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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 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

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 states that many data sets have a bias towards lower leading digits (about $30\%$ are 1s). There are numerous applications, from designing efficient computers to detecting tax, voter and image fraud. It's important to know…

Probability · Mathematics 2016-01-20 Victoria Cuff , Allison Lewis , Steven J. Miller

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

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 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…

Probability · Mathematics 2019-10-30 Rebecca F. Durst , Steven J. Miller

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

Infrared spectra of various polymers were treated statistically. It was established that for the absorbance spectra the Benford distribution of leading digits takes place, whereas the distribution of leading digits for transmittance spectra…

General Physics · Physics 2015-06-10 Ed. Bormashenko , E. Shulzinger , G. Whyman , Ye. Bormashenko

According to Benford's Law, many data sets have a bias towards lower leading digits (about $30\%$ are $1$'s). The applications of Benford's Law vary: from detecting tax, voter and image fraud to determining the possibility of match-fixing…

The occurrence of first significant digits of numbers in large data is often governed by a logarithmically decreasing distribution called Benford's law (BL), reported first by S. Newcomb (SN) and many decades later independently by F.…

Digital Libraries · Computer Science 2016-03-22 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

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

Benford's Law states that the frequency of first digits of numbers in naturally occurring systems is not evenly distributed. Numbers beginning with a 1 occur roughly 30\% of the time, and are six times more common than numbers beginning…

Social and Information Networks · Computer Science 2016-02-17 Jennifer Golbeck

The Newcomb-Benford law, also known as the first-digit law, gives the probability distribution associated with the first digit of a dataset, so that, for example, the first significant digit has a probability of $30.1$ % of being $1$ and…

Popular Physics · Physics 2021-08-25 Andrea Burgos , Andrés Santos

The occurrence of the nonzero leftmost digit, i.e., 1, 2, ..., 9, of numbers from many real world sources is not uniformly distributed as one might naively expect, but instead, the nature favors smaller ones according to a logarithmic…

Data Analysis, Statistics and Probability · Physics 2014-11-21 Lijing Shao , Bo-Qiang Ma

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…

Other Statistics · Statistics 2019-05-02 Mingshu Cong , Congqiao Li , Bo-Qiang Ma

We explain Kossovsky's generalization of Benford's law which is a formula that approximates the distribution of leftmost digits in finite sequences of natural data and apply it to six sequences of data including populations of US cities and…

Methodology · Statistics 2023-08-16 Alex E. Kossovsky , Wayne M. Lawton

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 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…

Statistics Theory · Mathematics 2007-06-13 Werner Hurlimann