Test Reliability Calculator
Estimate a test’s reliability with KR-21 from the mean and spread of scores, or Cronbach’s alpha from item and total variances, and its standard error of measurement.
Result
Enter mean total score and standard deviation of total scores to see the result.
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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 Test Reliability Calculator
- Choose KR-21 if your items are scored right or wrong and you have only the mean and standard deviation of the total scores. Choose Cronbach’s alpha if you have the variance of each item’s scores.
- Enter the number of items.
- For KR-21, enter the mean total score. For alpha, enter the sum of the item variances.
- Enter the standard deviation of the total scores, computed the same way as the item variances.
- Read the reliability coefficient and the standard error of measurement.
The formulas
KR-21 r = k ÷ (k − 1) × (1 − M × (k − M) ÷ (k × SD²))alpha α = k ÷ (k − 1) × (1 − Σ item variances ÷ SD²)SEM = SD × √(1 − r)k is the number of items, M the mean total score and SD the standard deviation of the total scores. KR-21 is equation 21 of Kuder and Richardson’s 1937 paper, “The theory of the estimation of test reliability”. Alpha is formula 2 of Cronbach’s 1951 paper, “Coefficient alpha and the internal structure of tests”. The standard error of measurement follows Harvill’s instructional module for the National Council on Measurement in Education (1991).
Entries that cannot happen
For items scored right or wrong, the variance of the total scores can never exceed M × (k − M), so KR-21 refuses a standard deviation larger than that. For any items, the total variance can never exceed k times the sum of the item variances, so alpha refuses that too. Either usually means a figure was typed in the wrong box or in the wrong units.
Reading the result
The page does not label a coefficient as acceptable or not. Guidance differs with the purpose of the test, and a short classroom quiz and a certification exam are held to different standards. KR-21 assumes equally difficult items, so on a test whose items vary it understates reliability; alpha from item variances avoids that assumption. The item analysis calculator looks at individual questions.
With your numbers
- KR-21 = 50 ÷ 49 × (1 − 35 × (50 − 35) ÷ (50 × 6²))
- SEM = 6 × √(1 − 0.723) = 3.16
- Reliability = 0.723
Quick reference
An estimate from one sitting of one test. KR-21 reads lower than KR-20 when items differ in difficulty.
| Example | Reliability coefficient |
|---|---|
| KR-21: 50 items, mean 35, SD 6 | 0.723 |
| KR-21: 40 items, mean 28, SD 5 | 0.681 |
| KR-21: 20 items, mean 14, SD 3.5 | 0.692 |
| KR-21: 100 items, mean 70, SD 12 | 0.863 |
| Alpha: 10 items, item variances 2.0, SD 3 | 0.864 |
| Alpha: 25 items, item variances 5.5, SD 7.2 | 0.931 |
| Alpha: 5 items, item variances 1.0, SD 1.53 | 0.716 |
Common questions about the Test Reliability Calculator
- What is KR-21?
- Kuder-Richardson formula 21, published by G. F. Kuder and M. W. Richardson in Psychometrika in 1937. It estimates a test’s reliability from just the number of items and the mean and standard deviation of the scores, by assuming every item is equally difficult. When items differ in difficulty it gives a lower figure than KR-20, which needs each item’s difficulty.
- How is Cronbach’s alpha calculated?
- Alpha = k ÷ (k − 1) × (1 − sum of the item variances ÷ variance of the total scores), where k is the number of items. Lee Cronbach set it out in Psychometrika in 1951. For items scored right or wrong, each item’s variance is p × (1 − p) and alpha equals KR-20.
- What is the standard error of measurement?
- An estimate of how far observed scores typically sit from the scores a student would get on a perfectly reliable test: SEM = SD × √(1 − reliability). It is in score points, so a teacher can read it against a pass mark or a grade boundary.
- Should I use the sample or population standard deviation?
- Either, as long as the item variances and the total standard deviation are computed the same way. Mixing them skews alpha, especially in a small class. KR-21 uses the standard deviation you enter throughout.
- Why is my coefficient negative?
- Alpha and KR-21 turn negative when the items vary more on their own than the total scores do together, which happens when items do not measure one thing, or when an entry is mistyped. The page reports the number and leaves out the standard error, which has no meaning for a negative coefficient.