Editing Benford's law (section)

==Tests with common distributions==
Benford's law was empirically tested against the numbers (up to the 10th digit) generated by a number of important distributions, including the [[uniform distribution (discrete)|uniform distribution]], the [[exponential distribution]], the [[normal distribution]], and others.<ref name=Formann2010 />

The uniform distribution, as might be expected, does not obey Benford's law. In contrast, the [[ratio distribution]] of [[Ratio distribution#Uniform ratio distribution|two uniform distributions]] is well-described by Benford's law.

Neither the normal distribution nor the ratio distribution of two normal distributions (the [[Cauchy distribution]]) obey Benford's law. Although the [[half-normal distribution]] does not obey Benford's law, the ratio distribution of two half-normal distributions does. Neither the right-truncated normal distribution nor the ratio distribution of two right-truncated normal distributions are well described by Benford's law. This is not surprising as this distribution is weighted towards larger numbers.

Benford's law also describes the exponential distribution and the ratio distribution of two exponential distributions well. The fit of chi-squared distribution depends on the [[degrees of freedom (statistics)|degrees of freedom]] (df) with good agreement with df = 1 and decreasing agreement as the df increases. The ''F''-distribution is fitted well for low degrees of freedom. With increasing dfs the fit decreases but much more slowly than the chi-squared distribution. The fit of the log-normal distribution depends on the [[mean]] and the [[variance]] of the distribution. The variance has a much greater effect on the fit than does the mean. Larger values of both parameters result in better agreement with the law. The ratio of two log normal distributions is a log normal so this distribution was not examined.

Other distributions that have been examined include the Muth distribution, [[Gompertz distribution]], [[Weibull distribution]], [[gamma distribution]], [[log-logistic distribution]] and the [[exponential power distribution]] all of which show reasonable agreement with the law.<ref name=Leemis2000 /><ref name="Dümbgen2008">{{cite journal|last1=Dümbgen|first1=L|last2=Leuenberger|first2=C|s2cid =2596996|year=2008|title=Explicit bounds for the approximation error in Benford's Law|journal =Electronic Communications in Probability|volume=13|pages=99–112|doi=10.1214/ECP.v13-1358 |arxiv=0705.4488}}</ref> The [[Gumbel distribution]] – a density increases with increasing value of the random variable – does not show agreement with this law.<ref name="Dümbgen2008"/>