Some Theory of Sampling. by William Edwards DemingReview by: Lester R, FrankelJournal of the American Statistical Association, Vol. 46, No. 253 (Mar., 1951), pp. 127-129Published by: American Statistical AssociationStable URL: http://www.jstor.org/stable/2280103 .

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BOOK REVIEWS Some Theory of Sampling. William Edwards Deming (Adviser in Sampling, Bureau of the Budget, Washington; Adjunct Professor of Statistics, Graduate School of Business Administration, New York University). New York: John Wiley & Sons, 1950. Pp. xvii, 602. $9.00.

LESTER R. FRANKEL, Alfred Politz Research, Inc., D URING the past two decades, the increased use of statistical methods in solving problems of government, industry, and commerce has been due

mainly to the development of survey sampling procedures. Much of the theory and techniques applicable to this phase of statistics has been pub- lished in scientific journals; other techniques in use have appeared in techni- cal appendixes of research reports. But a great deal of the theory of sampling has not been published at all, and because of the importance of statistical sampling and the dispersion of information on sampling, there has been for some years a growing demand for a full-length book dealing with the subject. This book satisfies the demand.

The author's aims are "to teach some theory of sampling as met in large- scale surveys . . . and to develop in the student some power and desire for originality in dealing with problems of sampling." In successfully achieving these aims, the author draws on his vast experience in government, in indus- try, as a consultant in marketing and as a teacher of statistics.

The book has five parts. The first four deal with existing applications of the theory of probability to problems of sampling. The fifth part is devoted to advanced statistical theory and is intended to provide enough background and knowledge to enable students to develop new techniques in sampling. Part I deals with the preliminary problems arising in connection with a large- scale survey. It points out that all operations involved in the execution of a survey should be considered together, and that sampling techniques as such cannot be separated. In all surveys, we are concerned with various types of errors and biases that may occur. Bias in a survey may enter not only from faulty sample design but also from non-statistical sources. It is pointed out that in some cases it may be desirable, in planning a survey, to allow an in- crease in the sampling error in order to make possible the reduction of these non-statistical errors. The goal is to minimize the total, over-all error. Part I clearly reflects the author's background and experience in survey and sam- pling procedures. This reviewer recalls that some 10 to 15 years ago, when sampling methods were first used to any extent in this country, many surveys did yield incorrect results, even though the sampling procedures were correct. These failures were often laid to sampling because, at the time, the existence of non-statistical errors was not considered.

Part II deals with the design of sample surveys after the initial specifica- tions have been set. In the entire discussion, the aim is to design a sample, applicable under certain administrative conditions, that will yield the small-

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128 AMERICAN STATISTICAL ASSOCIATION JOURNAL, MARCH 1951

est possible sampling error. The theory is first developed through a discussion of moments and expected values. Then the reader is introduced to problems of random sampling, both from the procedural aspect in the selection of random samples and from the statistical aspect in the derivation of variances of statistics derived from random samples. While unrestricted random sam- pling provides the simplest theory, it is seldom used because of many adminis- trative limitations and also because more efficient types of sampling may be employed. Multi-stage sampling, where the final selection of sampling units is obtained after two or more stages of sampling; ratio estimates; and cluster sampling are then discussed. In connection with cluster sampling, the close connection between theory and practice is clearly visible. The author de- velops the formula for the variance of a cluster sample, analyzes the various components, and suggests practical procedures designed to minimize the variance. In the following chapter, titled "Allocations in Stratified Sampling," proportionate, Neyman, and other types of allocation under various cost conditions are investigated. It is pointed out that the various sampling pro- cedures discussed previously may be applied within the framework of strati- fied sampling.

The discussion thus far in the book has centered around the use of sam- pling techniques to obtain an estimate of the results that would have been ob- tained had the same enumerative procedure been applied to every member of the universe. This type of study is known as an enumerative study and is usually associated with the concept of a finite population. In the next chapter, a distinction is drawn between this type of study and an analytic study, where interest is in the underlying cause system and where even a complete census may be regarded as a sample. The final chapter in this section makes use of this concept in discussing acceptance sampling, that phase of sampling which is a form of quality control.

The appraisal of survey results after the data have been collected and processed is discussed in Part III. The first chapter is concerned with the general problem of statistical inference. It emphasizes that the results of a sample survey provide a basis for action; in interpreting the findings, one must take into account the statistical problems of estimation and tests of hypothesis. The second chapter in this part describes methods of obtaining sampling errors of statistics derived from the survey from the actual obser- vations themselves. In the design of the sample (Part II), the statistician usually does not know the exact values of the various components of the statistical error. He draws on his experience and upon related data to obtain indications, and then proceeds to design a sample which he believes will yield minimum variance of the statistics being estimated. After the survey has been completed, procedures are available to obtain unbiased estimates of error.

Two applications of the material covered in the preceding chapters are given in Part IV. The first application is concerned with a sample survey of tire distributors in order to obtain an estimate of tire inventories. The second

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BOOK REVIEWS 129

application deals with a population sample for Greece. Both of these applica- tions appear almost verbatim in the 1946 and 1949 volumes of this Journal. It might perhaps have been more useful to shorten these two examples and add one or two others. Both of these surveys involved huge operations con- ducted on a nationwide scale. In addition, the designs used in both samples were conditioned by the fact that the government agencies responsible for the execution of the surveys had access to records and facilities not available to private practitioners. Statisticians in business would have appreciated a description of a commercial sample survey.

Despite the author's statement, in the Preface, that the book is not a text on mathematical statistics, Part V may be considered one. The author has assembled in this section a great deal of the statistical theory on such topics as the binomial and related distributions, the Beta and Gamma functions, and distribution theory.

The entire treatment of the subject matter in this book is on a high, pro- fessional level. The reader is assumed to have a knowledge of elementary statistical methods, college algebra, and some calculus. However, students who rely on this minimum requirement may have some difficulty in certain sections of the book. The author has made clever use of the devices of "exer- cises" and "remarks" to discuss and elaborate pertinent topics and sidelights as they appear, without interrupting the continuity of the text. These "exercises" and "remarks" are extremely important and a great deal of the theory of sampling will be missed by the reader who decides to skip them. The information in this book is so extensive that the presentation may appear bewildering at first glance. But once the reader gets acquainted with its contents, it becomes clear that the topics are developed logically and systematically. It seems likely that for some time to come this book will be the "bible" of sampling statisticians.

An Introduction to the Theory of Statistics. G. Udny Yule (formerly Reader in SUatistics, University of Cambridge) and M. G. Kendall (Professor of Statistics, University of London). New York: Hafner Publishing Company, 1950. Pp. xxiv, 701. $7.50.

rrHE Preface states that "this fourteenth edition is again a substantial lrevision.... Most of the alterations are additions, but the treatment of the theory of attributes, which in earlier editions occupied five chapters, has been condensed into three to make room for new material.

"The major additions fall into two groups. Chapters 21-23 expand the former treatment of small-sample theory and give an introduction to the practical problems of samplings. Chapters 25-27 give an account of index- numbers and the elementary theory of time-series. Chapter 13 on practical problems of correlation has also been re-written. Additions have been made in the remaining chapters to keep the treatment abreast of new discoveries, some of the examples have been modernised and some further exercises added. The list of references has been onmitted because a much more exten-

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