Benford's Law

Theory and Applications

Steven J. Miller, Steven Miller

• Publisher: Princeton University Press
• Year: 2015
• Print ISBN:
  9780691147611
• eBook ISBN:
  9781400866595
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

Benford's law states that the leading digits of many data sets are not uniformly distributed from one through nine, but rather exhibit a profound bias. This bias is evident in everything from electricity bills and street addresses to stock prices, population numbers, mortality rates, and the lengths of rivers. Here, Steven Miller brings together many of the world's leading experts on Benford's law to demonstrate the many useful techniques that arise from the law, show how truly multidisciplinary it is, and encourage collaboration.

Beginning with the general theory, the contributors explain the prevalence of the bias, highlighting explanations for when systems should and should not follow Benford's law and how quickly such behavior sets in. They go on to discuss important applications in disciplines ranging from accounting and economics to psychology and the natural sciences. The contributors describe how Benford's law has been successfully used to expose fraud in elections, medical tests, tax filings, and financial reports. Additionally, numerous problems, background materials, and technical details are available online to help instructors create courses around the book.

Emphasizing common challenges and techniques across the disciplines, this accessible book shows how Benford's law can serve as a productive meeting ground for researchers and practitioners in diverse fields.

PART I

General Theory I: Basis of Benford’s Law

Chapter One

A Quick Introduction to Benford’s Law

Steven J. Miller¹

The history of Benford’s Law is a fascinating and unexpected story of the interplay between theory and applications. From its beginnings in understanding the distribution of digits in tables of logarithms, the subject has grown enormously. Currently hundreds of papers are being written by accountants, computer scientists, engineers, mathematicians, statisticians and many others. In this chapter we start by stating Benford’s Law of digit bias and describing its history. We discuss its origins and give numerous examples of data sets that follow this law, as well as some that do not. From these examples we extract several explanations as to the prevalence of Benford’s Law, which are described in greater detail later in the book. We end by quickly summarizing many of the diverse situations in which Benford’s Law holds, and why an observation that began in looking at the wear and tear in tables of logarithms has become a major tool in subjects as diverse as detecting tax fraud and building efficient computers. We then continue in the next chapters with rigorous derivations, and then launch into a survey of some of the many applications. In particular, in the next chapter we put Benford’s Law on a solid foundation. There we explore several different categorizations of Benford’s Law, and rigorously prove that certain systems satisfy these conditions.

1.1 OVERVIEW

We live in an age when we are constantly bombarded with massive amounts of data. Satellites orbiting the Earth daily transmit more information than is in the entire Library of Congress; researchers must quickly sort through these data sets to find the relevant pieces. It is thus not surprising that people are interested in patterns in data. One of the more interesting, and initially surprising, is Benford’s Law on the distribution of the first or the leading digits.

In this chapter we concentrate on a mostly non-technical introduction to the subject, saving the details for later. Before we can describe the law, we must first set notation. At some point in secondary school, we are introduced to scientific notation: any positive number x may be written as S(x) · 10^k, where S(x) ∈ [1, 10) is the significand and k is an integer (called the exponent). The integer part of the significand is called the leading digit or the first digit. Some people prefer to call S(x) the mantissa and not the significand; unfortunately this can lead to confusion, as the mantissa is the fractional part of the logarithm, and this quantity too will be important in our investigations. As always, examples help clarify the notation. The number 1701.24601 would be written as 1.70124601 · 10³ in scientific notation. The significand is 1.70124601, the exponent is 3 and the leading digit is 1. If we take the logarithm base 10, we find log₁₀ 1701.24601 ≈ 3.2307671196444460726, so the mantissa is approximately .2307671196444460726.

There are many advantages to ...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Dedication Page
• Contents
• Foreword
• Preface
• Notation
• Part I. General Theory I: Basis of Benford’s Law
• Part II. General Theory II: Distributions and Rates of Convergence
• Part III. Applications I: Accounting and Vote Fraud
• Part IV. Applications II: Economics
• Part V. Applications III: Sciences
• Part VI. Applications IV: Images
• Part VII. Exercises
• Bibliography
• Index