score, income, recovery score, or productivity. A point-biserial correlation is ideal when the research question is really about how strongly group membership predicts or tracks a continuous result.
Mathematically, point-biserial correlation is not a different family of statistic from Pearson correlation. It is simply Pearson r applied to the special case where one variable is binary. That means if X is coded 0 and 1, Pearson r and are the same number. The interpretation is then immediate: positive values mean the group coded 1 tends to have higher Y values, negative values mean the group coded 1 tends to have lower Y values, and larger absolute values indicate stronger group separation.
Typical examples include item analysis in education, where correct or incorrect on a question is compared to total score, often as an item discrimination index; medical studies, where treatment or control is compared to a biomarker or recovery index; marketing, where purchase or non-purchase is compared to income; and HR analytics, where certification status is compared to performance score. One important distinction is that point-biserial correlation is for naturally dichotomous variables. If the binary variable is artificially dichotomous because a continuous variable was cut into high versus low groups, the biserial correlation is conceptually more appropriate.
| Symbol | Meaning |
|---|---|
| Mean of the continuous variable in the group coded 1 | |
| Mean of the continuous variable in the group coded 0 | |
| Population standard deviation of the continuous variable, using n in the denominator | |
| Number of observations in the group coded 1 | |
| Number of observations in the group coded 0 | |
| Total sample size, equal to n_1 + n_0 | |
| Proportion of observations in the group coded 1 | |
| Proportion of observations in the group coded 0 |
The main formula compares the mean of the continuous outcome in the group coded 1 against the mean in the group coded 0, then standardizes that gap by the population standard deviation of the continuous variable. The weighting term involving , , , and ensures that the coefficient reflects both the mean gap and the balance between the two groups.
This calculator uses the standard form with computed using in the denominator, not . That matches the conventional point-biserial formula. Once is known, you can convert it directly to the equivalent t statistic, recover it from a reported t test, and treat as an immediately readable effect size.
Identify Group 1 (binary = 1) and Group 0 (binary = 0), then list the continuous values in each group.
Compute Ȳ₁ and Ȳ₀ so you can see the raw mean difference before any standardization.
Compute the overall population standard deviation of the continuous variable using all observations.
Apply the rpb formula with the mean gap, the standard deviation, and the group proportions.
Convert rpb into the equivalent t-statistic and read the identical p-value you would get from an independent samples t-test.
A point-biserial correlation and an independent samples t-test are two ways to report the same comparison. Both ask whether the two binary groups differ on a continuous outcome. The t-test frames the question as a mean difference, while frames it as a standardized association between a binary grouping and a numeric variable. Because the two statistics are mathematically equivalent, they return the same p-value for the same data.
The choice between them depends on the story you want to tell. Report the t-test when your audience expects hypothesis testing about group means. Report when you want a compact effect size inside a correlation framework, especially alongside other coefficients such as Pearson or Spearman. In practice, point-biserial correlation is often the cleaner option because it gives you both significance and an immediately interpretable effect size in one statistic.
Point-biserial correlation is best interpreted as an effect size for the difference between two groups. The sign tells you whether the group coded 1 tends to score higher or lower than the group coded 0. The magnitude tells you how substantial that difference is. Because , the squared coefficient gives the proportion of variance in the continuous outcome explained by group membership.
A helpful cross-check is Cohen's d. Roughly speaking, is near for a small effect, is near for a medium effect, and is near for a large effect.
Use point-biserial correlation when the binary variable is genuinely binary. If both variables are continuous, switch to Pearson correlation. If the binary grouping was created by cutting an underlying continuous variable at a threshold, the biserial correlation is the theoretical alternative, but it depends on stronger assumptions and is far less common in applied software.
| Point-Biserial rpb | Pearson r | Biserial rb | |
|---|---|---|---|
| X variable type | True binary 0 or 1 | Continuous | Artificially dichotomized |
| Y variable type | Continuous | Continuous | Continuous |
| Typical use case | Pass or fail vs score | Height vs weight | High income vs low income after a cutoff |
| Relation to Pearson | Pearson special case | Base statistic | Adjusted under normality assumptions |
| Range | -1 to +1 | -1 to +1 | Can exceed ±1 |
| Recommended use | Natural binary grouping | Two continuous variables | Only for true artificial dichotomies |
| Link | Current page | /pearson-correlation/ | Conceptual comparison only |
A classic use of point-biserial correlation is item analysis: correct versus incorrect on a specific question against the total test score. In this setting it is commonly reported as an item discrimination index. Higher values mean the item separates stronger performers from weaker performers.
Clinical analysts often compare a treatment indicator coded 1 or 0 against a recovery score, biomarker level, or symptom index. Point-biserial correlation expresses that group difference as a standardized effect size.
Certification status versus productivity, purchase versus non-purchase against income, and yes or no outcomes versus continuous KPIs are all natural point-biserial use cases where a binary grouping is linked to a continuous measure.
A clean HR-style example with pass or fail group membership and a productivity score.
A clinical group comparison between treated patients and control patients.
Classic educational testing example: got the item correct or incorrect versus total exam score.
Point-biserial correlation (rpb) measures the strength and direction of the relationship between a naturally dichotomous binary variable coded 0 and 1 and a continuous variable. It is mathematically equivalent to Pearson r when one variable is binary, and it produces the same p-value as an independent samples t-test. Values range from -1 to +1.
Use point-biserial correlation when one variable is a true binary variable such as pass or fail, yes or no, treatment or control, or certified or not certified, and the other variable is continuous. Common applications include item analysis in educational testing, where it often acts as an item discrimination index, along with clinical group comparisons, marketing conversion analysis, and HR performance studies.
Point-biserial correlation and the independent samples t-test are mathematically equivalent and produce identical p-values for the same dataset. You can convert between them using rpb = √(t² / (t² + df)). Point-biserial is often preferred when you want to report a standardized effect size inside a correlation framework.
Using common effect-size rules of thumb, rpb around 0.10 is small, 0.24 is medium, and 0.37 or above is large. In educational item analysis, values above 0.30 are usually acceptable and values above 0.40 are often considered excellent discrimination.
Point-biserial correlation is used when the binary variable is naturally dichotomous, such as alive versus dead or passed versus failed. Biserial correlation is used when the binary variable is artificially dichotomous, meaning it was created by splitting an underlying continuous variable at a threshold. Biserial correlation relies on stronger normality assumptions and is less commonly used in practical software workflows.
First separate the continuous values into the group coded 1 and the group coded 0. Then compute each group mean, compute the overall standard deviation of the continuous variable using n in the denominator, and apply rpb = ((Ȳ₁ - Ȳ₀) / sY) × √(n₁n₀ / n²). This calculator shows each of those steps explicitly.
Yes. A negative point-biserial coefficient means the group coded as 1 has a lower mean on the continuous variable than the group coded as 0. The sign depends entirely on your coding choice, so swapping 0 and 1 flips the sign while keeping the effect size magnitude the same.
A total sample of about 20, with at least around 10 observations per group when possible, is a practical minimum for a stable first estimate. Larger samples are better for significance testing, and very unequal group sizes reduce power and limit the maximum possible rpb value.