The probability that a number in many naturally occurring tables of numerical data has first significant digit $d$ is predicted by Benford's Law ${\rm Prob} (d) = \log_{10} (1 + {\displaystyle{1\over d}}), d = 1, 2 >..., 9$. Illustrations of Benford's Law from both theoretical and real-life sources on both science and social science areas are shown in detail with some novel ideas and generalizations developed solely by the authors of this paper. Three tests, Chi-Square test, total variation distance, and maximum deviations are adopted to examine the fitness of the datasets to Benford's distribution. Finally, applications of Benford's Law are summarized and explored to reveal the power of this mathematical principle.
The diverse applications of the Benford law attract investigators working in various fields of physics, biology and sociology. At the same time, the groundings of the Benford law remain obscure. Our paper demonstrates that the Benford law arises from the positional (place-value) notation accepted for representing various sets of data. An alternative to Benford formulae to predict the distribution of digits in statistical data are derived. Application of these formulae to the statistical analysis of infrared spectra of polymers is presented. Violations of the Benford Law are discussed.
A simple method to derive parametric analytical extensions of Benford's law for first digits of numerical data is proposed. Two generalized Benford distributions are considered, namely the two-sided power Benford distribution and the new Pareto Benford distribution. The fitting capabilities of these generalized Benford distributions are illustrated and compared at some interesting and important integer sequences.
In this paper, we will see that the proportion of d as leading digit, d $\in$ 1, 9, in data (obtained thanks to the hereunder developed model) is more likely to follow a law whose probability distribution is determined by a specific upper bound, rather than Benford's Law. These probability distributions fluctuate around Benford's value as can often be observed in the literature in many naturally occurring collections of data (where the physical , biological or economical quantities considered are upper bounded). Knowing beforehand the value of the upper bound can be a way to find a better adju…
This is a detailed survey which mainly presents the Pinkham-Feller way. I added some new points to the first version [V2] and I suppressed "Examples" devoted to Gamma, Fréchet and Weibull laws. Theorem 2 is a bit more general (no assumption of density: this answers a question of T. Hill). Section 10 is new and devoted to an argument (Poincaré, Fewster) about the effect of high frequencies oscillations. Maybe many works, many efforts, have been devoted to the study of a sufficient condition of poor value: see Sections 7 and 11. The final Section gives some suggestions.
On the occasion of the 125-th anniversary of Newcomb's paper, a bibliography of academic work related to Benford's law from its year of origin 1881 to 2006 has been compiled.
Benford's law is frequently used to evaluate the likihood that data is misrepresentative. Typically statistical tests measure the likihood. Another method of employing Benford's law is to compare the frequency of leading digits to the probabilities of leading digits over a subset of the natural numbers. This paper proposes using the probabilities of leading digits from uniform, natural numbers to establish interval criteria for when to look more closely into the possibility of misrepresentative data.
This article provides a concise overview of the main mathematical theory of Benford's law in a form accessible to scientists and students who have had first courses in calculus and probability. In particular, one of the main objectives here is to aid researchers who are interested in applying Benford's law, and need to understand general principles clarifying when to expect the appearance of Benford's law in real-life data and when not to expect it. A second main target audience is students of statistics or mathematics, at all levels, who are curious about the mathematics underlying this surpr…
Benford's law describes a common phenomenon among many naturally occurring data sets and distributions in which the leading digits of the data are distributed with the probability of a first digit of $d$ base $B$ being $\log_{B}{\frac{d+1}{d}}$. As it often successfully detects fraud in medical trials, voting, science and finance, significant effort has been made to understand when and how distributions exhibit Benford behavior. Most of the previous work has been restricted to cases of independent variables, and little is known about situations involving dependence. We use copulas to investiga…
Feller's classic text 'An Introduction to Probability Theory and its Applications' contains a derivation of the well known significant-digit law called Benford's law. More specifically, Feller gives a sufficient condition ("large spread") for a random variable $X$ to be approximately Benford distributed, that is, for $\log_{10}X$ to be approximately uniformly distributed modulo one. This note shows that the large-spread derivation, which continues to be widely cited and used, contains serious basic errors. Concrete examples and a new inequality clearly demonstrate that large spread (or large s…
Benford's Law predicts that the first significant digit on the leftmost side of numbers in real-life data is proportioned between all possible 1 to 9 digits approximately as in LOG(1 + 1/digit), so that low digits occur much more frequently than high digits in the first place. The two essential prerequisites for data configuration with regards to compliance with Benford's Law are high order of magnitude and positive skewness with a tail falling to the right of the histogram, so that quantitative configuration is such that the small is numerous and the big is rare. A related topic in the study …
Benford's Law predicts that the first significant digit on the leftmost side of numbers in real-life data is proportioned between all possible 1 to 9 digits approximately as in LOG(1 + 1/digit), so that low digits occur much more frequently than high digits in the first place. For example, digit 1 occurs approximately 30.1% in the first place in random numbers, while digit 9 occurs only approximately 4.6%. In this article it is shown that a process where a large enough set of identical quantities constantly alternates between minuscule random consolidations (summing two randomly chosen values …
Kossovsky recently conjectured that the distribution of leading digits of a chain of probability distributions converges to Benford's law as the length of the chain grows. We prove his conjecture in many cases, and provide an interpretation in terms of products of independent random variables and a central limit theorem. An interesting consequence is that in hierarchical Bayesian models priors tend to satisfy Benford's Law as the number of levels of the hierarchy increases, which allows us to develop some simple tests (based on Benford's law) to test proposed models. We give explicit formulas …
Exponential growth occurs when the growth rate of a given quantity is proportional to the quantity's current value. Surprisingly, when exponential growth data is plotted as a simple histogram disregarding the time dimension, a remarkable fit to the positively skewed k/x distribution is found, where the small is numerous and the big is rare. Such quantitative preference for the small has a corresponding digital preference known as Benford's Law which predicts that the first significant digit on the left-most side of numbers in typical real-life data is proportioned between all possible 1 to 9 d…
The occurrence of digits 1 through 9 as the leftmost nonzero digit of numbers from real-world sources is distributed unevenly according to an empirical law, known as Benford's law or the first digit law. It remains obscure why a variety of data sets generated from quite different dynamics obey this particular law. We perform a study of Benford's law from the application of the Laplace transform, and find that the logarithmic Laplace spectrum of the digital indicator function can be approximately taken as a constant. This particular constant, being exactly the Benford term, explains the prevale…
Prime numbers seem to distribute among the natural numbers with no other law than that of chance, however its global distribution presents a quite remarkable smoothness. Such interplay between randomness and regularity has motivated sci- entists of all ages to search for local and global patterns in this distribution that eventually could shed light into the ultimate nature of primes. In this work we show that a generalization of the well known first-digit Benford's law, which addresses the rate of appearance of a given leading digit d in data sets, describes with astonishing precision the sta…
That the logarithmic distribution manifests itself in the random as well as in the deterministic (multiplication processes) has long intrigued researchers in Benford's Law. In this article it is argued that it springs from one common intrinsic feature of their density curves. On the other hand, the profound dichotomy between the random and the deterministic in the context of Benford's Law is noted here, acknowledging the need to distinguish between them. From its very inception, the field has been suffering from a profound confusion and mixing of these two very different logarithmic flavors, c…
Manipulation testing based on Benford's Law for discrete scores
Sapienza University of Rome（罗马大学）；GRANEM, University of Angers（昂热大学 GRANEM）；Department of Social and Economic Sciences, Sapienza University of Rome（罗马大学社会与经济科学系）；Department of Economics and Law, Sapienza University of Rome（罗马大学经济与法律系）
