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We address the task of identifying anomalous observations by analyzing digits under the lens of Benford's law. Motivated by the crucial objective of providing reliable statistical analysis of customs declarations, we answer one major and…

Methodology · Statistics 2025-07-14 Lucio Barabesi , Andrea Cerioli , Andrea Cerasa , Domenico Perrotta

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…

Statistics Theory · Mathematics 2019-01-04 Alex Ely Kossovsky

Many mathematical, man-made and natural systems exhibit a leading-digit bias, where a first digit (base 10) of 1 occurs not 11\% of the time, as one would expect if all digits were equally likely, but rather 30\%. This phenomenon is known…

Benford's law is widely used for fraud-detection nowadays. The underlying assumption for using the law is that a "regular" dataset follows the significant digit phenomenon. In this paper, we address the scenario where a shrewd fraudster…

Applications · Statistics 2021-05-21 Javad Kazemitabar

Benford's Law (BL) or the Significant Digit Law defines the probability distribution of the first digit of numerical values in a data sample. This Law is observed in many naturally occurring datasets. It can be seen as a measure of…

Machine Learning · Computer Science 2021-10-25 Surya Kant Sahu , Abhinav Java , Arshad Shaikh , Yannic Kilcher

The applicability of the Benford law for different data sets of atomic cross sections by ion impact is studied. We find that the data sets corresponding to theoretical ionization and capture cross sections of neutral targets by multicharged…

Atomic Physics · Physics 2021-06-14 J. E. Miraglia , M. S. Gravielle

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…

Statistics Theory · Mathematics 2019-01-03 Alex Ely Kossovsky

Nature and our world have a bias! Roughly $30\%$ of the time the number $1$ occurs as the leading digit in many datasets base $10$. This phenomenon is known as Benford's law and it arrises in diverse fields such as the stock market,…

Probability · Mathematics 2023-08-16 Irfan Durmić , Steven J. Miller

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…

Statistics Theory · Mathematics 2013-01-28 Aaron Carl Smith

Benford's law describes the distribution of the first digit of numbers appearing in a wide variety of numerical data, including tax records, and election outcomes, and has been used to raise "red flags" about potential anomalies in the data…

Social and Information Networks · Computer Science 2022-05-27 Tianyi Chen , Charalampos E. Tsourakakis

Benford's law states that the occurrence of significant digits in many data sets is not uniform but tends to follow a logarithmic distribution such that the smaller digits appear as first significant digits more frequently than the larger…

Data Analysis, Statistics and Probability · Physics 2011-11-07 Tariq Ahmad Mir

Benford's law is a famous law in statistics which states that the leading digits of random variables in diverse data sets appear not uniformly from 1 to 9; the probability that d (d=1,...,9) appears as a leading digit is given by…

Probability · Mathematics 2019-05-07 Kazufumi Ozawa

The use of statistical methods for the description of complex quantum systems was primarily motivated by the failure of a line-by-line interpretation of atomic spectra. Such methods reveal regularities and trends in the distributions of…

Atomic Physics · Physics 2021-05-03 Jean-Christophe Pain

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…

Statistics Theory · Mathematics 2015-11-19 G. Whyman , E. Shulzinger , Ed. Bormashenko

The Newcomb-Benford Law, which is also called the first digit phenomenon, has applications in diverse phenomena ranging from social and computer networks, engineering systems, natural sciences, and accounting. In forensics, it has been used…

Physics and Society · Physics 2018-06-19 Subhash Kak

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…

Physics and Society · Physics 2020-01-22 Alex Ely Kossovsky

A phenomenological law, called Benford's law, states that the occurrence of the first digit, i.e., $1,2,...,9$, of numbers from many real world sources is not uniformly distributed, but instead favors smaller ones according to a logarithmic…

High Energy Physics - Phenomenology · Physics 2010-04-22 Lijing Shao , Bo-Qiang Ma

Benford's law is the statement that in many real-world data sets, the probability of having digit \(d\) in base \(B\), where \(1 \leq d \leq B\), as the first digit is \(\log_{B}\left(\tfrac{d+1}{d}\right)\). We sometimes refer to this as…

Probability · Mathematics 2025-08-26 Bruce Fang , Ava Irons , Ella Lippelman , Steven J. Miller

Benford's law is an empirical ``law'' governing the frequency of leading digits in numerical data sets. Surprisingly, for mathematical sequences the predictions derived from it can be uncannily accurate. For example, among the first billion…

Probability · Mathematics 2020-04-28 Zhaodong Cai , Matthew Faust , A. J. Hildebrand , Junxian Li , Yuan Zhang

We show that the frequency distribution of the first significant digits of the numbers in the data sets generated from a large class of measures of quantum correlations, which are either entanglement measures, or belong to the…

Quantum Physics · Physics 2016-07-28 Titas Chanda , Tamoghna Das , Debasis Sadhukhan , Amit Kumar Pal , Aditi Sen De , Ujjwal Sen