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Some Generalizations of a Theoretical Distribution of Mental Test Scores

Keats, John A.
Publication Year:
Report Number:
ETS Research Bulletin
Document Type:
Page Count:
Subject/Key Words:
Cognitive Measurement, Hypergeometric Distribution, Mathematical Models, Scores, Test Theory


The negative hypergeometric distribution may be generalized to provide distributions involving 3, 4 or more parameters. It is shown that, in the case of the binomial error model, such distributions correspond to quadratic, cubic, etc., regressions of true scores on raw scores. Explicit formulas are given for the bivariate moments required to fit these regressions and so estimate the parameters of the generalized hypergeometric distribution. Equations are also developed for fitting the 3 and 4 parameter distributions directly, i.e, without reference to the model, and the method for generalizing these is indicated. The methods developed are applied to objective test data with satisfactory results.

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