Benford’s law

In 1881, the American mathematician Simon Newcomb noticed that in the logarithm tables used by his students, the pages with logarithms beginning with the digit “1” had more “dog-ears” than the following pages on which the logarithms with “2”, “3”, “4”, etc. came first. However, his mathematical description of this phenomenon in the American Journal of Mathematics was quickly forgotten.

It was not until 1937 that this regularity was rediscovered by the physicist Frank Benford (1883–1948), studied on the basis of over 20,000 data and systematically analyzed. The law subsequently named after him states — formulated mathematically — that when a number is randomly selected from a table of physical constants or statistical data, the probability that the first digit is a “1” is about 0.301. Thus, it is far greater than 0.1, the value one might expect if all digits were equally likely to occur. In general, Benford’s law says about the probability that the first digit is equal to “k“:


Figure 2: Benford’s publication

Here \log denotes the decadic logarithm and k can take the values 1, 2, …, 9. In detail thus results:

    \[P{1}=\log(1+1/1)=\log(2)\approx 0,301;\]

    \[P{2}=\log(1+1/2)=\log(3/2)\approx 0,176;\]

    \[P{3}=\log(1+1/3)=\log(4/3)\approx 0,125;\]

    \[P{4}=\log(1+1/4)=\log(5/4)\approx 0,097;\]

    \[P{5}=\log(1+1/5)=\log(6/5)\approx 0,079;\]

    \[P{6}=\log(1+1/6)=\log(7/6)\approx 0,067;\]

    \[P{7}=\log(1+1/7)=\log(8/7)\approx 0,058;\]

    \[P{8}=\log(1+1/8)=\log(9/8)\approx 0,05;\]

    \[P{9}=\log(1+1/9)=\log(10/9)\approx 0,046.\]

Consequently, Benford’s Law implies that a number with a “smaller” first digit is more likely to occur in a table of statistical data than a number with a “larger” first digit. Benford’s law is used in detecting fraud in the preparation of financial statements, falsification in settlements, generally for the rapid detection of glaring irregularities in accounting. In 2001, Benford’s Law was used to uncover the remarkably “creative” accounting system at ENRON, the largest American energy company at the time (with 22,000 employees), through which management had defrauded investors of their deposits in the amount of approximately 30 billion. Today, auditors, tax investigators, and even election observers use mathematical-statistical methods to detect falsification in cases of conspicuous deviations from the Benford distribution. <!-- /wp:paragraph -->  <!-- wp:image {"align":"center","id":7239,"sizeSlug":"large","linkDestination":"custom"} --> <div class="wp-block-image"><figure class="aligncenter size-large"><img src="" alt="" class="wp-image-7239"/><figcaption>Figure 1: The Benford's Law Exhibit</figcaption></figure></div> <!-- /wp:image -->  <!-- wp:paragraph --> By means of the exhibit in MATHEMATICS ADVENTURE LAND, Benford's law can be reproduced experimentally with frequencies of spheres lying on top of each other (see Figure 1 above). Suitable panels of random numbers are available for selection for the experiment. <!-- /wp:paragraph -->  <!-- wp:separator {"className":"is-style-wide"} --> <hr class="wp-block-separator is-style-wide"/> <!-- /wp:separator -->  <!-- wp:heading {"level":4} --> <h4>And now … the mathematics of it:</h4> <!-- /wp:heading -->  <!-- wp:paragraph --> The starting point for Benford's law is Newcomb's so-called <em>mantissa law</em>: <!-- /wp:paragraph -->  <!-- wp:paragraph --> <em>"The frequency of numbers is such that the mantissas of their logarithms are equally distributed."</em> <!-- /wp:paragraph -->  <!-- wp:paragraph --> Here, the mantissa of a positive number is understood to be its so-called <em>fractal part</em>. For a positive real numberx, the mantissa ofxis equal to\langle x\rangle\coloneqq x-\lfloor x\rfloor(e.g.,\langle 3.1415\rangle=3.1415-\lfloor 3.1415\rfloor=3.1415-3=0.1415

    ). <!-- /wp:paragraph --> <!-- wp:paragraph --> Now suppose that a set of randomly chosen numbers satisfies the above mantissa law and consider the following sets: <span class="ql-right-eqno">   </span><span class="ql-left-eqno">   </span><img src="" height="19" width="292" class="ql-img-displayed-equation quicklatex-auto-format" alt="\[E_i\coloneqq{x\lt 0|\text{ the leading digit of $x$ is $i$}}.\]" title="Rendered by"/> <!-- /wp:paragraph --> <!-- wp:paragraph --> Then, the following holds: the set

\mathbb R_+of <em>positive real numbers</em> is the union of the (element-unrelated) setsE_i

    , i.e. <span class="ql-right-eqno">   </span><span class="ql-left-eqno">   </span><img src="" height="17" width="187" class="ql-img-displayed-equation quicklatex-auto-format" alt="\[\mathbb R_+=E_1\cup E_2\cup\cdots\cup E_9.\]" title="Rendered by"/> <!-- /wp:paragraph --> <!-- wp:paragraph --> Thus, for the probability

P(<em>probability</em>) that any of the considered random numbersXbelongs to the setE_i$, we get:

    \[P(X\in E_i)=P(\langle\log(X)\rangle\in[\log(i),\log(i+1)))=\log(i+1)-\log(i)=\log(1+1/i).\]


[1] Benford, F.: The Law of Anomalous Numbers, Proceedings of the American Philosophical Society 78, S. 551–572, 1938.

[2] Glück, M.: Die Benford-Verteilung — Anwendung auf reale Daten der Marktforschung, Diplomarbeit TU Dresden, Betreuer: V. Nollau und H.-O. Müller, 2007.

[3] Newcomb, S.: Note on the Frequency of the Use of different Digits in Natural Numbers, American Journal of Mathematics 4, S. 39–40, 1881.