View source for Benford law

''significant-digit law, first-digit law''

A [[Probability distribution|probability distribution]] on the significant digits of real numbers named after one of the early researchers, [[#References|[a1]]]. Letting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b1102701.png" /> denote the (base-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b1102702.png" />) significant digit functions (on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b1102703.png" />), i.e.,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b1102704.png" /></td> </tr></table>

(so, e.g., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b1102705.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b1102706.png" />, etc.), Benford's law is the logarithmic probability distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b1102707.png" /> given by

1) (first digit law)

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b1102708.png" /></td> </tr></table>

2) (second digit law)

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b1102709.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027010.png" /></td> </tr></table>

3) (general digit law)

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027011.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027012.png" /></td> </tr></table>

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027016.png" />.

An alternate form of the general law 3) is

4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027017.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027018.png" />. Here, the mantissa (base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027019.png" />) of a positive real number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027020.png" /> is the real number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027021.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027022.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027023.png" />; e.g., the mantissas of both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027025.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027026.png" />.

More formally, the logarithmic [[Probability measure|probability measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027027.png" /> in 1)–4) is defined on the [[Measurable space|measurable space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027028.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027029.png" /> is the set of positive real numbers and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027030.png" /> is the (base-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027031.png" />) mantissa sigma algebra, i.e., the sub-sigma-algebra of the Borel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027032.png" />-algebra generated by the significant digit functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027033.png" /> (or, equivalently, generated by the single function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027034.png" />). In some combinatorial and number-theoretic treatises of Benford's law, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027035.png" /> is replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027036.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027037.png" /> by a finitely-additive [[Probability measure|probability measure]] defined on all subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027038.png" />.

Empirical evidence of Benford's law in numerical data has appeared in a wide variety of contexts, including tables of physical constants, newspaper articles and almanacs, scientific computations, and many areas of accounting and demographic data (see [[#References|[a1]]], [[#References|[a5]]], [[#References|[a6]]], [[#References|[a7]]]), and these observations have led to many mathematical derivations based on combinatorial (e.g., [[#References|[a2]]]), analytic ([[#References|[a3]]], [[#References|[a8]]]), and various urn-scheme arguments, among others (see [[#References|[a7]]] for a review of these ideas).

Benford's law <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027039.png" /> can also be characterized by several invariance properties, such as the following two. Say that a probability measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027040.png" /> on the mantissa space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027041.png" /> is scale-invariant if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027042.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027044.png" />, and is base-invariant if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027045.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027047.png" />. Letting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027048.png" /> denote the logarithmic probability distribution given in 1)–4), then (see [[#References|[a4]]])

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027049.png" /> is the unique probability on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027050.png" /> which is scale-invariant;

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027051.png" /> is the unique atomless probability on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027052.png" /> which is base-invariant.

A statistical derivation of Benford's law in the form of a central limit-like theorem (cf., e.g., [[Central limit theorem|Central limit theorem]]) characterizes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027053.png" /> as the unique limit of the significant-digit frequencies of a sequence of random variables generated as follows. First, pick probability distributions at random, and then take random samples (independent, identically distributed random variables) from each of these distributions. If the overall process is scale- or base-neutral (see [[#References|[a5]]]), the frequencies of occurrence of the significant digits approach the Benford frequencies 1)–4) in the limit almost surely (i.e., with probability one; cf. also [[Convergence, almost-certain|Convergence, almost-certain]]).

There is nothing special about the decimal base in 1)–4), and the analogue of Benford's law 4) for general bases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027054.png" /> is simply

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027055.png" /></td> </tr></table>

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027056.png" />.

Applications of Benford's law have been given to design of computers, mathematical modelling, and detection of fraud in accounting data (see [[#References|[a5]]], [[#References|[a7]]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F. Benford,   "The law of anomalous numbers"  ''Proc. Amer. Philos. Soc.'' , '''78'''  (1938)  pp. 551–572</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D. Cohen,   "An explanation of the first digit phenomenon"  ''J. Combinatorial Th. A'' , '''20'''  (1976)  pp. 367–370</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P. Diaconis,   "The distribution of leading digits and uniform distribution mod <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110270/b11027057.png" />"  ''Ann. of Probab.'' , '''5'''  (1977)  pp. 72–81</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  T. Hill,   "Base-invariance implies Benford's law"  ''Proc. Amer. Math. Soc.'' , '''123'''  (1995)  pp. 887–895</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  T. Hill,   "A statistical derivation of the significant-digit law"  ''Statistical Sci.'' , '''10'''  (1996)  pp. 354–363</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  S. Newcomb,   "Note on the frequency of use of different digits in natural numbers"  ''Amer. J. Math.'' , '''4'''  (1881)  pp. 39–40</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  R. Raimi,   "The first digit problem"  ''Amer. Math. Monthly'' , '''102'''  (1976)  pp. 322–327</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  P. Schatte,   "On mantissa distributions in computing and Benford's law"  ''J. Inform. Process. Cybern.'' , '''24'''  (1988)  pp. 443–445</TD></TR></table>