An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements

Abstract

Students in a science or engineering curriculum ought to be introduced early to the requirement that a meaningful measurement result should always be accompanied by a statement of its uncertainty. This book has been written specifically with this objective in mind. That the first edition has been successful in doing this is attested to by its popularity with both faculty and students, and its translation into six languages. This book is not a statistics text - nor was it intended to be - but an introdution to the mathematics required for the analysis of measurements at the level of a first-year laboratory course. Part 1 begins with uncertainty as a qualitative concept and builds slowly, using many numerical examples and exercises for the student, to develop methods for quantifying uncertainty, and ultimately relating it to the standard deviation of a statistical distribution. Along the way, Taylor develops the rules for expressing and combining (`propagating') uncertainties, and introduces the student to the gaussian (normal) distribution and some of its properties. Part 2 covers, with somewhat more mathematical rigor, specific topics such as data rejection criteria, the binomial and Poisson distributions, covariance and correlation, least-squares fitting, and the chi-squared test. I was not familiar with the first edition, and from a quick scan of the Preface I looked forward to reading this book and learning something about the state of statistical analysis in first-year university texts today. I was disappointed (in part with what the level of the book implies about the sad state of preparation of today's students). Although there are now two ISO publications ( International Vocabulary of Basic and General Terms in Metrology (VIM) and Guide to the Expression of Uncertainty in Measurement (GUM), Geneva, 1993), Taylor makes no mention of either, and never gives a formal definition of `uncertainty' (although he ultimately associates `random uncertainty' with the standard deviation of a gaussian distribution). The book also does not clearly define `error', or the distinction between error and uncertainty. The important point, that the `propagation of uncertainty' is additive in terms of variances is valid for any distributions with finite variance, is not emphasized; instead Taylor restricts the discussion solely to the normal distribution and or those that can be approximated by it. I also find it unfortunate that the book does not clearly distinguish between the variance of a sample , the variance of a distribution , and the sample estimate of the variance of the distribution ( or ). Instead, he accepts the fact that formulas for the variance with either N or N - 1 dividing the sum of the squares of the deviations from the mean exist in the literature and concludes simply: `Nevertheless, you need to be aware of both definitions. In the physics laboratory, using the more conservative... def- inition... is almost always best.' In spite of these shortcomings, the book is a significant contribution to a student laboratory reading list, and it is written at a level that facilitates a self-study program. It has an important message to deliver and it appears to be delivering it well.