Kuder and Richardson Formula 20

The Kuder and Richardson Formula 20 test checks the internal consistency of measurements with dichotomous choices. It is equivalent to performing the split half methodology on all combinations of questions and is applicable when each question is either right or wrong. A correct question scores 1 and an incorrect question scores 0. The test statistic is

Kuder-Richardson formula 20


k = number of questions

pj = number of people in the sample who answered question j correctly

qj = number of people in the sample who didn’t answer question j correctly

σ2 = variance of the total scores of all the people taking the test = VARP(R1) where R1 = array containing the total scores of all the people taking the test.

Values range from 0 to 1. A high value indicates reliability, while too high a value (in excess of .90) indicates a homogeneous test.

Example 1: A questionnaire with 11 questions is administered to 12 students. The results are listed in the upper portion of Figure 1. Determine the reliability of the questionnaire using Kuder and Richardson Formula 20.

Kuder Richardson Excel

Figure 1 – Kuder and Richardson Formula 20 for Example 1

The values of p in row 18 are the percentage of students who answered that question correctly – e.g. the formula in cell B18 is =B16/COUNT(B4:B15). The values of q in row 19 are the percentage of students who answered that question incorrectly – e.g. the formula in cell B19 is =1–B18. The values of pq are simply the product of the p and q values, with the sum given in cell M20.

We can calculate ρKR20 as described in Figure 2.

Kuder Richardson formulas

Figure 2 – Key formulas for worksheet in Figure 1

The value ρKR20 = 0.738 shows that the test has high reliability.

Real Statistics Function: The Real Statistics Resource Pack contains the following supplemental function:

KUDER(R1) = KR20 coefficient for the data in range R1.

Observation: For Example 1, KUDER(B4:L15) = .738.

Observation: Where the questions in a test all have approximately the same difficulty (i.e. the mean score of each question is approximately equal to the mean score of all the questions), then a simplified version of Kuder and Richardson Formula 20 is Kuder and Richardson Formula 21, defined as follows:


where μ is the population mean score (obviously approximated by the observed mean score).

For Example 1,  μ = 69/12 = 5.75, and so


Note that ρKR21 typically underestimates the reliability of a test compared to ρKR20 .

81 Responses to Kuder and Richardson Formula 20

  1. Marc says:

    Dear Charles,
    Thanks a million for these excellent explanations. I conducted an online poll for my MBA Thesis. Now I tried to compute the Cronbach’s Alpha. As it turned out the variance in my case is 0 (cell B21 in your example) because it was not possible to give a wrong answer, e.g. all participants answered all questions.
    As it is not possible to divide through 0 I get an invalid result for the Alpha. My suggestion is to state that the questionnaire provides a perfect reliability. What would be the best way to interpret this fact or did I misunderstand something?
    Thanks in advance,

    • Charles says:

      If there aren’t any wrong answers, it is not clear to me why you even want to use Cronbach’s alpha. I guess in this case you could consider the questionnaire reliable.

  2. applefabela says:

    Sir Charle
    Can’t I ask you about kuder richard formula 20
    what exact meaning about kuder richrd formula 20 thanks

  3. joseph says:

    Hello Charles,
    You are doing a good job. Keep it up. God bless you.

  4. ayu says:

    I have exercises that consist of multiple choice, true-false question and short answear to analyze the scores of this test which is better split-half question or K-20, I tried both I used of split-half Test odd-even and the result unreliable, later I tried to K-20 results are reliable, how come this happened?

    test instruments is valid valid but unreliable , why this condition can occur?

    pls help me with this..thankyou

    • Charles says:

      It is not too surprising in general that two tests would give different results. This is especially true with small samples.
      If you send me an Excel file with your data I can try to understand why you are getting such different results in this case. You can find my email address at Contact Us.

      • Ayu says:

        Thankyou fr yr response,
        I already sent you my excel file to your contact, pls be considerable to check it

        • Charles says:

          I have now looked at the spreadsheet that you sent me. The value I calculated for KR20 = .0677, which agrees with the value you calculated. I calculated a value of -.052 for split-half (odd-even). These values are both very low, indicating unreliability.

  5. charl arl says:

    hi sir charles.

    if i am going to solve it on paper what formula should i use for KR20? because there is so many formula on net and i dont know what to use.. i just need a simple example and process of solving using KR20 for my report, tommorow. and kindly give me another example because your sample is easier to understand than other source on net.. hope that you can help me.. thanks a lot.. God bless.

    • Charles says:

      Hi, the example on the referenced webpage provides all the information you need to calculate KR20. You can also download the Excel spreadsheet for this example from Examples Workbooks so that you can see all the formulas better.

