Kuder-Richardson 20 (KR-20) & 21 (KR-21)

Statistics Definitions > Kuder-Richardson Formula 20 (KR-20)

What is the Kuder-Richardson 20?

KR-20/KR-20 are measures of test reliability,
Kuder-Richardson Formula 20, or KR-20, is a measure reliability for a test with binary variables (i.e. answers that are right or wrong). Reliability refers to how consistent the results from the test are, or how well the test is actually measuring what you want it to measure.

The KR20 is used for items that have varying difficulty. For example, some items might be very easy, others more challenging. It should only be used if there is a correct answer for each question — it shouldn’t be used for questions with partial credit is possible or for scales like the Likert Scale.

  • If all questions in your binary test are equally challenging, use the KR-21 (see below).
  • If you have a test with more than two answer possibilities (or opportunities for partial credit), use Cronbach’s Alpha instead.

KR-20 Scores

The scores for KR-20 range from 0 to 1, where 0 is no reliability and 1 is perfect reliability. The closer the score is to 1, the more reliable the test. Just what constitites an “acceptable” KR-20 score depends on the type of test. In general, a score of above .5 is usually considered reasonable.

Apply the following formula once for each item:

KR-20 is [n/n-1] * [1-(Σp*q)/Var]


  • n = sample size for the test,
  • Var = variance for the test,
  • p = proportion of people passing the item,
  • q = proportion of people failing the item.
  • Σ = sum up (add up). In other words, multiple Each question’s p by q, and then add them all up.If you have 10 items, you’ll multiply p*q ten times, then you’ll add those ten items up to get a total.

As this can quickly get tedious for tests with a large amount of items, it’s usually calculated with some type of software like Excel. Dr. Katrina Korb put together a great PowerPoint showing the steps in Excel for calculating the Kuder-Richardson 20. You can download it RS 12 Calculating Reliability of a Measure.


The KR-21 is similar, except it’s used for a test where the items are all about the same difficulty.
The formula is [n/(n-1) * [1-(M*(n-M)/(n*Var))]

  • n= sample size,
  • Var= variance for the test,
  • M = mean score for the test.

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