KuderRichardson formulas

The Kuder–Richardson formulas (KR-20 and KR-21) are measures of internal consistency reliability for tests with dichotomous responses, introduced by Kuder and Richardson in 1937. KR-20 is a special case of Cronbach's alpha, designed for dichotomous data. It ranges from 0 to 1, with higher values indicating better reliability. While a high KR-20 suggests a homogeneous test, this homogeneity is an assumption, not a conclusion. The formula accounts for the number of items (K), proportions of correct and incorrect responses (p_i and q_i), and the variance of total scores.

KR-21 is a simplified version used when all items have equal difficulty, calculated using the average item difficulty. It tends to be more conservative than KR-20, which in turn is more conservative than Cronbach's alpha. Both formulas provide reliability estimates but require assumptions about item homogeneity or difficulty. Adjustments can be made for unbiased estimates by considering degrees of freedom and sample size adjustments.