Point-Biserial Correlation Calculator

Point-biserial correlation measures the strength and direction of association between one genuinely dichotomous variable and one continuous variable. The binary variable has two categories coded as zero and one, while the continuous variable contains a numeric measurement for the same observations. Typical examples include pass or fail paired with study time, treatment or control paired with an outcome score, and purchased or did not purchase paired with customer age. The coefficient is conventionally written rpb and ranges from -1 to 1. The calculation compares the mean continuous score for the observations coded one with the mean for those coded zero. Their difference is divided by the population standard deviation of all continuous scores and multiplied by the square root of p times q, where p is the proportion coded one and q is the proportion coded zero. This adjustment accounts for the relative sizes of the two groups. A positive coefficient means the group coded one has the higher mean; a negative coefficient means the group coded zero has the higher mean. The sign depends entirely on coding. Reversing every zero and one leaves the magnitude unchanged but reverses the sign, so reports should explain what each code represents. The magnitude has the same interpretation as a Pearson correlation: values closer to zero indicate weak linear association between group membership and the continuous outcome, while values closer to either extreme indicate stronger separation. Context matters, and universal cutoffs such as small, medium, or large should be used cautiously. Point-biserial correlation is mathematically equivalent to Pearson correlation when the dichotomous variable is represented by zeros and ones. It is also closely related to the independent-samples t test. Both methods assess separation between two group means, but they present the effect in different forms. The correlation coefficient is a standardized effect-size measure, whereas a t test combines effect and sampling uncertainty to test a null hypothesis. This calculator returns the coefficient and group details, not a p-value or confidence interval. The paired lists must contain the same number of observations and remain in matching order. The first binary code belongs to the first continuous value, the second code to the second value, and so forth. Both binary groups must appear, and the continuous data must vary; otherwise the correlation is undefined. Missing values should be removed as complete pairs rather than from only one list, because independent deletion would misalign observations and produce a meaningless result. Interpretation also relies on sound data collection. A high coefficient does not establish that group membership caused the numeric difference, and confounding variables may explain an observed relationship. Outliers can strongly affect group means and standard deviation. Inspect distributions, sample sizes, and unusual measurements before drawing conclusions. For an artificially dichotomized continuous variable, a biserial correlation or a model using the original continuous predictor may be more appropriate, because splitting a natural continuum loses information.