This book is the second volume of the Sigma Series in Stochastics published by Heldermann Verlag (the other series of introductory books is Applied Mathematics and Pure Mathematics). The purpose of the series is “to establish stochastics as an independent, mathematically-based science, the subject of which is chance and uncertainty” (from the preface by the series editors). E. von Collani is a professor at Würzburg University and Karl Baur works in the energy industry (E.ON Kernkraft, Hanover) and has been interested in quality problems for a long time. Perhaps everybody understands the notion of ‘quality’ that appears in the works of Aristotle, Thomas of Aquinas, Descartes and others but the number of definitions is continually increasing. Five different approaches to quality can be distinguished, namely transcendent, product-based, user-based, manufacturing-based and value-based. The first section, entitled What is the quality?, comprises a deep discussion of various approaches from the historical development of the Second World War, with its need for top-quality weapons, to the immediate post-war reaction of quality gurus (W. E. Denning, J. M. Juran, A. W. Feigenbauer, Kaoru Ishikawa and Genichi Taguchi) up to the Six sigma movement of the 80s.

The second section, Quantification and definition of quality, contains (after an historical review) an attempt to develop a stochastic model of quality originally proposed by Collani in the first volume of the Sigma Series in Stochastics, Defining the Science of Stochastics. The model is then used in the third section, Solutions of problems, to discuss problems encountered in the first section. The most important mission of the book is to show that not only physical quantities like temperature and length but also such a dark notion as quality can be quantified. This conclusion also has important consequences relating to recent science and the ways of thinking resulting from it, as well as to human behaviour. This conclusion constitutes the content of the fourth section entitled Message. The book can be heartily recommended to all students and scientists understanding that stochastic approaches to the real world can be extremely fertile.