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.

The first five sections of the paper describe the Bayesian paradigm for statistics and its relationship with other attitudes towards inference. Section 1 outlines Wald's major contributions and explains how they omit the vital consideration of coherence. When this point is included the Bayesian view results, with the main difference that Waldean ideas require the concept of the sample space, whereas the Bayesian approach may dispense with it, using a probability distribution over parameter space instead. Section 2 relates statistical ideas to the problem of inference in science. Scientific inference is essentially the passage from observed, past data to unobserved, future data. The roles of models and theories in doing this are explored. The Bayesian view is that all this should be accomplished entirely within the calculus of probability and Section 3 justifies this choice by various axiom systems. The claim is made that this leads to a quite different paradigm from that of classical statistics and, in particular, problems in the latter paradigm cease to have importance within the other. Point estimation provides an illustration. Some counter-examples to the Bayesian view are discussed. It is important that statistical conclusions should be usable in making decisions. Section 4 explains how the Bayesian view achieves this practicality by introducing utilities and the principle of maximizing expected utility. Practitioners are often unhappy with the ideas of basing inferences on one number, probability, or action on another, an expectation, so these points are considered and the methods justified. Section 5 discusses why the Bayesian viewpoint has not achieved the success that its logic suggests. Points discussed include the relationship between the inferences and the practical situation, for example with multiple comparisons; and the lack of the need to confine attention to normality or the exponential family. Its extensive use by nonstatisticians is documented. The most important objection to the Bayesian view is that which rightly says that probabilities are hard to assess. Consequently Section 6 considers how this might be done and an attempt is made to appreciate how accurate formulae like the extension of the conversation, the product law and Bayes rule are in evaluating probabilities.

This article traces the history of the problem of estimating the variance, $\sigma^2$, based on a random sample from a normal distribution with mean $\mu$ unknown. Considered are both the point estimation and confidence interval cases. We see that improvement over both usual estimators follows a remarkably parallel development and stemmed from the innovative ideas presented in Stein (1964). We examine developments through the most recent dealing with improved confidence intervals and conditional evaluations of interval estimators.

The driving force behind the development of modern statistics has been the need to solve practical problems. Historically, statisticians have achieved signal advances in theory and methods as they worked on applications in many fields, and in turn, statistical thinking and methodology have greatly influenced the development of virtually all areas of science. In 1985 the National Science Foundation funded a proposal by the Institute of Mathematical Statistics (IMS) to assess the current status of cross-disciplinary statistical research and to make recommendations for its future. The IMS formed a panel to carry out the project. The panel endorses the principles that advances in substantive knowledge and in statistical theory and methods are virtually inseparable and that the continued health of statistics depends strongly on continuing cross-disciplinary research in many fields. Its report reviews past successes in collaborative research among statisticians and scientists in other fields and also important current problems that would greatly benefit from close collaboration among statisticians and other scientists to push forward the frontiers of theory, methods and knowledge. Yet the panel finds that constrained resources and the existing infrastructure within government, academia and industry thwart the growth and development of needed cross-disciplinary work. The panel's report presents recommendations for concrete steps to help rectify this situation and to promote and encourage cross-disciplinary research in the statistical sciences. The panel addresses issues concerning support--general support for large scale projects, support for research by statisticians in primarily advisory roles, support for effective transfer of technical results. The place and role of statistics in universities, industry and national laboratories are reviewed. The role of professional societies and suggested changes are discussed. Finally, the panel recommends the establishment of an Institute for Statistical Sciences that will serve to encourage cross-disciplinary research between statistics and other fields.

More than 30 years ago, Charles Stein discovered that in three or more dimensions, the ordinary estimator of the vector of means of a multivariate normal distribution is inadmissible. This article examines Stein's paradox from the perspective of an earlier century and shows that from that point of view the phenomenon is transparent. Furthermore, this earlier perspective leads to a relatively simple rigorous proof of Stein's result, and the perspective can be extended to cover other situations, such as the simultaneous estimation of several Poisson means. The relationship of this perspective to other earlier work, including the empirical Bayes approach, is also discussed.