Corrected Point-Biserial Item Analysis Calculator
Enter the item count, correct and incorrect group means, the total-score standard deviation, and the item value. The calculator reports difficulty, raw discrimination, corrected discrimination, and a practical review flag.
Corrected point-biserial correlation0.31
- Item difficulty index
- 0.6%
- Difficulty 95% confidence interval
- 50.20% to 69.06%
- Raw point-biserial correlation
- 0.39
- Rest-of-test standard deviation
- 4.83
- Item review guidance
- Strong: 0.30 or higher, while mastery items still need content judgment.
Uses the inclusive total-score means and standard deviation from the same scored cohort.
Applies to zero-or-full-credit items, including weighted binary items, not partial-credit item scores.
How to use this calculator
- Enter the number of scored examinees and the number who answered the item correctly.
- Enter both group means using the same inclusive whole-test score scale.
- Enter the sample standard deviation for the same whole-test scores.
- Set the item point value, then read the corrected discrimination and review guidance.
How the corrected point-biserial is calculated
The item difficulty index is the proportion correct, p = C / N. This calculator also reports a 95% Wilson confidence interval for that proportion, which is usually more stable than the simple plus-or-minus normal interval when a class is small or the item is very easy or hard.
The raw point-biserial is reconstructed from summary statistics. Let N be the number scored, C the number correct, q = 1 - p, M1 the inclusive total-score mean for examinees who answered correctly, M0 the corresponding mean for examinees who did not, and sT the sample standard deviation of inclusive total scores. The finite-sample factor is k = sqrt(Npq / (N - 1)), so rRaw = ((M1 - M0) / sT) * k.
That raw value is inflated because the item score is part of the total score. To remove the part-whole overlap, the calculator treats the item score X as a binary score worth w points and derives sX = w * k. It computes covXT = rRaw * sX * sT, then defines the rest score as R = T - X. The rest-score variance is sR^2 = sT^2 + sX^2 - 2 * covXT, the item-rest covariance is covXR = covXT - sX^2, and the corrected point-biserial is rCorrected = covXR / (sX * sqrt(sR^2)).
What moves the result
The gap between the correct-group and incorrect-group means drives discrimination. A larger gap raises both the raw and corrected correlations. The item difficulty matters too: an item answered correctly by almost everyone or almost no one has little item-score variance, so it cannot separate examinees reliably. The item point value affects the part-whole correction because a heavier item contributes more to the inclusive total score.
What this calculator leaves out
The calculation summarizes a dichotomously scored item under classical test theory. It does not inspect distractors, check content alignment, test for differential item functioning, model item response theory parameters, or decide whether a mastery item should discriminate strongly. Missing responses must already be scored according to the assessment policy before these aggregate inputs are prepared.
Worked example
Suppose 100 examinees took a test, 60 answered the item correctly, the correct group averaged 18 total-score points, the incorrect group averaged 14 points, the inclusive total-score sample standard deviation was 5 points, and the item was worth 1 point.
The difficulty is 60.00%, with a Wilson 95% interval of 50.20% to 69.06%. The raw point-biserial is 0.3939. After removing the item from the total score, the rest-of-test standard deviation is 4.8273 and the corrected point-biserial is 0.3060, which falls in the strong review band for many item analyses.
Common questions
What does a negative point-biserial correlation mean?
A negative value means examinees with higher total scores were less likely to answer the item correctly than examinees with lower total scores. That pattern can come from a miskeyed answer, ambiguous wording, content that was not taught, or a mismatch between the item and the construct being tested.
Why is the corrected point-biserial lower than the raw value?
The raw item-total correlation uses a total score that already includes the item. That shared part inflates the relationship, especially when the item is worth several points or the test is short. The corrected value correlates the item with the rest of the test after algebraically removing the item score.
What is a good point-biserial value for a test item?
Common practical bands treat values below 0.10 as weak, 0.10 to 0.19 as minimal, 0.20 to 0.29 as good, and 0.30 or higher as strong. Those bands are not universal rules. Item purpose, test length, sample size, and the examinee population all matter.
Is item difficulty the percentage correct or the percentage incorrect?
In classical item analysis, the difficulty index usually means the proportion correct. A value of 0.60 means 60% of scored examinees answered correctly. Some reports discuss difficulty in the everyday sense of percentage incorrect, so check the definition before comparing systems.
Can I use this for weighted or partial-credit questions?
You can use it for zero-or-full-credit binary items that are worth more than one point by entering the full-credit point value. Do not use this version for partial-credit or polytomous items because their item-score variance and covariance are not captured by a simple correct count.