Kuder–Richardson formulas

Kuder–Richardson Formula 20 (KR-20)

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.

It is a special case of Cronbach's α, computed for dichotomous scores. 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

${\displaystyle {\sigma }_X^2=\frac{\sum _{i=1}^n(X_i-\overline{X}{)}^2}{n}.}$

where n is the total sample size, X__i is the sum of items correct for the ith respondent and ${\displaystyle \overline{X}}$ is the mean of X__i values.

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 $n/(n-1).$