Kuder-Richardson Formula 20

In statistics, the Kuder-Richardson Formula 20 (KR-20) is a measure of internal consistency reliability for measures with dichotomous choices, first published in 1937. It is analogous to Cronbach's α, except Cronbach's α is used for non-dichotomous (continuous) measures. A high KR-20 coefficient (e.g., >0.90) indicates a homogeneous test.

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 KR20 is impacted by difficulty, spread in scores and 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).


 * $$\alpha={K\over{K-1}}{[{1-{\sum_{i=1}^N{p_{i}q_{i}}\over\sigma^{2}_{X}}}]}$$

Note that variance for KR-20 is


 * $$\sigma^{2}_{X} = {\sum_{i=1}^N{(X_i-\bar{X})^{2}}\over{N}}$$

If it is important to use unbiased operators then the Sum of Squares should be divided by degrees of freedom (N &minus; 1) and the probabilities are multiplied by


 * $${N}\over{N-1}$$

Since Cronbach's α was published in 1951, there has been no known advantage to KR-20 over Cronbach. KR-20 is seen as a derivative of the Cronbach formula, with the advantage to Cronbach that it can handle both dichotomous and continuous variables.

References & related material

 * Statistical analysis of multiple choice exams
 * Quality of assessment chapter in Illinois State Assessment handbook (1995)