Kuder–Richardson formulas

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 ${\textstyle n/(n-1).}$

Often discussed in tandem with KR-20, is Kuder–Richardson Formula 21 (KR-21).[4] 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:

${\displaystyle r={\frac {K}{K-1}}\left[1-{\frac {Kp(1-p)}{\sigma _{X}^{2}}}\right]}$

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 item difficulty across the entirety of the test. KR-21 tends to be a more conservative estimate of reliability than KR-20, which in turn is a more conservative estimate than Cronbach's α.[4]

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