KR-20 Test Reliability Calculator
Enter item-level p × q totals and the population spread of total scores to estimate KR-20 reliability. The calculator also gives a standard error of measurement and a bounded interval around one observed score when the coefficient is valid.
KR-20 reliability0.7
- Standard error of measurement
- 1.64
- Observed-score confidence interval
- 10.79 to 17.21 score points
- Estimated error variance
- 2.68
- Estimated true-score variance
- 6.32
KR-20 uses dichotomously scored items and the population standard deviation of total scores.
How to use this calculator
- Enter the number of right-or-wrong items included in every student’s total score.
- Add p × q for every item, where p is the proportion correct and q is one minus p.
- Enter the population standard deviation of total scores from the same group, using STDEV.P rather than STDEV.S.
- Enter an observed raw score and choose the confidence level for its measurement-error interval.
- Review any result note before interpreting a negative coefficient or a coefficient above one.
How KR-20 is calculated
KR-20 estimates internal consistency for a test whose items are scored right or wrong. It asks whether the items, taken together, distinguish among examinees consistently in the group whose results you entered. Unlike KR-21, it uses each item’s observed difficulty through the p × q term.
The calculation is r = [k / (k - 1)] × [1 - S / variance(X)], where S = sum(p × q) and variance(X) = SD(X)^2.
- k is the number of dichotomously scored items.
- p is the proportion of examinees who answered one item correctly.
- q is the proportion who answered that item incorrectly, or
1 - p. - S is the sum of every item’s p × q value.
- SD(X) is the population standard deviation of total test scores.
Use population, not sample, variance: in a spreadsheet that is usually VAR.P or STDEV.P. The standard error of measurement is SEM = SD(X) × sqrt(1 - r). For a coefficient from zero through one, the calculator applies the selected normal multiplier to SEM, then limits the score interval to zero through the number of items.
What moves the estimate
The total-score variance and the item p × q total move KR-20 most. Items near 50% correct have p × q near 0.25, while very easy or very hard items contribute less. A test can have low reliability when its scores are tightly clustered, even if the content is sound, because it has little observed variation to separate examinees. Adding comparable items can help, but only when they measure the same construct.
How to interpret unusual values
A negative KR-20 is displayed rather than hidden. It can arise from miscoding, items measuring different things, errors in the p × q total, or noisy results from a small or unusual sample. A value above one means the entered summaries are internally inconsistent, so the calculator does not estimate SEM or a score interval. Neither result should be rounded away.
What this calculator leaves out
KR-20 is not evidence that a test is valid, fair, unidimensional, or suitable for a decision. It does not diagnose missing responses, partial-credit scoring, speededness, local item dependence, differential item functioning, or changes across testing occasions. Review item analyses and the intended use of scores alongside this one reliability estimate.
Worked example
Suppose a 20-item test has a sum of item p × q values of 3.00 and a population total-score standard deviation of 3.00. Total-score variance is 9.00, so KR-20 is [20 / 19] × [1 - 3 / 9], or 0.702. The estimated error variance is 2.68 and estimated true-score variance is 6.32.
For a student with an observed score of 14, the SEM is 1.64 points. At 95%, multiplying SEM by 1.960 gives a margin of about 3.21 points, so the displayed interval is 10.79 to 17.21 score points. It stays within the feasible zero-to-20 score range.
Common questions
What does KR-20 measure?
KR-20 estimates the internal consistency of a test made of dichotomously scored items, such as correct-or-incorrect questions. For the examined group, a higher value generally means less random measurement error in total scores. It does not establish that the test measures the intended construct.
How do I calculate p and q for each question?
Divide the number of examinees answering an item correctly by the number who answered it to get p. Then calculate q as one minus p and multiply p by q. Add that product across all included items before using this calculator.
Should total-score variance use VAR.P or VAR.S?
Use VAR.P, or STDEV.P if you enter the standard deviation, for the population formula used here. If your examinees are instead a sample used to estimate a larger population, document that choice and use a consistent reliability procedure. Changing the variance convention changes the coefficient.
What is the difference between KR-20 and KR-21?
KR-20 uses the observed p and q for every item. KR-21 uses only test length, the mean score, and total-score variance, so it makes a stronger similar-item-difficulty assumption. KR-20 is generally preferable when item-level response summaries are available.
Is KR-20 the same as Cronbach’s alpha?
For dichotomously scored items, KR-20 is algebraically equivalent to Cronbach’s alpha. The names point to the same internal-consistency calculation under that scoring condition. Alpha itself still does not prove unidimensionality or validity.
Can KR-20 be negative?
Yes. A negative result means the input summaries produce more inconsistency than the model permits for a useful reliability coefficient. Check scoring keys and data handling, then consider whether the items are heterogeneous or the sample is too unstable for this estimate.