ESTIMATING TRUE MEASUREMENTS FROM FALLIBLE MEASUREMENTS (BINOMIAL CASE)–EXPANSION IN A SERIES OF BETA DISTRIBUTIONS *

In mental‐test theory, a useful mathematical model specifies the relation of the examinee's observed score, x, to his “true score,” ζ. The present paper is concerned with a model for the number‐right scores on a test composed of n questions or “items.” This model is completely specified by the assertion that the conditional frequency distribution of x when ζ is fixed is the binomial distribution . The basic problem in the use of this model may be thought of as the problem of estimating the unknown frequency distribution of true scores. Once this is done, the bivariate distribution of ζ and x has also been estimated. All important properties of the test score can thus be investigated. Although it might at first seem otherwise, the model has empirical verifiable implications. The first section of the present paper discusses various methods for estimating g(ζ). The second section outlines the method to be reported here. The third section presents and discusses the results of applying this method to Widely different sets of mental test data.