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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…

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 describes the finding that the distribution of leading (or leftmost) digits of innumerable datasets follows a well-defined logarithmic trend, rather than an intuitive uniformity. In practice this means that the most common…

Data Analysis, Statistics and Probability · Physics 2013-11-20 Aaron D. Slepkov , Kevin B. Ironside , David DiBattista

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

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

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…

Statistics Theory · Mathematics 2007-06-13 Zhipeng Li , Lin Cong , Huajia Wang

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 the statement that in many real world data sets, the probability of having digit $d$ in base $B$ as the first digit is \log_{B}\!\left(\frac{d+1}{d}\right) for all $1 \leq d \leq B$. We sometimes refer to this as weak…

Probability · Mathematics 2026-03-06 Bruce Fang , 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

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 the occurrence of the $n^{\mathrm{th}}$ digit of numbers in datasets originating from various sources of the world, ranging from financial data to atomic spectra. It is intriguing that although many features of…

Popular Physics · Physics 2014-09-11 T. Alexopoulos , S. Leontsinis

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

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

The uneven distribution of digits in numerical data, known as Benford's law, was discovered in 1881. Since then, this law has been shown to be correct in copious numerical data relating to economics, physics and even prime numbers. Although…

Discrete Mathematics · Computer Science 2009-07-28 Oded Kafri

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 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

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

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…

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…

Probability · Mathematics 2018-06-13 Stéphane Blondeau da Silva

Benford's Law is an empirical law which predicts the frequency of significant digits in databases corresponding to various phenomena, natural or artificial. Although counter intuitive at the first sight, it predicts a higher occurrence of…

Data Analysis, Statistics and Probability · Physics 2014-06-30 Gaurav Bhole , Abhishek Shukla , T. S. Mahesh
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