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This book provides the first comprehensive treatment of Benford's law, the surprising logarithmic distribution of significant digits discovered in the late nineteenth century. Establishing the mathematical and statistical principles that underpin this intriguing phenomenon, the text combines up-to-date theoretical results with overviews of the law's colorful history, rapidly growing body of empirical evidence, and wide range of applications.An Introduction to Benford's Law begins with basic facts about significant digits, Benford functions, sequences, and random variables, including tools from the theory of uniform distribution. After introducing the scale-, base-, and sum-invariance characterizations of the law, the book develops the significant-digit properties of both deterministic and stochastic processes, such as iterations of functions, powers of matrices, differential equations, and products, powers, and mixtures of random variables. Two concluding chapters survey the finitely additive theory and the flourishing applications of Benford's law.Carefully selected diagrams, tables, and close to 150 examples illuminate the main concepts throughout. The text includes many open problems, in addition to dozens of new basic theorems and all the main references. A distinguishing feature is the emphasis on the surprising ubiquity and robustness of the significant-digit law. This text can serve as both a primary reference and a basis for seminars and courses.

Distribution (Probability theory) --- Probability measures. --- Measures, Normalized --- Measures, Probability --- Normalized measures --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Bayesian models. --- Benford distribution. --- Benford distributions. --- Benford functions. --- Benford sequences. --- Benford's law. --- First-digit law. --- Significant-digit law. --- base-invariance property. --- computations. --- computer science. --- diagnostics. --- differential equations. --- exponential growth. --- finitely additive probability. --- fraud detection. --- functions. --- geometric Brownian motion. --- linear processes. --- logarithmic distribution. --- mathematical hypotheses. --- mathematical theory. --- multi-dimensional models. --- natural phenomena. --- one-dimensional deterministic systems. --- one-dimensional difference. --- pedagogical tool. --- polynomial growth. --- random matrices. --- random processes. --- random variables. --- scale-invariance property. --- sequences. --- significand functions. --- significands. --- significant digits. --- statistical distribution. --- stochastic process. --- sum-invariance property. --- superexponential growth. --- surveys. --- uniform distribution.

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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.

Distribution (Probability theory) --- Probability measures. --- 1938 paper. --- 2009 presidential elections. --- Benford analysis. --- Benford distribution. --- Benford property. --- Benford test. --- Benford's law geometry. --- Benford's law limit. --- Benford's law. --- Benford-good system. --- Eric Poehlman. --- European statistics. --- FSDs. --- Fourier analysis. --- Frank Benford. --- Fundamental Equivalence. --- Greek statistics. --- Iranian election. --- Iranian presidential elections. --- Lvy processes. --- PV effect. --- Partial Volume effect. --- Poisson Summation Formula. --- Poisson Summation. --- Simon Newcomb. --- Standard Condition. --- VAR. --- Value at Risk. --- accounting programs. --- accounting students. --- accounting. --- auditing. --- authentic data sets. --- behavioral approaches. --- bias. --- complaint data sets. --- computer systems. --- cumulative distribution. --- data sets. --- data-adaptive methods. --- decision-making research. --- densities. --- deterministic processes. --- digit bias. --- direct applications. --- econometric regression. --- economics. --- election fraud. --- elections. --- empirical economics research. --- empirical economics. --- error detection. --- explicit error bounds. --- explicit error estimates. --- exponential Lvy processes. --- finance. --- financial reports. --- financial statistics. --- first significant digits. --- first-digit analysis. --- first-digit frequency. --- fixed odds. --- forecasts. --- fraud detection. --- fraud. --- fraudulent data sets. --- functions. --- gambling. --- generic potential applications. --- geometry. --- government deficit. --- information-theoretic methods. --- local boostrap model. --- logarithms. --- lottery. --- macroeconomic data. --- managing risk. --- mathematical theory. --- meaningful numbers. --- medical tests. --- misreporting. --- natural data. --- natural sciences. --- non-uniformity. --- normalized functionals. --- number lottery games. --- numbers games. --- origins. --- parametric distributions. --- probability distributions. --- psychology. --- random processes. --- replication. --- scale invariance. --- scientific data sets. --- scientific misconduct. --- second digits. --- significand. --- significant digits. --- small number. --- social statistics. --- statistical relationship. --- statistics education. --- statistics. --- tampering. --- tax filing. --- tax fraud. --- total variation. --- uniform distribution. --- vote counts. --- voting.