Kuder-Richardson Formula 20 (Definition & Example)

The Kuder-Richardson Formula 20, often abbreviated KR-20, is used to measure the internal consistency reliability of a test in which each question only has two answers: right or wrong.

The Kuder-Richardson Formula 20 is as follows:

KR-20 = (k / (k-1)) * (1 – Σpjqj / σ2)


  • k: Total number of questions
  • pj: Proportion of individuals who answered question j correctly
  • qj: Proportion of individuals who answered question j incorrectly
  • σ2: Variance of scores for all individuals who took the test

The value for KR-20 ranges from 0 to 1, with higher values indicating higher reliability.

The following example shows how to calculate the value for KR-20 in practice.

Example: Calculating Kuder-Richardson Formula 20

Suppose we administer a test with 7 questions to 10 students.

The results of the test are listed below in Excel, with 1 indicating a correct answer and 0 indicating an incorrect answer:

The following screenshot shows how to calculate the KR-20 value for this test:

Here are the formulas used in various cells:

  • B13: =SUM(B2:B11) / 10
  • B14: =1-B13
  • B15: =B13*B14
  • B17: =COUNTA(B1:H1)
  • B18: =SUM(B15:H15)
  • B19: =VAR.S(I2:I11)
  • B20: =(B17/(B17-1))*(1-B18/B19)

The KR-20 value turns out to be 0.0603.

Since this value is extremely low, this indicates that the test has low reliability.

This means the questions may need to be re-written or re-phrased in such a way that the reliability of the test can be increased.

Additional Resources

The following tutorials provide explanations of terms commonly used when assessing the validity of tests and questionnaires:

What is Internal Consistency?
What is Reliability Analysis?

2 Replies to “Kuder-Richardson Formula 20 (Definition & Example)”

  1. thank you for providing this source. i am chemistry teacher without statistic background. this page help me a lot!

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