A binary variable against a continuous one, in all three units at once.
A binary column against a continuous
r = 0.9467916, identical to Pearson on the same two columns. Cohen’s d is 5.2623 on the same comparison — the two are one finding in two units.
5 zeros, 5 ones, 10 pairs
r = 0.9467916
Cohen's d on the same comparison is 5.2623, t = 8.3205 on 8 degrees of freedom. Pearson's r computed independently on the same two columns is 0.9467916 — the same number, because the point-biserial correlation IS Pearson's r with the binary variable coded 0/1.
Point-biserial r
0.9467916
agrees with Pearson to 3.3e-16
Cohen's d
5.2623
same comparison, no ceiling
t
8.3205
df = 8
Group means
2.4000 → 8.4000
pooled SD 1.1402
The same effect, at six different splits
Cohen’s d held at 1.9799 while only the proportion in group 1 changes. The gap between the groups and the spread within them are identical in every row.
Point-biserial r at six splits for a fixed Cohen’s d of 1.9799
Split
Cohen’s d
Point-biserial r
Share of the 50/50 value
50 / 50
1.9799
0.707107
100.0%
40 / 60
1.9799
0.699854
99.0%
30 / 70
1.9799
0.675664
95.6%
20 / 80
1.9799
0.624695
88.3%
10 / 90
1.9799
0.514496
72.8%
2 / 98
1.9799
0.269630
38.1%
Nothing about the effect changed across those rows, and r fell by 62%. A point-biserial correlation is not comparable between studies whose groups split differently — Cohen’s d is.
The same t as an independent-samples t-test Exactly convertible to and from Cohen’s d Do not compare it across unequal splits
What this tool shows
Hold Cohen’s d fixed at 1.9799 and change nothing but how the binary variable splits the sample: r falls from 0.707107 at 50/50 to 0.269630 at 2/98. The gap between the groups and the spread within them are identical in every row — a 62% drop in the correlation with no change to the effect. The tool prints r, d and t together so the three views of one comparison cannot be mistaken for three findings.
The point-biserial correlation, with Pearson’s r on the same columns computed independently
Cohen’s d on the same comparison — the unit that does not move with the split
The t statistic and degrees of freedom, identical to an independent-samples t-test
Group means, group sizes and the pooled standard deviation
A split table: r at six proportions for one fixed effect size
The √(pq) factor that makes r depend on the split, stated rather than hidden
r, d and t Pearson identity checked Split table Exact d conversion
Updated 12 September 2026 · Works in any browser, no installation
The point-biserial correlation is Pearson’s r with one variable coded 0 and 1 — the same number, not an analogue. No separate formula is being applied; the special name exists only because one of the variables is dichotomous. It measures the same thing an independent-samples t-test tests, and converts exactly to and from Cohen’s d.
At a glance
Formula shown
r_pb = [(M₁ − M₀)/sₙ]·√(pq), where M₁ and M₀ are the group means, sₙ is the standard deviation of the whole continuous variable (dividing by n), and p and q are the proportions in the two groups. The √(pq) term is the whole story: it peaks at 0.5 for an even split and collapses toward zero as the split becomes extreme, which is why the same mean difference produces a smaller r in an unbalanced sample. In t terms, r_pb = t/√(t² + df).
Scenario support
Item analysis in testing — how well one right/wrong item predicts the total score; pass/fail against a continuous predictor; treatment versus control on a measured outcome; gender, membership or exposure against any quantity; and any two-group comparison where the effect is wanted as a correlation rather than a mean difference.
Educational estimate
Planning support from the values you enter — not professional advice.
The correlation moves with the split; the effect does not
This is the property that makes point-biserial correlations dangerous to compare, and it follows directly from the √(pq) factor in the formula.
Fix the group gap and the within-group spread, then change only the proportion: Cohen’s d stays at 1.9799 in every row of the tool’s table while r runs 0.707107, 0.699854, 0.675664, 0.624695, 0.514496, 0.269630.
At 2/98 the correlation has fallen to 38% of its 50/50 value. Nothing about the underlying difference changed. A reader comparing r = 0.27 from one study against r = 0.71 from another would conclude the second effect is far stronger, and be wrong.
√(pq) is at its maximum of 0.5 when the split is even and falls away on both sides — 0.4899 at 40/60, 0.4 at 20/80, 0.14 at 2/98. That single term is the entire difference between the columns.
So report d when comparing across studies, and r when describing one. The tool prints both for exactly this reason, and the conversion between them is exact rather than approximate.
It is Pearson’s r, and the tool proves it rather than claiming it
Textbooks present a distinct-looking formula for the point-biserial correlation, which leads people to treat it as a different coefficient. It is not.
Code the binary variable 0 and 1 and run an ordinary Pearson correlation: you get the identical number, to every digit. The engine computes both paths and the tool reports the agreement.
Which means all of Pearson’s properties carry over. It is bounded by −1 and 1, it is invariant to linear rescaling of the continuous variable, and r² is the share of variance explained — the same interpretation, not an analogous one.
It also means the sign is arbitrary until you say which group is 1. Swapping the coding flips the sign and changes nothing else, so a negative point-biserial correlation is a labelling fact rather than a finding.
The choice of 0 and 1 specifically does not matter either. Any two distinct numbers give the identical r, because Pearson’s r is invariant to linear transformation of either variable.
Point-biserial is not the biserial correlation
The names differ by one word and the coefficients differ by a great deal. Using the wrong one is a common and consequential mix-up.
The point-biserial applies when the binary variable is genuinely dichotomous — alive or dead, passed or failed, treatment or control. There is no underlying continuum being cut.
