Teachers & educators
KR-20 calculator.
Kuder-Richardson Formula 20 from item difficulties and the spread of total scores.
Inputs
Result
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How is KR-20 calculated?
Kuder-Richardson Formula 20 estimates the internal-consistency reliability of a test whose items are scored right or wrong. KR-20 = (k ÷ (k − 1)) × (1 − Σpq ÷ σ²), where k is the number of items, p is the share of students who got an item right, q = 1 − p, and σ² is the variance of total scores. The Real Statistics page uses the population variance and notes either version works if you do not mix them; this tool defaults to population and converts a sample SD for you. It is the same as Cronbach's alpha for right/wrong items. Values above 0.90 can mean a very homogeneous test.
Worked example. Five items with p of 0.667, 0.667, 0.5, 0.333 and 0.333 have Σpq = 1.139. If the total-score variance is 2.917 (SD 1.708), KR-20 = 1.25 × (1 − 0.390) = 0.762.
Frequently asked questions
What is a good KR-20 score?
The University of Arizona guide says above 0.50 is acceptable with at least 30 students, 50 items and one correct answer per item. Short tests and small classes give less stable values.
Where do I get the item difficulties?
Each is the number of students who got that item right divided by the number who answered. The item analysis calculator on this site computes them from a grid of 1s and 0s.
Is KR-20 the same as Cronbach's alpha?
For items scored 1 or 0, yes. Alpha generalizes the idea to partial-credit items.
Why is my KR-20 negative or above 1?
Inputs that do not come from the same test, such as a variance that is too small for the item difficulties, can give odd values. Recheck the spread of total scores.
Sources
Checked
- 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)
- University of Arizona College of Medicine Phoenix: Item Analysis
Formula per Kuder and Richardson (1937) and Real Statistics. The 0.50 threshold is from the University of Arizona guide. Runs entirely in your browser. Nothing you type is sent or stored.
Scope
KR-20 is a property of one set of scores from one group, not of the test forever. Education arithmetic, not professional measurement advice.