In psychometrics, the Kuder–Richardson formulas, first published in 1937, are a measure of internal consistency reliability for measures with dichotomous choices. They were developed by Kuder and Richardson. 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 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 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 Kuder-Richardson Formula 21 (KR-21) Often discussed in tandem with KR-20, is Kuder-Richardson Formula 21 (KR-21). KR-21 is a simplified version of KR-20, which can be used when the difficulty of all items on the test are known to be equal. Like KR-20, KR-21 was first set forth as the twenty-first formula discussed in Kuder and Richardson's 1937 paper. The formula for KR-21 is as such: Similarly to KR-20, K is equal to the number of items. Difficulty level of the items (p), is assumed to be the same for each item, however, in practice, KR-21 can be applied by finding the average