Kuder–Richardson Formula 20

In psychometrics, the Kuder–Richardson Formula 20 (KR-20), first published in 1937, is a measure of internal consistency reliability for measures with dichotomous choices. It was developed by Kuder and Richardson. The name of this formula stems from the fact that is the twentieth formula discussed in Kuder and Richardson's seminal paper on test reliability.[1]

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 p_i is the proportion of correct responses to test item i, q_i is the proportion of incorrect responses to test item i (so that p_i + q_i = 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. (18 February 2010). Understanding a Widely Misunderstood Statistic: Cronbach's "Alpha". Annual meeting of the Southwest Educational Research Association. New Orleans.
4. "Kuder and Richardson Formula 20 | Real Statistics Using Excel". Retrieved 8 March 2019.

External links

• Quality of assessment chapter in Illinois State Assessment handbook (1995)