KR-21 Test Reliability Calculator
Enter the number of items, mean score, and score standard deviation to estimate KR-21 reliability. The calculator also shows SEM intervals for one observed score and a Spearman-Brown estimate for reaching a target reliability.
KR-21 reliability coefficient0.68
- Standard error of measurement
- 2.82
- Approximate 68% score interval
- 27.18 to 32.82 points
- Approximate 95% score interval
- 24.47 to 35.53 points
- Estimated total items for target reliability
- 75
- Estimated additional items needed
- 35
- Projected reliability after adding items
- 0.8
- Result note
- The added-item estimate assumes comparable parallel items.
KR-21 assumes dichotomously scored items with broadly similar difficulty.
How to use this calculator
- Enter the number of dichotomously scored items on the test.
- Enter the group mean and standard deviation for total test scores.
- Enter one observed score to see approximate measurement-error bands.
- Choose the reliability target for the added-item estimate.
- Review the notes before using the Spearman-Brown projection.
How KR-21 is calculated
KR-21 is a summary-statistics reliability estimate for tests made of right-or-wrong items. It uses only the number of items, the mean total score, and the variance of total scores, so it is useful when you do not have item-level response data.
The formula is:
r = [k / (k - 1)] x [1 - M(k - M) / (k s^2)]
- k is the number of test items.
- M is the mean total score.
- s is the standard deviation of total scores.
- s^2 is the total-score variance.
After the coefficient is estimated, the standard error of measurement is SEM = s x sqrt(1 - r). The score bands shown by the tool are the observed score plus or minus one SEM for an approximate 68% interval and plus or minus 1.96 SEMs for an approximate 95% interval. Displayed endpoints are limited to the possible score range, but the underlying normal approximation can extend below zero or above the maximum score.
What changes the result most
KR-21 rises when total scores vary more relative to the expected binomial item-score spread implied by the mean. A longer test can also raise reliability, but only if the added items behave like parallel measures of the same construct. A very easy or very hard test can show lower reliability for a particular group because scores have less room to separate examinees.
How the added-item estimate works
When the current coefficient is positive and below the target, the calculator applies the Spearman-Brown prophecy formula. The length multiplier is L = t(1 - r) / [r(1 - t)], where t is the target reliability. The estimated target length is ceil(k x L), and the added items are that rounded length minus the current item count.
What this calculator leaves out
KR-21 does not inspect individual items. It cannot show whether one question is miskeyed, whether distractors are weak, whether item difficulties vary widely, or whether the test has more than one dimension. KR-20, coefficient alpha, item response theory, and item-analysis reports can answer some of those questions when item-level data are available.
Worked example
Suppose a 40-item test has a mean score of 28, a total-score standard deviation of 5, and a student observed score of 30. The KR-21 estimate is 0.681, and the SEM is about 2.82 points. The approximate 68% score interval is 27.18 to 32.82 points, and the approximate 95% interval is 24.47 to 35.53 points.
With a target reliability of 0.80, the Spearman-Brown length multiplier is about 1.873. Rounding up gives an estimated total length of 75 comparable items, or 35 additional items. Using the rounded length, the projected reliability is about 0.800.
Common questions
What does a KR-21 reliability coefficient mean?
It estimates how consistently a right-or-wrong test ranks examinees in the group represented by the entered scores. Values closer to 1 mean less random measurement error, while values near zero or below mean the summary data do not support a useful reliability estimate.
What is the difference between KR-20 and KR-21?
KR-20 uses item-level difficulty information for each question. KR-21 uses only the test length, mean, and total-score standard deviation, so it is easier to compute but depends on stronger assumptions about similar item difficulty.
When is KR-21 appropriate for a test?
KR-21 is appropriate for dichotomously scored tests when items are intended to measure one construct and have broadly similar difficulty. It is best treated as a quick screening estimate rather than a full item analysis.
Why can KR-21 be lower than KR-20?
KR-21 assumes item difficulties are similar. When a real test has a wider mix of easy and hard items, that assumption can pull the estimate downward compared with KR-20, which uses each item difficulty directly.
How is standard error of measurement related to reliability?
SEM combines the observed score standard deviation with the reliability coefficient. For the same score spread, higher reliability gives a smaller SEM, which means narrower approximate score intervals around an observed score.
How many questions are needed to improve test reliability?
The Spearman-Brown estimate gives a test-length target based on the current reliability and desired reliability. It assumes added questions are parallel in quality and content, so weak or off-topic questions may not improve reliability as predicted.