Editing Benford's law (section)

==Range of applicability==
===Distributions known to obey Benford's law===
Some well-known infinite [[integer sequence]]s {{not a typo|provably}} satisfy Benford's law exactly (in the [[asymptotic limit]] as more and more terms of the sequence are included). Among these are the [[Fibonacci number]]s,<ref>{{cite journal | last1 = Washington | first1 = L. C. | year = 1981 | title = Benford's Law for Fibonacci and Lucas Numbers | journal = [[The Fibonacci Quarterly]] | volume = 19 | issue = 2| pages = 175–177| doi = 10.1080/00150517.1981.12430109 }}</ref><ref>{{cite journal | last1 = Duncan | first1 = R. L. | year = 1967 | title = An Application of Uniform Distribution to the Fibonacci Numbers | journal = [[The Fibonacci Quarterly]] | volume = 5 | issue = 2 | pages = 137–140| doi = 10.1080/00150517.1967.12431312 }}</ref> the [[factorial]]s,<ref>{{cite journal | last1 = Sarkar | first1 = P. B. | year = 1973 | title = An Observation on the Significant Digits of Binomial Coefficients and Factorials | journal = Sankhya B | volume = 35 | pages = 363–364}}</ref> the powers of&nbsp;2,<ref name=powers>In general, the sequence ''k''<sup>1</sup>, ''k''<sup>2</sup>, ''k''<sup>3</sup>, etc., satisfies Benford's law exactly, under the condition that log<sub>10</sub> ''k'' is an [[irrational number]]. This is a straightforward consequence of the [[equidistribution theorem]].</ref><ref name=":0">That the first 100 powers of&nbsp;2 approximately satisfy Benford's law is mentioned by Ralph Raimi. {{cite journal | last1 = Raimi | first1 = Ralph A. | year = 1976 | title = The First Digit Problem | journal = [[American Mathematical Monthly]] | volume = 83 | issue = 7| pages = 521–538 | doi=10.2307/2319349| jstor = 2319349}}</ref> and the powers of almost any other number.<ref name=powers />

Likewise, some continuous processes satisfy Benford's law exactly (in the asymptotic limit as the process continues through time). One is an [[exponential growth]] or [[exponential decay|decay]] process: If a quantity is exponentially increasing or decreasing in time, then the percentage of time that it has each first digit satisfies Benford's law asymptotically (i.e. increasing accuracy as the process continues through time).

===Distributions known to disobey Benford's law===
The [[square root]]s and [[Multiplicative inverse|reciprocal]]s of successive natural numbers do not obey this law.<ref name=Raimi1976>{{cite journal |last=Raimi |first=Ralph A. |date=Aug–Sep 1976 |title=The first digit problem |journal=[[American Mathematical Monthly]] |volume=83 |issue=7 |pages=521–538 |doi=10.2307/2319349|jstor=2319349}}</ref> Prime numbers in a finite range follow a Generalized Benford’s law, that approaches uniformity as the size of the range approaches infinity.<ref>{{Cite web|last1=Zyga|first1=Lisa|last2=Phys.org|title=New Pattern Found in Prime Numbers|url=https://phys.org/news/2009-05-pattern-prime.html|access-date=2022-01-23|website=phys.org|language=en}}</ref> Lists of local telephone numbers violate Benford's law.<ref>{{Cite journal| issn = 0003-1305| volume = 61| issue = 3| pages = 218–223| last1 = Cho| first1 = Wendy K. Tam| last2 = Gaines| first2 = Brian J.| title = Breaking the (Benford) Law: Statistical Fraud Detection in Campaign Finance| journal = The American Statistician| accessdate = 2022-03-08| date = 2007| doi = 10.1198/000313007X223496| url = https://www.jstor.org/stable/27643897| jstor = 27643897| s2cid = 7938920}}</ref> Benford's law is violated by the populations of all places with a population of at least 2500 individuals from five US states according to the 1960 and 1970 censuses, where only 19 % began with digit 1 but 20 % began with digit 2, because truncation at 2500 introduces statistical bias.<ref name=Raimi1976/> The terminal digits in pathology reports violate Benford's law due to rounding.<ref name=Beer2009>{{cite journal |last1=Beer |first1=Trevor W. |s2cid=206987736 |year=2009 |title=Terminal digit preference: beware of Benford's law |journal=[[J. Clin. Pathol.]] |volume=62 |issue=2 |page=192 |doi=10.1136/jcp.2008.061721|pmid=19181640}}</ref>

Distributions that do not span several orders of magnitude will not follow Benford's law. Examples include height, weight, and IQ scores.<ref name=Formann2010/><ref>Singleton, Tommie W. (May 1, 2011). "[https://www.isaca.org/resources/isaca-journal/past-issues/2011/understanding-and-applying-benfords-law Understanding and Applying Benford’s Law]", ''ISACA Journal'', [[ISACA|Information Systems Audit and Control Association]]. Retrieved Nov. 9, 2020.</ref>

===Criteria for distributions expected and not expected to obey Benford's law===
A number of criteria, applicable particularly to accounting data, have been suggested where Benford's law can be expected to apply.<ref name=Durtschi2004>{{cite journal | last1 = Durtschi | first1 = C | last2 = Hillison | first2 = W | last3 = Pacini | first3 = C | year = 2004 | title = The effective use of Benford's law to assist in detecting fraud in accounting data | journal = J Forensic Accounting | volume = 5 | pages = 17–34}}</ref>

;Distributions that can be expected to obey Benford's law
* When the mean is greater than the median and the skew is positive
* Numbers that result from mathematical combination of numbers: e.g. quantity × price
* Transaction level data: e.g. disbursements, sales

;Distributions that would not be expected to obey Benford's law
* Where numbers are assigned sequentially: e.g. check numbers, invoice numbers
* Where numbers are influenced by human thought: e.g. prices set by psychological thresholds ($9.99)
* Accounts with a large number of firm-specific numbers: e.g. accounts set up to record $100 refunds
* Accounts with a built-in minimum or maximum
* Distributions that do not span an order of magnitude of numbers.

===Benford’s law compliance theorem===
Mathematically, Benford’s law applies if the distribution being tested fits the "Benford’s law compliance theorem".<ref name="dspguide"/> The derivation says that Benford's law is followed if the [[Fourier transform]] of the logarithm of the probability density function is zero for all integer values. Most notably, this is satisfied if the Fourier transform is zero (or negligible) for ''n''&nbsp;≥&nbsp;1. This is satisfied if the distribution is wide (since wide distribution implies a narrow Fourier transform). Smith summarizes thus (p.&nbsp;716): 
<blockquote>
Benford's law is followed by distributions that are wide compared with unit distance along the logarithmic scale. Likewise, the law is not followed by distributions that are narrow compared with unit distance&nbsp;… If the distribution is wide compared with unit distance on the log axis, it means that the spread in the set of numbers being examined is much greater than ten.
</blockquote>

In short, Benford’s law requires that the numbers in the distribution being measured have a spread across at least an order of magnitude.