Editing Benford's law (section)

===Multiple probability distributions===
[[Anton Formann]] provided an alternative explanation by directing attention to the interrelation between the [[Probability distribution|distribution]] of the significant digits and the distribution of the [[dependent variable|observed variable]]. He showed in a simulation study that long-right-tailed distributions of a [[random variable]] are compatible with the Newcomb–Benford law, and that for distributions of the ratio of two random variables the fit generally improves.<ref>{{cite journal |last1=Formann |first1=A. K. |year=2010 |title=The Newcomb–Benford law in its relation to some common distributions |journal=PLOS ONE |volume=5 |issue=5 |page=e10541 |doi=10.1371/journal.pone.0010541 |pmid=20479878 |pmc=2866333 |bibcode=2010PLoSO...510541F |doi-access=free}}</ref> For numbers drawn from certain distributions ([[IQ score]]s, human heights) the Benford's law fails to hold because these variates obey a normal distribution, which is known not to satisfy Benford's law,<ref name=Formann2010>{{Cite journal
| last1 = Formann | first1 = A. K.
| title = The Newcomb–Benford Law in Its Relation to Some Common Distributions
| doi = 10.1371/journal.pone.0010541
| journal = PLOS ONE
| volume = 5
| issue = 5
| pages = e10541
| year = 2010
| pmid = 20479878
| pmc = 2866333
| editor1-last = Morris
| editor1-first = Richard James
| bibcode = 2010PLoSO...510541F
| doi-access = free
}}</ref> since normal distributions can't span several orders of magnitude and the [[Significand]] of their logarithms will not be (even approximately) uniformly distributed. However, if one "mixes" numbers from those distributions, for example, by taking numbers from newspaper articles, Benford's law reappears. This can also be proven mathematically: if one repeatedly "randomly" chooses a [[probability distribution]] (from an uncorrelated set) and then randomly chooses a number according to that distribution, the resulting list of numbers will obey Benford's law.<ref name=Hill1995>{{Cite journal
 | author = Theodore P. Hill
 | author-link = Theodore P. Hill
 | title = A Statistical Derivation of the Significant-Digit Law
 | journal = Statistical Science
 | volume = 10
 | issue = 4
 | pages = 354–363
 | year = 1995
 | url = http://digitalcommons.calpoly.edu/cgi/viewcontent.cgi?article=1042&context=rgp_rsr
 | doi = 10.1214/ss/1177009869
 | mr = 1421567
 | doi-access = free
}}</ref><ref name=Hill1998>{{Cite journal
 | author = Theodore P. Hill
 | author-link = Theodore P. Hill
 | title = The first digit phenomenon
 | journal = [[American Scientist]]
 | volume = 86
 | date = July–August 1998
 | page = 358
 | url = http://people.math.gatech.edu/~hill/publications/PAPER%20PDFS/TheFirstDigitPhenomenonAmericanScientist1996.pdf
 | bibcode = 1998AmSci..86..358H
 | doi = 10.1511/1998.4.358
 | issue = 4| s2cid = 13553246
 }}</ref> A similar probabilistic explanation for the appearance of Benford's law in everyday-life numbers has been advanced by showing that it arises naturally when one considers mixtures of uniform distributions.<ref>{{cite journal | last1 = Janvresse | first1 = Élise | last2 = Thierry | year = 2004 | title = From Uniform Distributions to Benford's Law | url = http://lmrs.univ-rouen.fr/Persopage/Delarue/Publis/PDF/uniform_distribution_to_Benford_law.pdf | journal = Journal of Applied Probability | volume = 41 | issue = 4 | pages = 1203–1210 | doi = 10.1239/jap/1101840566 | mr = 2122815 | access-date = 13 August 2015 | archive-url = https://web.archive.org/web/20160304125725/http://lmrs.univ-rouen.fr/Persopage/Delarue/Publis/PDF/uniform_distribution_to_Benford_law.pdf | archive-date = 4 March 2016 | url-status = dead }}</ref>