  6. Grace says:

    Hi Sir Charles,

    1. What reliability test can be used for essay type of exam?
    2. If the set of exam is mixed of different types of test (e.g. I. Multiple Choice, II. True or False, III. Fill in the blanks, IV. Enumeration, V. Essay), will it be ok to still use the KR-20? How will it be done, will it be per type of exam or will it be as total of all question? If not, what reliability test is more appropriate to use?
    3. For enumeration type of exam, how will the KR-20 be done? (e.g. Q1 enumerate 5 things, Q2 enumerate 3 things, Q3 enumerate 4 things) Will k = 3 or k = 12?

    Thanks in advance.

    • Charles says:

      1. You need to be able to assign a score of 1 for correct and 0 for incorrect you can use KR20 or any of the other commonly used reliability tests.
      2. As long as all the questions are testing the same thing and can be coded 0 or 1 as described above, you can use KR20. If you have 20 questions that test one thing and 20 questions that test something else, then you should create two separate KR20 measurements
      3. It depends on how you code correct vs. incorrect. If for example, a person is given 5 words to recall and you define “correct” as they recall all five (or 4 out of 5), then this counts as one question. If they get one point for each word they correctly recall then this counts as 5 questions. In this latter case, you have the further problem that the order of the words will be relevant, plus you have the issue of how to score a response of a word that wasn’t even on the original list.

  7. Adrian says:

    i got the result of my KR21 is 0.983
    what does it mean?
    do you have a list to show whether the score has high/medium/low reliability?
    it would be very helpful..thank you, Charles.

    • Charles says:

      The result indicates extremely high reliability (almost the maximum value), but this is not necessarily good since a value this high indicates homogeneity of questions (i.e. there isn’t much difference between the questions).

  8. Josef says:

    100 Schüler
    14 Fragen
    1123 positive Antworten
    sd² = 2,1587
    KR-21 = -0,0316

    Wie interpretiert man negatives Ergebnis von KR-21?

    • Charles says:

      Negative value for KR-21, KR-20 and Cronbach’s Alpha can happen, although in practice they are not so common. These are generally signs that reliability is very very poor.

  9. Yijun says:

    I’ve tried your formula, but it turns out the KR-20 can be negative if I randomly generate students’ correctness. Can you explain why? According to Wiki, KR-20 range from 0-1, my data is as follow:
    Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11
    1 0 0 0 1 0 0 1 1 0 0 0 3
    2 0 1 1 0 0 0 1 1 0 0 1 5
    3 0 0 1 1 0 0 0 1 1 1 1 6
    4 1 1 0 0 0 0 1 1 1 1 0 6
    5 1 0 1 1 0 0 0 1 0 0 1 5
    6 1 0 0 1 0 1 0 0 1 0 1 5
    7 1 1 1 1 0 1 0 0 0 0 1 6
    8 0 0 0 0 1 1 1 0 0 0 0 3
    9 1 1 1 1 0 0 0 0 0 0 0 4
    10 1 0 0 1 0 1 0 0 1 0 0 4
    11 1 0 1 1 1 1 0 0 0 0 1 6
    12 0 0 0 0 1 0 1 0 0 1 1 4
    Total 7 4 6 8 3 5 5 5 4 3 7 57

    • Charles says:

      The simple answer is that KR-20 can indeed be negative. This is not so common in practice but it can happen.

  10. Asad says:

    Hello Dear Charles,

    I would like to ask you that can we calculate KR 20 by using SPSS?kind provide some reference.

  11. erie says:

    is KR-20 just for multiple choice question?

    • Charles says:

      KR-20 is not just for multiple choice questions, but it is just for questions which take only two values (e.g. 1 = correct, 0 = incorrect). This includes True/False questions as well.

  12. Justin says:

    What do I do when I only have a single question which is like this?

    • Charles says:

      Cronbach’s alpha can be used even when the questions take a value 1 (say for correct) and 0 (say for incorrect). This would cover True/False and multiple choice questions. The problem is that you can’t use Cronbach’s alpha with only one question. After all, Cronbach’s alpha measures internal consistency, but withonly one question there is no internal consistency to measure. The same is true for Kuder and Richardson.

      • Asad says:

        Hello Dear Charles,

        Kindly give reference of this

        • Charles says:

          There are lots of references for the KR20 test. Most statistic textbooks as well as many websites on the Internet. The original paper is

          Kuder, G. F., & Richardson, M. W. (1937). The theory of the estimation of test reliability. Psychometrika, 2(3), 151–160.


  13. vangie says:

    Hello sir. Thank you for the above info. Very helpful indeed. Please help me how you came up with variance or 02 of 6.52083. I just cant figure it out. Thank you sir!

    • Charles says:

      The formula to calculate 6.52083 is shown in Figure 2. I use the VARP function for all the variances. If I had used VAR I would get the same result.