The biserial correlation applies when a continuous variable has been dichotomised — an exam score cut at a pass mark, blood pressure split at a threshold. It estimates what the correlation would have been before the cut.
The biserial is always larger in magnitude, typically by about 25% at an even split and far more at an extreme one, because it corrects for the information the dichotomy threw away. It can even exceed 1, which is a sign the normality assumption behind it has failed.
This tool computes the point-biserial. If your binary variable was made by cutting a continuum and you want the pre-cut relationship, the biserial is the coefficient to look up — and it rests on an assumption of normality that the point-biserial does not need.
Item analysis is where this coefficient earns its keep
In testing, the point-biserial between one item (right or wrong) and the total score is the standard item discrimination index, and it is read as a diagnostic rather than as a finding.
A high value means the item separates strong candidates from weak ones, which is what an item is for. Above about 0.3 is usually considered acceptable, above 0.4 good.
A near-zero value means the item is not discriminating. Everyone gets it right, everyone gets it wrong, or the answer is essentially random with respect to ability.
A NEGATIVE value is the one that matters. Stronger candidates are getting it wrong more often than weaker ones, which almost always means a miskeyed answer or a question that rewards a misconception. It is the same signal a negative item-total correlation gives in Cronbach’s alpha.
The split caveat applies here too, and hard. A very easy or very difficult item has an extreme split, so its point-biserial is capped low by √(pq) regardless of how well it discriminates. Comparing discrimination across items of very different difficulty is the same mistake as comparing r across studies.
Testing it is the t-test you already ran
The significance test for a point-biserial correlation and an independent-samples t-test on the same two groups are the same test, and they return the same t.
t = r√(n−2) / √(1−r²), with n−2 degrees of freedom, and running a Student’s t-test on the two groups gives the same value to machine precision.
So the assumptions are the t-test’s assumptions. Independent observations, roughly normal within each group, and comparable variances — the last of which is worth a Levene test when the groups are unbalanced.
An extreme split makes the test fragile rather than merely small. Two observations in one group gives 18 degrees of freedom but almost no information about that group’s variance, and the p-value inherits that.
Reporting both r and t is not double-counting. One is the size of the effect and the other is the evidence that it is not zero, and a large sample can make a tiny r highly significant.
Reporting a point-biserial correlation
Four numbers, and the group sizes are the one that makes the correlation readable.
Give both group sizes, not just the total. Without them the reader cannot tell whether an r of 0.27 is a weak effect or a strong one measured at an extreme split.
Give Cohen’s d alongside r. It is the comparable unit, it is one line, and the conversion is exact so nothing is being estimated twice.
Say which group was coded 1. The sign carries no meaning without it.
And give a confidence interval on r. Point-biserial correlations from small, unbalanced samples are unstable, and the interval is usually wider than the point estimate suggests.
Sources and methodology
References for the point-biserial correlation and its relatives.
Method. The point-biserial correlation is computed from the group means and the population standard deviation, and Pearson’s r is computed independently on the same two columns with the binary variable as 0/1 — so the identity between them is checked on every input rather than asserted once. Cohen’s d and t come from the same two groups, which makes the exact conversion r = t/√(t²+df) a verified property of the engine rather than a formula quoted at you. The split table is arithmetic on a fixed gap and spread: the rows share a Cohen’s d of 1.9799 to four decimal places and differ only in √(pq). Fewer than three pairs, a single group, and a constant continuous column all return no result rather than a number. That engine is verified on every change against 69 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Correlation CoefficientReports Pearson, Spearman and Kendall together with the scatter plot, and ships Anscombe's quartet built in — four datasets with an identical r of 0.816 that Spearman tells apart.
Effect SizeCohen d, Hedges g and the overlap between groups, with a sample-size control that moves the p-value while leaving the effect size fixed — the same d gives t = 1.29 at n=30 and 23.57 at n=10,000.
t-testOne-sample, two-sample and paired t-tests defaulting to Welch, with Student's pooled version printed beside it — and a warning when the two disagree on the verdict.
Phi CoefficientPhi for a 2x2 table printed against the ceiling its marginals impose: on [10, 40, 0, 50] every case is exposed, a complete association, and phi is 0.3333333, which is exactly max phi.
Cronbach's AlphaAlpha with alpha-if-item-dropped, item-total correlations and the mean inter-item correlation, plus the length table: at a mean inter-item correlation of 0.05, a 100-item scale still reports 0.8403.
Spearman CorrelationComputes rho correctly as Pearson on the midranks, and beside it the 6Σd²/(n(n²−1)) shortcut every textbook teaches — which is exact only when no two values tie, and overstates the correlation when they do.
An educational tool. The point-biserial correlation depends on how the binary variable splits the sample as well as on the effect itself, so values are not comparable between samples with different group proportions — and this is the point-biserial, not the biserial correlation used when a continuous variable has been dichotomised.
Published a binary-versus-continuous correlation that computes Pearson's r on the same two columns independently, so the identity between them is checked on every input rather than asserted once in the text.
Measured the property that makes point-biserial correlations dangerous to compare: holding Cohen's d at 1.9799 and changing only the split, r falls from 0.707107 at 50/50 to 0.269630 at 2/98. The gap between the groups and the spread within them are identical in every row, and the correlation drops 62%.
Printed r, Cohen's d and t together, since they are three views of one comparison and the exact conversions between them mean nothing is being estimated twice.
Set out the distinction from the biserial correlation, which applies when a continuum has been dichotomised and which is always larger — about 25% larger at an even split.
Verified against 69 assertions including the Pearson identity, the r = t/sqrt(t^2+df) relation, and the constant-column and single-group cases that must return no result.
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