Item Difficulty & Discrimination Calculator: p-Value, D Index and Point-Biserial
Item analysis for one test question: difficulty index (p) from all students or the upper and lower groups, discrimination index D with unequal group sizes, the highest D the item could reach, optional point-biserial correlation, chance level for multiple-choice items and a keep / review / revise verdict.
How it works
difficulty p = students correct ÷ students · discrimination D = upper correct ÷ upper size − lower correct ÷ lower size (= (U − L) ÷ n for equal groups) · highest D = share of the U + L correct answers that fit in the upper group − the rest in the lower group · point-biserial r_pb = (M_right − M_wrong) ÷ SD × √(p × (1 − p)) · chance level = 1 ÷ number of options
Item analysis judges each test question on two axes. The difficulty index p is the share of students who answered correctly; despite the name, a higher p means an easier item. The discrimination index D compares how often the top scorers and the bottom scorers got the item right. Sort the papers by total score, take the top and bottom groups (traditionally 27% each, or halves in a small class), and subtract the lower group's proportion correct from the upper group's. A positive D means stronger students did better, as they should; a D near zero means the item does not separate them, and a negative D usually points to a wrong answer key or a misleading question. D is capped by difficulty: a very easy or very hard item cannot reach a high D because almost everyone, or almost no one, answers it correctly. The calculator shows the highest D possible for this item so you can judge it fairly. For multiple-choice items it also shows the chance level (0.25 for four options); an item with p near chance may be working no better than guessing. If you have total scores, the point-biserial correlation (under More options) uses the whole class rather than just the two extreme groups. Rules of thumb used here: D of 0.40 or more is excellent, 0.30–0.39 good, 0.20–0.29 fair and below 0.20 poor; a point-biserial of 0.20 or more is good. Use them as guides, not strict cut-offs, especially with small classes.
Worked example
Of 100 students, 65 answered correctly, so p = 65 ÷ 100 = 0.650 (moderate). In the top 27 scorers 24 were correct (0.889) and in the bottom 27, 10 were correct (0.370), so D = (24 − 10) ÷ 27 = 0.519, excellent. The 34 correct answers in the two groups could at most all sit in the upper group (27) with 7 in the lower, giving a highest possible D of (27 − 7) ÷ 27 = 0.741, so this item reaches 70% of it. On a four-option item the chance level is 0.25, well below 0.65, and the verdict is keep.
Sources: Assessment Systems: classical item difficulty (p-value) bands · Assessment Systems: item analysis, point-biserial guidelines and distractors · Classical test theory (item difficulty and item-total correlation) · Point-biserial correlation coefficient
Frequently asked questions
Does a high difficulty index mean a hard question?
No, it means an easy one. The difficulty index is the proportion who answered correctly, so p = 0.90 is an easy item and p = 0.30 a hard one. Some books call it the facility index for that reason.
What discrimination index is good?
A common guide is 0.40 or more excellent, 0.30–0.39 good, 0.20–0.29 fair and below 0.20 poor. A negative D is a red flag: weaker students outperformed stronger ones, which often means a miskeyed or misleading item.
How big should the upper and lower groups be?
The traditional choice is the top and bottom 27% of total scores, a balance between group size and how different the groups are. In small classes the top and bottom halves are often used so each group has enough students.
What is the ideal difficulty for a multiple-choice question?
Items near the middle discriminate best. A simple guide is the midpoint between the chance level and 1: about 0.63 for four options, 0.67 for three and 0.75 for true/false. Mastery tests deliberately use easier items.
What is the difference between D and the point-biserial correlation?
D uses only the top and bottom groups. The point-biserial correlation compares the item with total scores for every student, so it uses more information. Values of 0.20 or more are usually good, and a negative value is a warning, just as with D.
Why can an easy item not have a high D?
If 95% of students get an item right, nearly everyone in both groups does, so the difference between the groups must be small. The highest-D result shows the ceiling set by the item's difficulty.
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