COMPREHENSIVE SUMMARY OF THE BENFORD\'S LAW PHENOME

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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. Errata(s) Errata (177 KB) Sample Chapter(s) Sample of Section I: The Digits Phenomenon 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.