Benford's Law: Theory, the General Law of Relative Quantities, and Forensic Fraud Detection Applications

Description

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 gives 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.* * *Benford, Frank. “[The Law of Anomalous Numbers](https://isidore.co/misc/Physics%20papers%20and%20books/Zotero/storage/ZEBWDL73/Benford%20-%201938%20-%20The%20Law%20of%20Anomalous%20Numbers.pdf).” *Proceedings of the American Philosophical Society* 78, no. 4 (1938): 551–72. ref:9.384 discusses the scale-invariance of Benford's Law. (Related to my question of why 1 mi. = 5280 ft. have different numbers of significant figures; is a ft. more precise than a mi.?) The log distribution is the only one with such property. cf. ref:12.171 ("The Scale Invariance Principle"). (This reminds me of the [Buckingham π theorem](https://isidore.co/calibre#panel=book_details&book_id=6104) / dimensional analysis, which shows that there are dimensionless / scale-invariant quantites.)* * *Someone claimed this violates Benford's law, which is used for fraud detection in accounting ([its *Encylo. of Math.* article](https://encyclopediaofmath.org/wiki/Benford_law) discusses the interesting function D *n* ( *x* )= *n* th significant digit of *x* ) ~~, but it seems to be a simple application of the Central Limit Theorem (each new ballot/sample coming in could be distributed according to any distribution, but the overall distribution is Gaussian)~~ : [](https://nitter.dark.fail/MrAnthonyRogers/status/1325575157256232960)Update/correction: I think what's being plotted is first digit frequencies of election data. Biden's doesn't follow Bedford's logarithmic distribution like Trump's does. (When I wrote what I ~~struck out~~ above, I thought they were plotting votes counted vs. days after election.)§2 "Forensic Digital Analysis & Fraud Detection", §§"The part and type of data applicable to forensic testing" (p. 94 / PDF p. 117; EPUB ref:10.55):> Election results closely conform to Benford’s Law. This is so since electoral results are simply manipulated fractional population data; as in 31% of population voting for candidate A promising policy bundle X, 27% voting for candidate B promising policy bundle Y, and the remaining 42% of the population simply not voting in order to preserve their human dignity and freedom by not participating in such a ridiculous charade, patiently awaiting the triumphant arrival of direct and participatory democracy instead. Since population data itself conforms strongly to the law, by the scale invariance principle, the same conformity should be found here for fractional population values. A more careful examination of the underlying statistical process reveals that this is ultimately a Random Linear Combination from random logarithmic data. This confers election results an even more logarithmic aura than mere population data. A cursory look at details of election data reveals this, as it typically reads: 43.78% Democratic vote in Rockland County, New York; 50.01% in Palm Beach County, Florida; 23.95% in Mariposa County, California; and so forth. A few fascinating electoral studies using Benford’s Law to detect potential electoral fraud have been published recently, with better results obtained by way of the second-order-digit distribution.Update #2: Check out [cjph8914's 2020_benfords repo](https://github.com/cjph8914/2020_benfords), an analysis of first digit distributions of real election data.Another useful package is [benford_py.](https://github.com/milcent/benford_py)A little [first-two-digits analysis on Allegheny Co., PA's and Maricopa Co., AZ's data](https://isidore.co/jupyter/notebooks/benford_py/BenfordFTD_2020election.ipynb):Allegheny: Trump χ² = 201, Biden χ² = 671 Maricopa: Trump χ² = 127, Biden χ² = 297 (95% conf. int. critical χ² = 112)⅔ of election-day voters in Maricopa Co. voted for Trump! Roughly 100k more mail-in ballots for Biden than Trump, too; are there any non-fraud explanations for this?