Teachers & educators
Item analysis calculator.
Difficulty, discrimination (top and bottom 27%) and KR-20 from a student by item grid of right and wrong answers.
Inputs
Result
Enter your numbers.
How do I do an item analysis of a test?
Difficulty (p) is the proportion of students who answered the item correctly, so a higher p means an easier item. The University of Arizona guide treats 0.30 to 0.70 as generally acceptable for items meant to discriminate, and 0.80 to 1.00 as acceptable for mastery items. Discrimination (D) compares the top and bottom 27% of students by total score: D = (correct in upper group − correct in lower group) ÷ group size. It runs from −1 to +1, and 0.30 or more is called highly discriminating. A BYU open textbook treats D below 0.20 as poor. Students are ranked by total score and ties are split by row order, so tied totals can shift D slightly. KR-20 is the reliability coefficient for right/wrong items: (k ÷ (k − 1)) × (1 − Σpq ÷ variance of total scores), where q = 1 − p and the variance divides by the number of students.
Worked example. Six students and five items with totals 5, 4, 3, 2, 1 and 0 have a total-score variance of 2.917 and Σpq of 1.139, so KR-20 = 1.25 × (1 − 1.139 ÷ 2.917) = 0.762.
Frequently asked questions
What is a good discrimination index?
The University of Arizona guide calls 0.30 or more highly discriminating. A BYU open textbook treats under 0.20 as poor, and a negative value means low scorers did better than high scorers on that item, so check the key and the wording.
What does the difficulty index mean?
It is the share of students who got the item right. 0.30 to 0.70 is the usual range for items meant to separate students, and 0.80 to 1.00 is acceptable for mastery items.
Why 27% for the upper and lower groups?
Item analysis guides commonly compare the top 27% and bottom 27% of scorers. With a small class the tool uses at least 1 student per group.
What KR-20 value is acceptable?
The Arizona guide says above 0.50 is acceptable with at least 30 students, 50 items and one correct answer per item. Small classes give unstable values.
Can I use this for essay or partial-credit items?
No. KR-20 and these indices assume each item is scored right or wrong.
Sources
Checked
- University of Arizona College of Medicine Phoenix: Item Analysis
- BYU Open Textbooks: Item Statistics and Analysis (difficulty and discrimination)
- Real Statistics: Kuder and Richardson Formula 20
- Kuder, G. F. and Richardson, M. W. (1937), The theory of the estimation of test reliability, Psychometrika 2(3)
Difficulty and discrimination thresholds are conventions from the sources above, not universal standards. The KR-20 formula is from Kuder and Richardson (1937); variance uses the population (N) denominator. Runs entirely in your browser. Nothing you type is sent or stored.
Scope
Small classes give unstable statistics. Use item analysis to flag items for review, not to throw questions out automatically. Education arithmetic, not professional measurement advice.