Abstract
ABSTRACTThis report covers part of a continuing project to devise and try out methods for estimating examinees' true scores from their observed scores on a mental test. Although a method for doing this has significant practical applications, these are not discussed here.The basic mathematical model used is a quite unrestrictive one, representing the frequency distribution of true scores for a population of examinees as the unknown function in an integral equation. By solving this equation, the unknown true‐score distribution can be determined from the frequency distribution of observed test scores in the population of examinees, but not uniquely. A suitable unique solution can be obtained by requiring that the true‐score distribution be “smooth” in some sense. The calculus of variations yields a formula for finding a unique “smooth” true‐score distribution from a population distribution of observed test scores.Attempts to substitute a sample distribution of observed test scores for the population distribution in the formula so obtained yield absurd results. An alternative method of using the sample observed‐score distribution to estimate the population true‐score distribution is developed, leading to a different type of integral equation. Practical procedures for solving this integral equation are presented.The method has been tried out on 60 empirical distributions of observed test scores to determine how well the model fits the data. An extension of the model has been used to predict certain bivariate distributions of observed scores from the corresponding marginal distributions. The effectiveness of the model in making such predictions is reported for 26 different sets of data. The model is judged to be satisfactory for practical applications to many sets of data. It seems likely, however, that the estimation method can be improved on.
Published Version
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