Benford's Law: Theory and Applications

Benford’s law is the curious observation that the leading digits in many naturally-occurring sets of numbers appear with the relative frequencies \(\log_{10}\left(\frac{d+1}{d}\right)\) for each possible digit \(d=1,\ldots,9\) rather than (as many people would be inclined to assume at first) with equal frequencies. Since my first time hearing of this phenomenon as a student, I’ve always thought that this was a delightfully quirky mathematical idea and deserves to be more widely known. Sadly, unlike other famous results of probability theory such as the law of large numbers and the central limit theorem, Benford’s law is usually relegated to little more than footnote status, and one finds precious little discussion of it in textbooks — perhaps owing to the fact that it is hard to formulate a single clean mathematical theorem explaining its ubiquity in nature.

Happily, this book contains a great collection of essays exploring both the mathematical underpinnings of Benford’s law and its applications, which are heavily centered around fraud detection in accounting, finance, economics, politics, and scientific research. The book is edited by Steven J. Miller and contains contributions from a number of experts from a variety of disciplines. Strangely for an edited collection of articles, I could not find a list of contributors anywhere; the authors of each chapter are listed only at the beginning of that chapter and not mentioned in the table of contents next to the chapter titles. This does the contributors a disservice, and I hope that a list of their names will be added at least on the book’s home page on the publisher’s and editor’s websites, and that this oversight will be corrected in a future printing.

Roughly the first half of the book covers the mathematical theory behind Benford’s law. The essential theoretical reason for the appearance of Benford’s law in connection with many phenomena is fairly simple (in a nutshell: scale invariance), but a fairly large literature has developed with the goal of formulating precise conditions for various situations leading to Benford-distributed behavior, and studying additional theoretical aspects such as explicit rate of convergence estimates. This theoretical framework is discussed in chapters 1–6. This part of the book at times delves into theoretical minutiae that will appeal only to a fairly small audience, but much of it will be of interest to a general audience of probabilists and statisticians.

The part of the book that I found most enjoyable was its second half, in which applications are discussed. I have long known that Benford’s law was used in detection of tax and financial fraud, but it was entertaining to read the practical details behind this idea, including discussions of real-life case studies. These applications are a lot more diverse than I had realized, pertaining not just to the area of accounting but also to detection of election fraud, forged data in scientific research, and the manipulation by certain countries of the macroeconomic and fisheries data they report to international governance and monitoring bodies. The detection methodologies also involve some interesting non-mathematical ideas, often from the realm of psychology, and I picked up fascinating tidbits of information, such as that embezzlers who steal money from their employers often begin by stealing small amounts and over the years work their way up towards more substantial sums; this provides a useful pattern of behavior that can be used by auditors and criminal investigators. I will be sure to keep this in mind!

To summarize, this book will serve as an excellent reference for mathematicians with an interest in probability and applied mathematics, statisticians, and accountants. I also recommend it warmly to fraudsters, embezzlers, dishonest scientists, corrupt politicians, dictators, and government officials of countries involved in manipulation of macroeconomic data.

Dan Romik is a professor of mathematics at the University of California, Davis. His book The Surprising Mathematics of Longest Increasing Subsequences was published in 2015 by Cambridge University Press.