Editing Benford's law (section)

===Invariance===
In a list of lengths, the distribution of first digits of numbers in the list may be generally similar regardless of whether all the lengths are expressed in metres, yards, feet, inches, etc. The same applies to monetary units.

This is not always the case. For example, the height of adult humans almost always starts with a 1 or 2 when measured in metres and almost always starts with 4, 5, 6, or 7 when measured in feet. But in a list of lengths spread evenly over many orders of magnitude—for example, a list of 1000 lengths mentioned in scientific papers that includes the measurements of molecules, bacteria, plants, and galaxies—it is reasonable to expect the distribution of first digits to be the same no matter whether the lengths are written in metres or in feet.

When the distribution of the first digits of a data set is [[scale-invariant]] (independent of the units that the data are expressed in), it is always given by Benford's law.<ref name=Pinkham>{{cite journal | last1 = Pinkham | first1 = Roger S. | year = 1961 | title = On the Distribution of First Significant Digits | url = http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.aoms/1177704862 | journal = Ann. Math. Statist. | volume = 32 | issue = 4 | pages = 1223–1230 | doi = 10.1214/aoms/1177704862 | doi-access = free }}</ref><ref name="wolfram">{{Cite web |url=https://mathworld.wolfram.com/BenfordsLaw.html |title=Benford's Law |first=Eric W. |last=Weisstein |website=mathworld.wolfram.com}}</ref>

For example, the first (non-zero) digit on the aforementioned list of lengths should have the same distribution whether the unit of measurement is feet or yards. But there are three feet in a yard, so the probability that the first digit of a length in yards is 1 must be the same as the probability that the first digit of a length in feet is 3, 4, or&nbsp;5; similarly, the probability that the first digit of a length in yards is 2 must be the same as the probability that the first digit of a length in feet is 6, 7, or&nbsp;8. Applying this to all possible measurement scales gives the logarithmic distribution of Benford's law.

Benford's law for first digits is [[radix|base]] invariant for number systems. There are conditions and proofs of sum invariance, inverse invariance, and addition and subtraction invariance.<ref>{{Cite web |last=Jamain |first=Adrien |date=September 2001 |title=Benford's Law |url=https://wwwf.imperial.ac.uk/~nadams/classificationgroup/Benfords-Law.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://wwwf.imperial.ac.uk/~nadams/classificationgroup/Benfords-Law.pdf |archive-date=2022-10-09 |url-status=live |access-date=2020-11-15 |website=Imperial College of London}}</ref><ref>{{Cite journal |last=Berger |first=Arno |date=June 2011 |title=A basic theory of Benford's Law |url=https://projecteuclid.org/download/pdfview_1/euclid.ps/1311860830 |journal=Probability Surveys |volume=8 (2011) |pages=1–126}}</ref>