  14. Godspower says:

    Pls Charles, can you break it down for me, pls, if possible use figure frm the table.
    How did you work VARP(M4:M15)?

  15. Godspower says:

    Pls i stil don’t understand how the variance was calculated. i.e B24

  16. Lizzy says:

    Thanks for your explanation on KR20. Please I calculated the reliability of my research instrument and I got KR20 = 0.65. Can I call this a high reliability? If no, what should be done to the instrument.
    Thanks. I will appreciate your quick response please.

    • Charles says:


      There isn’t complete agreement as to what constitutes “high reliability”, but here are my observations about Cronbach’s alpha. These apply to KR20 as well (since the index produced are identical):

      A commonly-accepted rule of thumb is that an alpha of 0.7 (some say 0.6) indicates acceptable reliability, and 0.8 or higher indicates good reliability. Very high reliability (0.95 or higher) is not necessarily desirable, as this indicates that the items may be entirely redundant. These are only guidelines and the actual value of Cronbach’s alpha will depend on many things. E.g. as the number of items increases, Cronbach’s alpha tends to increase too even without any increase in internal consistency.


  17. Tony says:

    Charles, I was wondering if you could look at a screen-shot I have provided and help explain the results to me. I have many different content areas to work on, but in the following example, I have a 10 question dichotomous exam which was given to about 11,300 students. I am using all the results from these exams to find p, which, is quite low. What I do not understand, is that the higher my variance is, the better my p value, or, the lower my Σpq the higher my p.

    KR-20 Example

    Obviously, I have a bias; and am sure that our questions are reliable, but, the numbers appear to say otherwise. The only other thing I can think of, is that our exam is random. So while we have 10 questions per exam, Q1 may be different than the next student’s Q1, although the question is designed to measure the same knoweldge of the given topic. Do I need to use this formula with a static exam? Where Q1 is the exact same question for all student’s?

    Your assistance is much appreciated, Thanks

    • Charles says:

      KR20 and Cronbach’s Alpha assume that Q1 is the same for all the students. If this is not the case, then I wouldn’t rely on the KR20 or Cronbach’s alpha measures.

      • Tony says:

        Thank you Charles. Due to the fact, that we provide randomization w/ our questions, would you be able to provide a plausible approach for measuring the reliability/consistency of our examinations? When we run a t-test; our results indicate a high probability that our results correlate. Aside from Pearson’s R/Spearman (which can change based on how we collect/sort data) is there another method we can use to prove internal consistency?

  18. Mark Campbell says:

    Question should read “What does it mean?”

  19. Mark Campbell says:

    We gave the exact same 100 question exam to students in consecutive classes (1000 students in both 2013 and 2014). The calculated KR21 values for each class were equal to within 3 digits (0.8763 vs 0.8765). I assume this tells me something about the student populations. Does does it mean?

    • Charles says:

      The results seem to show high internal consistency of the exam. If the 1,000 students are independent then the similar KR21 scores is not surprising.

  20. Umut says:

    Dear Dr. Charles,
    First of all I would like to express my gratitude for your precious efforts. This page is a gold mine for people like me.

    I wish to use Kuder and Richardson Formula for a number of tests for language learning. Do you think I should use any other formulas or would KR-20 fit for my purposes.

    Here are my tests:

    Test A: Translation
    1- DOG: …………….. (Write the meaning in your first language)

    Test B: Multiple Choice
    Bow Wow – What animal is this?
    a) CAT b)DOG c)…… etc.

    Test C: Fill in the blank
    A …………. barks.

    • Charles says:

      KR20 only supports dichotomous questions (e.g. 1 for correct answer and 0 for incorrect answer). This seems to be the case with the three questions you are posing, and so KR20 could be used. You can also use Cronbach’s Alpha.

      Whether KR20 (or Cronbach’s Alpha) is a fit for your purposes depends of course on what you are trying to do (which is not stated in your comment).


      • Umut says:

        Thank you Charles, I really appreciate your response. I aim to measure learners existing vocabulary knowledge on some technical engineering vocabulary the above examples have been given to show you my test formats.

  21. Jarvis says:

    Thanks Charles for this information. I do have a question; is it possible to calculate the KR20 using the split-half method? When I attempt, I’m always getting a number higher than 0-1 ..i.e… 1.9, etc.

    • Charles says:

      I don’t understand what it means to calculate KR20 using the split-half method or why you would want to do so. If you calculate KR20 as described on the referenced webpage you should get a value no higher than 1.

