Kuder Richardson Coefficient of Reliabilityfor Binary Data Explained
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The Kuder Richardson Coefficient of reliability (K-R 20) is used to test the reliability of binary measurements such as exam questions, to see if the items within the instruments obtained the same binary (no/yes, right/wrong) results over a population of testing subjects.

The formula for the coefficient can easily be obtained from Wikipedia on the Internet.

Please Note that the K-R 20 was developed in 1937. Hoyt in 1940 modified the formula so that it can be applied to measurements that are not binary. Hoyt's modification eventually was popularised and is now known as Cronbach's Alpha.

Cronbach's Alpha, when applied to binary data, will produce the same value as K-R 20. As Alpha has wider applicability, it has increasingly replaced K-R 20 as a measurement of agreement or internal consistency.

### Data entry

The program Kuder_Pgm.php provides a single analysis of Kuder Richardson's coefficients.

The input data is a matrix of values, the columns representing questions or items in a batch of tests, and the rows representing the testing subjects. Each cell contains either 0 or 1, representing correct/incorrect, yes/no, and so on. All cells in the table must contain data.

A coefficient of 0.9 or more indicates a homogenous set of data.

### An Example

The use of the method and interpretation of results are best demonstrated by the example used for the default template data in Kuder_Pgm.php.

T1T2T3T4
Student 1wrongcorrectcorrectwrong
Student 2correctcorrectcorrectcorrect
Student 3wrongcorrectwrongwrong
Student 4wrongwrongcorrectwrong
Student 5correctcorrectcorrectcorrect

We have 4 multiple choice questions (T1 to T4), administered to 5 students. 0 represents wrong answer and 1 correct answer, as shown in the table on the left.

 1 1 1 1 0 1 0 0 0 0 1 0 1 1 1 1

The data set to be used is as shown in the table to the right, and the results are K-R 20 = 0.7536

### References

Kuder, G. F. ; M. W. Richardson (1937)The theory of the estimation of test reliability. Psychometrika, 2: 151-60

Brunning JL, Kintz BL (1996) Computational Handbook of Statistics 4th Edition.Addison-Wesley Educational Publishers Inc, New York. ISBN 0-673-99085-0 p 78-81.

Feldt LS. The approximate sampling distribution of Kuder-Richardson reliability coefficient twenty. Psychometrika 1965;30:357-371.

Andrich D. (1982) An Index of Person Separation in Latent Trait Theory, the Traditional KR-20 Index, and the Guttman Scale Response Pattern. Education Research and Perspectives, 9:1, 95-104.

For those who want to write their own program but have difficulties in getting hold of old journal articles or textbooks, the formulae are on the www, K-R 20 on Wiki