Editing Benford's law (section)

==Definition==
[[File:Logarithmic scale.svg|thumb|upright=1.7|alt=Rectangle with offset bolded axis in lower left, and light gray lines representing logarithms|A [[logarithmic scale]] bar. Picking a random ''x'' position [[Uniform distribution (continuous)|uniformly]] on this number line, roughly 30% of the time the first nonzero digit of the number will be 1.]]
A set of numbers is said to satisfy Benford's law if the leading digit&nbsp;{{mvar|d}} ({{math|{{var|d}}&nbsp;∈&nbsp;{{mset|1,&nbsp;...,&nbsp;9}}}}) occurs with [[probability]]<ref name="Miller2015" />
: <math>P(d) = \log_{10}(d + 1) - \log_{10}(d) = \log_{10}\left(\frac{d + 1}{d}\right) = \log_{10}\left(1 + \frac{1}{d}\right).</math>

The leading digits in such a set thus have the following distribution:
{| class="wikitable"
! {{nobold|{{mvar|d}}}} !! {{tmath|P(d)}} !! Relative size of {{tmath|P(d)}}
|-
| 1 ||{{right}} {{bartable|30.1|%|10}}
|-
| 2 ||{{right}} {{bartable|17.6|%|10}}
|-
| 3 ||{{right}} {{bartable|12.5|%|10}}
|-
| 4 ||{{right}} {{bartable| 9.7|%|10}}
|-
| 5 ||{{right}} {{bartable| 7.9|%|10}}
|-
| 6 ||{{right}} {{bartable| 6.7|%|10}}
|-
| 7 ||{{right}} {{bartable| 5.8|%|10}}
|-
| 8 ||{{right}} {{bartable| 5.1|%|10}}
|-
| 9 ||{{right}} {{bartable| 4.6|%|10}}
|}
The quantity {{tmath|P(d)}} is proportional to the space between {{mvar|d}} and {{math|{{var|d}}&nbsp;+&nbsp;1}} on a [[logarithmic scale]]. Therefore, this is the distribution expected if the ''logarithms'' of the numbers (but not the numbers themselves) are [[Uniform distribution (continuous)|uniformly and randomly distributed]].

For example, a number {{mvar|x}}, constrained to lie between 1 and 10, starts with the digit 1 if {{math|1&nbsp;≤&nbsp;{{var|x}}&nbsp;<&nbsp;2}}, and starts with the digit 9 if {{math|9&nbsp;≤&nbsp;{{var|x}}&nbsp;<&nbsp;10}}. Therefore, {{mvar|x}} starts with the digit 1 if {{math|log&nbsp;1&nbsp;≤&nbsp;log&nbsp; {{var|x}}&nbsp;<&nbsp;log&nbsp;2}}, or starts with 9 if {{math|log&nbsp;9&nbsp;≤&nbsp;log&nbsp;''x''&nbsp;<&nbsp;log&nbsp;10}}. The interval {{math|[log&nbsp;1,&nbsp;log&nbsp;2]}} is much wider than the interval {{math|[log&nbsp;9,&nbsp;log&nbsp;10]}} (0.30 and 0.05 respectively); therefore if log {{mvar|x}} is uniformly and randomly distributed, it is much more likely to fall into the wider interval than the narrower interval, i.e. more likely to start with 1 than with 9; the probabilities are proportional to the interval widths, giving the equation above (as well as the generalization to other bases besides decimal).

Benford's law is sometimes stated in a stronger form, asserting that the [[fractional part]] of the logarithm of data is typically close to uniformly distributed between 0 and 1; from this, the main claim about the distribution of first digits can be derived.<ref name=BergerHill2020/>

===In other bases===
[[File:Benford_law_bases.svg|thumb|200px|Graphs of ''P''(''d'') for initial digit ''d'' in various bases.<ref>They should strictly be bars but are shown as lines for clarity.</ref> The dotted line shows ''P''(''d'') were the distribution uniform. (In [http://upload.wikimedia.org/wikipedia/commons/1/14/Benford_law_bases.svg the SVG image], hover over a graph to show the value for each point.)]]
An extension of Benford's law predicts the distribution of first digits in other [[radix|bases]] besides [[decimal]]; in fact, any base {{math|''b''&nbsp;≥&nbsp;2}}. The general form is<ref>{{Cite web |last=Pimbley |first=J. M. |date=2014 |title=Benford's Law as a Logarithmic Transformation |url=http://www.maxwell-consulting.com/Benford_Logarithmic_Transformation.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.maxwell-consulting.com/Benford_Logarithmic_Transformation.pdf |archive-date=2022-10-09 |url-status=live |access-date=2020-11-15 |website=Maxwell Consulting, LLC}}</ref>
: <math>P(d) = \log_b(d + 1) - \log_b(d) = \log_b\left(1 + \frac{1}{d}\right).</math>
For {{math|1=''b'' = 2, 1}} (the [[Binary numeral system|binary]] and [[Unary numeral system|unary]]) number systems, Benford's law is true but trivial: All binary and unary numbers (except for 0 or the empty set) start with the digit 1. (On the other hand, the [[#Generalization to digits beyond the first|generalization of Benford's law to second and later digits]] is not trivial, even for binary numbers.<ref>{{Cite book |last=Khosravani |first=A. |title=Transformation Invariance of Benford Variables and their Numerical Modeling |publisher=Recent Researches in Automatic Control and Electronics |year=2012 |isbn=978-1-61804-080-0 |pages=57–61}}</ref>)