# Kuder–Richardson Formula 20

In psychometrics, the Kuder–Richardson Formula 20 (KR-20), first published in 1937,[1] is a measure of internal consistency reliability for measures with dichotomous choices. It was developed by Kuder and Richardson. It is a special case of Cronbach's α, computed for dichotomous scores.[2][3] It is often claimed that a high KR-20 coefficient (e.g., > 0.90) indicates a homogeneous test. However, like Cronbach's α, homogeneity (that is, unidimensionality) is actually an assumption, not a conclusion, of reliability coefficients. It is possible, for example, to have a high KR-20 with a multidimensional scale, especially with a large number of items.

Values can range from 0.00 to 1.00 (sometimes expressed as 0 to 100), with high values indicating that the examination is likely to correlate with alternate forms (a desirable characteristic). The KR-20 may be affected by difficulty of the test, the spread in scores and the length of the examination.

In the case when scores are not tau-equivalent (for example when there is not homogeneous but rather examination items of increasing difficulty) then the KR-20 is an indication of the lower bound of internal consistency (reliability).

The formula for KR-20 for a test with K test items numbered i=1 to K is

${\displaystyle r={\frac {K}{K-1}}\left[1-{\frac {\sum _{i=1}^{K}p_{i}q_{i}}{\sigma _{X}^{2}}}\right]}$

where pi is the proportion of correct responses to test item i, qi is the proportion of incorrect responses to test item i (so that pi + qi = 1), and the variance for the denominator is

${\displaystyle \sigma _{X}^{2}={\frac {\sum _{i=1}^{n}(X_{i}-{\bar {X}})^{2}\,{}}{n}}.}$

where n is the total sample size.

If it is important to use unbiased operators then the sum of squares should be divided by degrees of freedom (n − 1) and the probabilities are multiplied by

${\displaystyle {\frac {n}{n-1}}}$

## References

1. ^ Kuder, G. F., & Richardson, M. W. (1937). The theory of the estimation of test reliability. Psychometrika, 2(3), 151–160.
2. ^ Cortina, J. M., (1993). What Is Coefficient Alpha? An Examination of Theory and Applications. Journal of Applied Psychology, 78(1), 98–104.
3. ^ Ritter, Nicola L. (2010-02-18). Understanding a Widely Misunderstood Statistic: Cronbach's "Alpha". Annual meeting of the Southwest Educational Research Association. New Orleans.