Abstract. For 20 different studies, Table 1 tabulates numerical averages of opinions on quantitative meanings of 52 qualitative probabilistic expressions. Populations with differing occupations, mainly students, physicians, other medical workers, and science writers, contributed. In spite of the variety of populations, format of question, instructions, and context, the variation of the averages for most of the expressions was modest, suggesting that they might be useful for codification. One exception was possible, because it had distinctly different meanings for different people. We report new data from a survey of science writers. The effect of modifiers such as very or negation (not, un-, im-, in-) can be described approximately by a simple rule. The modified expression has probability meaning half as far from the appropriate boundary (0 or 100) as that of the original expression.This paper also reviews studies that show stability of meanings over 20 years, mild effects of translation into other languages, context, small order effects, and effects of scale for reporting on extreme values.The stem probability with modifiers gives a substantial range 6% to 91% and the stem chance might do as well if tried with very. The stems frequent, probable, likely, and often with modifiers produce roughly equivalent sets of means, but do not cover as wide a range as probability. Extreme values such as always and certain fall at 98% and 95%, respectively, and impossible and never at 1%.The next step will be to offer codifications and see how satisfactory people find them.

Qualitative expressions of probability, such as "likely," have different numerical meanings to different people, which can lead to misunderstanding among physicians and between physicians and patients. In a study conducted through a nationwide interactive computer network based at Massachusetts General Hospital, we gathered information on the meaning of common expressions of probability. Three groups of medical professionals assigned percentage values to 12 expressions of the probability that a given symptom would appear in a patient with an unspecified disease. The median values assigned to these expressions by physicians, medical students, and other professionals were almost the same. Comparisons of the means for 7 of these 12 expressions with those found in an earlier study by other investigators showed that they were quantified in the same order, although they had not been assigned the same numerical values. This degree of agreement among professionals and between studies is encouraging for the future prospects of codifying the meaning of such expressions. The variation among five studies in the mean values assigned to 37 expressions in the medical literature and the variation among individual opinions show that such codification is necessary. In the meantime, the average numerical values presented here for various qualitative expressions of probability could well be used to enhance communication among medical professionals.

The meanings of 18 verbal probability expressions were studied in 3 ways: (a) frequency distributions of what single number best represented each expression; (b) word-to-number acceptability functions from what range of numbers from 0% to 100% best represented each expression; and (c) number-to-word acceptability functions from which expressions were appropriate for multiples of 5% from 5% to 95%. These results agreed highly with others and were highly consistent across methods. Expressions incorporating the stem probable were quantitatively synonymous with expressions incorporating the stem likely. Except for expressions using the word chance, positive expressions (e.g., likely) were closer to 50% in meaning than corresponding negative expressions (e.g., unlikely). This method proved very useful in deriving fuzzy-set membership functions for probability words, encouraging us in our ongoing codification effort.

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JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Biometrika Trust is collaborating with JSTOR to digitize, preserve and extend access to Biometrika. SUMMARY We present a table of the Freeman-Tukey variance stabilizing arc-sine transformation for the binomial distribution together with properties of the transformation. Entries in the table are O=2(arcsinJ(X1)?arcsinJ(+ 1 where n is the sample size and x is the number of successes observed in a binomial experiment. Values of 0 are given in degrees, to two decimal places, for n = 1 [1] 50 and x = 0 [1] n. In addition, for completeness, we give a table of the corresponding square-root transformation to two decimal places for use with Poisson counts. The observed count is x (x = 0 [1] 50) and the transformed values are g = VX+V(X+ 1); the squares of the transformed values are also given for use in analysis of variance computations.