Item Analysis Calculator
Work out a test question’s difficulty and discrimination index from how many students in the top and bottom groups answered it correctly.
Result
Enter students who took the test, correct answers in the upper group and correct answers in the lower group to see the result.
https://calcgallery.com/education/item-analysis-calculator/
Academic results are arithmetic estimates based on the values you enter. Your syllabus, school policies, transcript and official degree audit control actual grades and progress. Confirm important figures with your instructor, adviser or registrar. Read the full disclaimer
How to use the Item Analysis Calculator
- Enter how many students took the test.
- Rank the papers by total score. Keep the top 27% as the upper group and the bottom 27% as the lower group; the page tells you how many that is. Or choose your own group size.
- For the question you are checking, count the correct answers in the upper group and in the lower group.
- Optionally enter the correct answers in the whole class, for the difficulty index of everyone.
- Read the discrimination index D and the difficulty index p.
The indices
Classical item analysis describes each test question with two numbers. The difficulty index is the share of students who got it right. The discrimination index compares the top of the class with the bottom:
group size = 27% of the students, rounded to a whole studentp upper = correct in the upper group ÷ group sizep lower = correct in the lower group ÷ group sizeD = p upper − p lowerdifficulty p = correct in the whole class ÷ studentsA question that stronger students answer correctly and weaker students miss has a D near 1. One that everyone gets right, or everyone gets wrong, has a D near 0, whatever its quality.
Kelley’s 27 percent
The upper and lower groups come from Truman L. Kelley’s paper “The selection of upper and lower groups for the validation of test items” in the Journal of Educational Psychology (1939). His conclusion was that groups of 27% from each end of the score distribution are optimal for studying items, provided the differences in score within each group are not used. For a class of 30, that is 8 students at each end.
Reading the numbers
The page shows the numbers without labels such as good or poor. Published cut-offs differ between textbooks, and a single test in a small class is a noisy sample: in a group of eight, one student changes p upper by 0.125. Looking at the question itself, its wording and its distractors, tells you more than any cut-off. The test reliability calculator covers the test as a whole.
With your numbers
- Group size = 27% of 30, rounded = 8
- p upper = 7 ÷ 8 = 0.875; p lower = 3 ÷ 8 = 0.375
- D = 0.875 − 0.375 = 0.500
Quick reference
A summary of one item on one test. Small classes make both indices jump with a single answer.
| Example | Discrimination index D |
|---|---|
| 30 students, upper 7, lower 3 | 0.500 |
| 40 students, upper 9, lower 4 | 0.455 |
| 25 students, upper 7, lower 0 | 1.000 |
| 100 students, upper 20, lower 22 | -0.074 |
| 20 students, groups of 10, upper 8, lower 5 | 0.300 |
| 60 students, upper 14, lower 8, 35 right | 0.375 |
Where the numbers come from
The fixed figures this calculator relies on, and the published source each one was checked against. How the calculators are built and checked
1 source
Kelley TL, "The selection of upper and lower groups for the validation of test items", Journal of Educational Psychology 30(1):17-24 (1939), doi:10.1037/h0057123, p. 24, conclusion 2: "Upper and lower groups consisting of twenty-seven per cent from the extremes of the criterion score distribution are optimal for the study of test items, provided the differences in criterion scores among the members of each group are not utilized"; p. 23 calls 25 to 27 per cent serviceable. Read from the Internet Archive scan of the January 1939 issue, https://archive.org/details/sim_journal-of-educational-psychology_1939-01_30_1
Used for
- Share of test takers placed in each of the upper and lower groups for item analysis.
Common questions about the Item Analysis Calculator
- How do I calculate the discrimination index?
- Rank the students by total score and take equal-sized groups from the top and bottom. For one question, divide the number correct in the upper group by the group size, do the same for the lower group, and subtract: D = p upper − p lower. D runs from −1 to 1.
- Why 27 percent?
- In 1939 Truman Kelley showed that, for a normally distributed score, upper and lower groups of 27% from each end are the best trade-off between having groups that clearly differ and having enough students in each. He also found 25% to 27% serviceable in general. With a small class the page rounds 27% to a whole number of students; you can choose your own group size instead.
- What is the difficulty index?
- The share of students who answered the question correctly, p. Despite the name, a higher p means an easier question. With the whole-class count, the page gives p for everyone; from the groups alone it gives p for the two groups combined, which leaves out the middle of the class.
- What does a negative D mean?
- More students in the lower group answered correctly than in the upper group. That is worth a second look at the question and its answer key. The page reports the number and does not grade the item.
- Can I get the point-biserial correlation here?
- No. The point-biserial needs each student’s total score and whether they got the item right. The page works from group counts, which is what a teacher tallying by hand has.