Topics in Benford S Law

This book provides an in-depth discussion of several key topics in the field of Benford's Law, highlighting recent advances in forensic data analysis while also drawing surprising links to diverse areas of mathematics.

Benford's Law predicts that the first digit on the left-most side of any number occurring in the physical world is proportioned between all possible digits from 1 to 9 according to the distribution LOG(1 + 1/digit), so that occurrences of low digits such as {1, 2, 3} in the first position are much more frequent than occurrences of high digits such as {7, 8, 9}. The digit 1 is allocated a proportion of LOG(1 + 1/1), about 30.1%, whereas the digit 9 is allocated a proportion of LOG(1 + 1/9), about 4.6%. Remarkably, Benford's Law is found to be valid for nearly all types of real-world statistics, including data relating to physics, chemistry, astronomy, geology, biology, economics, finance, accounting, engineering, and governmental census information. As such, Benford's Law constitutes a unique common thread running through and uniting data sets in all existing scientific disciplines.

Beyond a thorough treatment of the fundamental theory of Benford's Law, the book examines cutting-edge developments in its application to forensic data analysis and explores the subject's many unexpected connections to other areas of mathematics, including prime numbers, quantitative partition models, and exponential growth series. In particular, the book addresses the resultant quantitative and digital configurations of an arithmetical mix of random variables, such as addition processes which are known to yield the symmetrical Anti-Benford Normal Distribution as predicated by the Central Limit Theorem, as well as multiplication processes which are known to yield the skewed Pro-Benford Lognormal Distribution. The involvement of various additive and multiplicative terms within a single algebraic expression of a random process constitutes a tug of war between these two arithmetical operations.