Measuring the vague meanings of probability terms.

Abstract

Can the vague meanings of probability terms such as doubtful, probable, or likely be expressed as membership functions over the [0, 1] probability interval? A function for a given term would assign a membership value of /ero to probabilities not at all in the vague concept represented by the term, a membership value of one to probabilities definitely in the concept, and intermediate membership values to probabilities represented by the term to some degree. A modified pair-comparison procedure was used in two experiments to empirically establish and assess membership functions for several probability terms. Subjects performed two tasks in both experiments: They judged (a) to what degree one probability rather than another was better described by a given probability term, and (b) to what degree one term rather than another better described a specified probability. Probabilities were displayed as relative areas on spinners. Task a data were analyzed from the perspective of conjoint-measurement theory, and membership function values were obtained for each term according to various scaling models. The conjoint-measurement axioms were well satisfied and goodness-of-fit measures for the scaling procedures were high. Individual differences were large but stable. Furthermore, the derived membership function values satisfactorily predicted the judgments independently obtained in task b. The results support the claim that the scaled values represented the vague meanings of the terms to the individual subjects in the present experimental context. Methodological implications are discussed, as are substantive issues raised by the data regarding the vague meanings of probability terms. Most people, including expert forecasters, generally prefer communicating their uncertain opinions with nonnumerical terms such as doubtful, probable, slight chance, very likely, and so forth, rather than with numerical probabilities. On anecdotal grounds, the imprecision of nonnumerical terms is preferred to the precision of probability numbers for at least two reasons: First, opinions are generally not precise and therefore, the claim goes, it would be misleading to represent them precisely. For example, commenting that numbers denote authority and a precise understanding of relations, a committee of the U.S. Na-This research was supported by Contract MDA 903-83-K-0347 from the U.S. Army Research Institute for the Behavioral and Social Sciences to the L. L. Thurstone Psychometric Laboratory, University of North Carolina at Chapel Hill. The views, opinions, and findings contained in this paper are those of the authors and should not be construed as an official Department of the Army position, policy, or decision. Barbara Forsyth is now at Ohio University. We thank Samuel Fillenbaum for numerous helpful discussions throughout the course of the work, and James Cox, Brent Cohen, Samuel Fillenbaum, and Jaan Valsiner for comments on a previous draft of this article.

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