An Improved Internal Consistency Reliability Estimate

Kuder-Richardson Formula 20 was derived on the assumption that the average covariance between items on different forms is the same as the average covariance on the same form. This formula is considered an underestimate of the parallel form reliability because the average covariance of items with identical difficulty is expected to be higher than the average covariance of items of different difficulty.

The proposed coefficient is derived by assuming instead that the average Goodman-Kruskal gamma between items of identical difficulty would be the same as the average for items of different difficulty. Given the average gamma for the present items, this permits an estimate of the covariance between items of identical difficulty. This leads directly to an estimate of the correlation between scores on the present test and those on a form that had an identical distribution of difficulty. Some examples are given.