A Comprehensive Summary of the Benford's Law Phenomenon
Alex Ely Kossovsky
World Scientific Publishing Company 
Naturwissenschaften, Medizin, Informatik, Technik / Wahrscheinlichkeitstheorie, Stochastik, Mathematische Statistik
Beschreibung
Numbers are written in our digital language system by conveniently and efficiently utilizing the ten digits 0 to 9 in much the same way as sentences and books are written in the English language system by conveniently utilizing the 26 letters A to Z. Surprisingly, and against all common sense or intuition, the spread of these ten digits within numbers of random data is not uniform, but rather highly uneven. Benford's Law predicts that the first digit on the left-most side of numbers is proportioned between all possible digits 1 to 9 approximately according to 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}. Remarkably, Benford's Law is found to be valid in almost all real life statistics, from data relating to physics, astronomy, chemistry, geology, and biology to data relating to economics, accounting, finance, engineering, and governmental census information. Therefore, Benford's Law stands as the only common thread running through and uniting all scientific disciplines!
This book represents an intense and concentrated effort by the author to narrate this digital, numerical, and quantitative story of the Benford's Law phenomenon as briefly and as concisely as possible, while still ensuring a comprehensive coverage of all its aspects, results, causes, explanations, and perspectives. The most recent research results and discoveries in this field are included within this book in such a way as to be comprehensible and engaging to readers of all proficiencies.
Contents:
- About the Author
- Introduction
- The Digits Phenomenon:
- The First Digit on the Left Side of Numbers
- Benford's Law and the Predominance of Low Digits
- Second-Digits and Third-Digits Distributions
- The Quantitative Origin of the Digital-Numerical Phenomenon
- The Scale Invariance Principle
- The Base Invariance Principle
- Physical Order of Magnitude of Data
- Robust Measure of Physical Order of Magnitude
- Two Essential Requirements for Benford Behavior
- Sum of Squared Deviations Measure
- The Mistaken Use of the Chi-Square Test in Benford's Law
- Causes and Explanations:
- Multiplication Processes Lead to Positive Skewness and Often to Benford
- Addition Processes Lead to the Symmetrical Normal away from Benford
- The Multiplicative Central Limit Theorem and Lognormal Distribution
- Multiplications are More Prevalent than Additions in Real-Life Data
- Tugs of War between Addition and Multiplication
- Partitions Typically Lead to Positive Skewness and Often to Benford
- One-Dimensional Random Staged Partition
- One-Dimensional Chaotic Repeated Partition
- One-Dimensional Random Real Partition
- Two-Dimensional Random Partition
- The General Requirements for Partitions to Converge to Benford
- Benford Model for Planet and Star Formations
- Consolidation and Fragmentation Processes
- Random Exponential Growth Leads to Positive Skewness and Benford
- Data Aggregation Leads to Positive Skewness and Often to Benford
- Chains of Statistical Distributions and Benford's Law
- Meta-Explanation or the Explanation of all Explanations
- The Logarithmic Perspective:
- Benford's Law as Uniformity of Mantissa
- Rising or Falling Mantissa Distributions
- Uniqueness of k/x Distribution and Its Central Role in Benford's Law
- Related Log Conjecture
- The Random and Deterministic Flavors in Benford's Law
- The Great Prevalence of the Digital Development Pattern in Data
- The Absence of the Digital Development Pattern in k/x Distribution
- Benford's Law in Its Purest Form
- Constant Base Raised to a Random Power
- General Results:
- General Results in Benford's Law
- First Two Digits versus Last Two Digits
- The Law of Relative Quantities:
- The Relating Concepts of Digits, Numbers, and Quantities
- The Arbitrariness of our Positional Number System
- Two Radically Different Interpretations of the Benford Phenomenon
- The Quest for a Universal and Number-System-Invariant Measure
- The Shape and Nature of Histograms are Number-System Invariant
- Constructing a Three-Bin Histogram Signifying Small, Medium, and Big
- Constructing a Set of Infinitely Expanding Histograms
- Numerical Consistency in Bin Schemes for 15 Real-Life Data Sets
- The Postulate on Relative Quantities
- Application of the Postulate via Generic Bin Scheme on k/x
- The Infinite Sequence Result for the Bin Scheme on k/x
- The General Law of Relative Quantities
- Benford's Law as a Special Case and Direct Consequence of GLORQ
- The Universal Law of Relative Quantities
- Benford Second-Order Digits Interpreted as an Irregular Bin Scheme
- Concluding Historical and Conceptual Perspectives
- Appendices:
- Infinite Sequence Reduction
- Data Sets
- Glossary of Frequently Used Abbreviations
- First Two Digits versus Last Two Digits
- Index
Readership: This book is suitable for expert mathematicians, statisticians, and scientists, as well as university students of these disciplines. This book is also suitable for the layman, the non-expert, and the educated general public, who are not necessarily proficient in mathematics, statistics, and the sciences, but who have enough interest to still be able to understand the topic on the whole.
Alex Ely Kossovsky is an independent scholar and the author of the books Benford's Law: Theory, the General Law of Relative Quantities, and Forensic Fraud Detection Applications (World Scientific Publishing Company, 2014); Small is Beautiful: Why the Small is Numerous but the Big is Rare in the World (Kindle Direct Publishing, 2017); and Studies in Benford's Law: Arithmetical Tugs of War, Quantitative Partition Models, Prime Numbers, Exponential Growth Series, and Data Forensics (Kindle Direct Publishing, 2019). Kossovsky is the inventor of a patented mathematical algorithm in data fraud detection analysis, registered at the US Patent Office. He specialized in Applied Mathematics and Statistics at the City University of New York, and in Physics and Pure Mathematics at the State University of New York at Stony Brook.
