# Kuder–Richardson formulas – Wikipedia

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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.[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.

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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 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

${displaystyle sigma _{X}^{2}={frac {sum _{i=1}^{n}(X_{i}-{bar {X}})^{2},{}}{n}}.}$

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

${displaystyle {frac {n}{n-1}}}$

## Kuder-Richardson Formula 21 (KR-21)

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:

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

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]

## References

1. ^ Kuder, G. F., & Richardson, M. W. (1937). The theory of the estimation of test reliability. Psychometrika, 2(3), 151–160.
2. ^ Cortina, J. M., (1993). What Is Coefficient Alpha? An Examination of Theory and Applications. Journal of Applied Psychology, 78(1), 98–104.
3. ^ Ritter, Nicola L. (18 February 2010). Understanding a Widely Misunderstood Statistic: Cronbach’s “Alpha”. Annual meeting of the Southwest Educational Research Association. New Orleans.
4. ^ a b “Kuder and Richardson Formula 20 | Real Statistics Using Excel”. Retrieved 8 March 2019.

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