  22. madan says:

    dear admin,
    i just want to ask how to calculate the varices just give me the example

  23. judy says:

    good evening.. just wana ask ..how did you get the answer in the example shown above , where VAR is equal to 6.52083.. please. thankyou so much :) :)

  24. Raphaela says:

    Hello Charles,
    How could I know if the coeficient KR20 is high or low? could you send me a reference from any author?

  25. Salima says:

    hi, thanks, the information was worthy.
    What if I want to test the reliability of a self-designed questionnaire to test participant’s knowledge and attitude on some theme/subject like hepatitis, questionnaire has multiple choices however no choice is right.

    • Charles says:

      If no choice is right, how do you evaluate the answers given? Please provide more information.

  26. azlina says:

    evening dr charles, thank you for your great guidance on KR20. it helps.
    But, I would like to ask, to test realibility, there is another test on Pearson correlation to test on exam scores reability. Which is the better test ? -thank you

    • Charles says:

      I don’t completely understand your question, but Cronbach’s alpha is often used instead of KR20 to test releiability.

  27. farida shullai says:

    Hi, can I usd KR20 for calculate a test which is someehat like a likert scale.? If yes how do I go about it.

  28. Robert says:

    Good day, just wanna ask, at what value of KR 20 would we start to interpret as having low or no correlation? Is there a value that we can say this items have low, moderate or high correlation?

    • Charles says:

      Hello Robert,

      I have usually seen that a value of at least.70 is desired for most exams. The following are the criteria used by Imran Zafar in http://com.ksau-hs.edu.sa/eng/images/DME_Fact_Sheets/fs_24.doc


      Reliability Interpretation
      .90 and above Excellent reliability; at the level of the best standardized tests
      .80 – .90 Very good for a classroom test
      .70 – .80 Good for a classroom test; in the range of most. There are probably a few items which could be improved.
      .60 – .70 Somewhat low. This test needs to be supplemented by other measures (e.g., more tests) to determine grades. There are probably some items which could be improved.
      .50 – .60 Suggests need for revision of test, unless it is quite short (ten or fewer items). The test definitely needs to be supplemented by other measures (e.g., more tests) for grading.
      .50 or below Questionable reliability. This test should not contribute heavily to the course grade, and it needs revision.

  29. zuraidah nordin says:

    can i use KR20 for ‘fill in the blank questionnaire’, which could be given either right or wrong answer.

  30. Christine says:

    Thank you so much for the information. That is a great help!

  31. Helen says:

    Can you direct me on the use of LDC? Apparently it is used when the KR20 is low (i.e. below .80). I would like a reference to read, some understanding of the assumptions upon which it is based, and why it is used.

  32. badrus says:

    hi, there could i ask about the different spesific usage between: Kr20 & Kr21
    Cz im so confusedabout that ?canu help me? please reply @ my e,mail badrus_barokah21@yahoo.com Or twitter : @bad_lucky

    • Charles says:

      Hi Badrus,
      I have now updated the Kuder and Richardson Formula 20 webpage to also show how to calculate KR21 and when it can be used. Please look at that page for help in answering your question. Frankly KR21 is simply an approximation of KR20, and so it is of limited use.

  33. Tatiana says:

    Hello, Thank you very much for providing this example.
    Above you note that:
    A high value indicates reliability, while too high a value (in excess of .90) indicates a homogeneous test.

    What do you mean by ‘homogeneous test’ is this a problem? If you are aware of a source explaining this issue, could you please provide a reference?
    Thank you.

    • Charles says:

      By homogeneous test I am referring to tests which are likely testing the same thing. E.g. if I design a questionnaire with two questions whose sample data has a correlation of .98, it may be presumed that I don’t need to retain both questions since both are pretty much testing the same thing. In this case I might be justified in dropping one of the two questions from the questionnaire. The Cronbach’s Alpha page of the website (http://www.real-statistics.com/reliability/cronbachs-alpha/) also provides some information about this issue.

  34. jeeha says:

    under what condtions kudar richardson formula is applied and what are its different forms?

    • Charles says:

      Dear Jeeha,
      This test is used when there two choices for each answer: correct and incorrect. It can be applied for example for multiple choice tests even with say 4 choices. Here the wrong answers count as 0 and the right answer counts as 1. If you have non-dichotomous questions (e.g. with a Likert scale) you should use Cronbach’s alpha instead.

  35. abbey says:


    k 7
    Epq 0.888888889
    var 2.888888889
    P 0.807692308

    Does this mean that we have high reliability? thank you!

  36. Tycho says:

    Hi admin!
    if pj = number of people in the sample who answered question j “correctly”
    Would this be applicable to test with likert scale as scoring too? Since there will be no correct answers if this would be the case.
    Thank you 😀

  37. Mike says:

    Thank you for this. It will be a great help if you provide the Excel file in the example as well.

Leave a Reply

Your email address will not be published. Required fields are marked *