# Benford's Law

## Theory, the General Law of Relative Quantities, and Forensic Fraud Detection Applications

## Alex Ely Kossovsky

- 672 pages
- English
- ePUB (mobile friendly)
- Available on iOS & Android

# Benford's Law

## Theory, the General Law of Relative Quantities, and Forensic Fraud Detection Applications

## Alex Ely Kossovsky

## About This Book

Contrary to common intuition that all digits should occur randomly with equal chances in real data, empirical examinations consistently show that not all digits are created equal, but rather that low digits such as {1, 2, 3} occur much more frequently than high digits such as {7, 8, 9} in almost all data types, such as those relating to geology, chemistry, astronomy, physics, and engineering, as well as in accounting, financial, econometrics, and demographics data sets. This intriguing digital phenomenon is known as Benford's Law.

This book represents an attempt to give a comprehensive and in-depth account of all the theoretical aspects, results, causes and explanations of Benford's Law, with a strong emphasis on the connection to real-life data and the physical manifestation of the law. In addition to such a bird's eye view of the digital phenomenon, the conceptual distinctions between digits, numbers, and quantities are explored; leading to the key finding that the phenomenon is actually quantitative in nature; originating from the fact that in extreme generality, nature creates many small quantities but very few big quantities, corroborating the motto "small is beautiful", and that therefore all this is applicable just as well to data written in the ancient Roman, Mayan, Egyptian, and other digit-less civilizations.

Fraudsters are typically not aware of this digital pattern and tend to invent numbers with approximately equal digital frequencies. The digital analyst can easily check reported data for compliance with this digital law, enabling the detection of tax evasion, Ponzi schemes, and other financial scams. The forensic fraud detection section in this book is written in a very concise and reader-friendly style; gathering all known methods and standards in the accounting and auditing industry; summarizing and fusing them into a singular coherent whole; and can be understood without deep knowledge in statistical theory or advanced mathematics. In addition, a digital algorithm is presented, enabling the auditor to detect fraud even when the sophisticated cheater is aware of the law and invents numbers accordingly. The algorithm employs a subtle inner digital pattern within the Benford's pattern itself. This newly discovered pattern is deemed to be nearly universal, being even more prevalent than the Benford phenomenon, as it is found in all random data sets, Benford as well as non-Benford types.

Contents:

- Benford's Law
- Forensic Digital Analysis & Fraud Detection
- Data Compliance Tests
- Conceptual and Mathematical Foundations
- Benford's Law in the Physical Sciences
- Topics in Benford's Law
- The Law of Relative Quantities

Readership: Researchers in probability and statistics, forensic data analysis. Key Features:

- The book is a concise account of all known aspects in practical applications of the phenomenon to fraud detection. It also corrects several errors committed in the field where mistaken applications are used
- The perceptive reader such as an accountant, an auditor or an official at any governmental tax authority worldwide, interested in knowing about the use of this digital law in fraud detection, would be able to learn about it with ease and with a minimal amount of effort and time, instead of searching through literally hundreds of various small articles on the topic
- The book provides numerous new theoretical points of view of the phenomenon, new methods for testing data for compliance, and fuses many different aspects of the law into a singular explanation

## Frequently asked questions

## Information

**DIGITS VERSUS NUMBERS**

**ā for the statistician to even consider and analyze. Is there indeed a particular statistical law supposedly governing digital proportions? In addition, it seems doubtful that there would be any use or consequence in looking into this digital language proportion in the first place. Are there any applications that can exploit the examination of these digital proportions?**

*too random***TO FIND FRAUD, SIMPLY EXAMINE ITS DIGITS!**

**Figure 1.1**Hypothetical Accounting Data for Five Companies

**Figure 1.2**1st Digits Proportions of the Data of Five Companies

**FIRST LEADING DIGITS**

^{N}with N being an integer and A being a real number such that 1 ā¤ |A| < 10. For such representation of numbers, the integral part of A (excluding the fractional part), and with the positive or negative sign ignored, is what we consider the first leading digit. For example, the number 311.75 is scientifically written as 3.1175*10

^{2}and digit 3 leads the number. Naturally, when digit d appears first in a number composed of several digits, we call d the

**āleaderā**, as it leads all the other digits trailing behind it to the right.

**EMPIRICAL EVIDENCE FROM REAL-LIFE DATA ON DIGIT DISTRIBUTION**

*lowest*digit 1 (shown in large and bold font), and only five start with the

*highest*digit 9 (shown within circles), namely a ratio of 4:1 roughly. This result is surprising yet approximately compatible with the digital results seen in the example with stock prices and volume data. In this digital analysis the numbers 1, 10, and 100 are grouped together under the same category since all of them are being led by digit 1. Digital proportions here are {21%, 17%, 13%, 14%, 8%, 9%, 6%, 7%, 5%}.

**Figure 1.3**Price and Volume of Stocks Traded on the NYSE