Benford's Law Strikes Back: No Simple Explanation in Sight For
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Benford’s Law Strikes Back: No Simple Explanation in Sight for Mathematical Gem ARNO BERGER AND THEODORE P. HILL he widely known phenomenon called Benford’s Law branch of mathematics is it so easy for experts to blunder as continues to defy attempts at an easy derivation. This in probability theory’’. As the reader as well will see, this is TT article briefly reviews recurring flaws in ‘‘back-of- all too true for Benford’s Law, also known as the First-Digit the-envelope’’ explanations of the law, and then analyzes Phenomenon. in more detail some of the recently published attempts, Benford’s Law, abbreviated henceforth as BL, is one of many of which replicate an apparently unnoticed error in the gems of statistical folklore. It is the observation that Feller’s classic 1966 text An Introduction to Probability in many collections of numbers, be they mathematical Theory and Its Applications. Specifically, the claim by Feller tables, real-life data, or combinations thereof, the lead- and subsequent authors that ‘‘regularity and large spread ing significant digits are not uniformly distributed, as implies Benford’s Law’’ is fallacious for any reasonable might be expected, but are heavily skewed toward the definitions of regularity and spread (measure of disper- smaller digits. More precisely, BL says that the significant sion). The fallacy is brought to light by means of concrete digits in many datasets follow a very particular logarithmic examples and a new inequality. As for replacing the wrong distribution. In its most common formulation, the special assertions by an equally simple explanation which is valid, case of first significant decimal (i.e., base 10) digits, BL now—that is a task for the future. reads ÀÁ Prob D d log 1 dÀ1 ; for all d 1; ...; 9; It’s All About Digits ð 1 ¼ 1Þ¼ 10 þ 1 1 ¼ The eminent logician, mathematician, and philoso- ðBL1Þ pher C.S. Peirce once observed [Ga, p.273] that ‘‘in no other here D1 denotes the first significant decimal digit, e.g., scientists use it to identify voter fraud, and engineers to pffiffiffi detect altered digital images. As Raimi already observed D1ð 2Þ¼D1ð1:414...Þ¼1; some 25 years ago [R1, p.512], À1 D1ðp Þ¼D1ð0:3183...Þ¼3; This particular logarithmic distribution of the first digits, p D1ðe Þ¼D1ð23:14...Þ¼2: while not universal, is so common and yet so surprising at first glance that it has given rise to a varied literature, A crucial part of the content of (BL1), of course, is an among the authors of which are mathematicians, appropriate formulation or interpretation of ‘‘Prob’’. For statisticians, economists, engineers, physicists, and sequences of real numbers or real datasets, for example, amateurs. Prob usually refers to the proportion (or relative frequency) The online database [BH] now contains more than 600 of entries for which an event such as D1 = 1 occurs, articles on the subject. Many of these articles, including whereas for a random variable, Prob is simply the proba- some of the very recent ones, purport to provide easy bility on the underlying probability space. Figure 1 derivations or proofs of BL. The present article sets out to illustrates several of these settings, including mathematical identify some of the prevalent fallacies in those arguments. sequences such as the powers of 2, and real-life data from Benford’s original paper as well as recent census statistics. Simple Explanations? Are You Sure? In a form more complete than (BL1), BL is a statement Let’s start with the purely mathematical framework. Many about the joint distribution of all decimal digits: For every familiar sequences, including the Fibonacci numbers, the natural number n, this version states that powers of 2, and the factorial sequence (n!), all follow BL ProbðÞðD ; D ; ...; D Þ¼ðd ; d ; ...; d Þ exactly; in particular, exactly a proportion of log 2 ¼ 1 2 Xn 1 2 n 10 n À1 0:3010..., that is, approximately 30.1% of all entries of those ¼ log 1 þ 10nÀj d ðBL2Þ 10 j¼1 j sequences begin with the decimal digit 1. Similarly, start with any positive number and multiply by 3 repeatedly (i.e., holds for all n-tuples ðd1; d2; ...; dnÞ, where d1 is an integer iterate the function x 7! 3x), or multiply alternately by 3 and in 1; 2; ...; 9 and where for j [ 1, dj is an integer in by 4, or iterate the function x 7! 2x þ 1. Each of these iter- 0; 1; ...; 9. Here D2; D3; D4, etc. represent the second, third, ations results in a sequence that follows BL exactly, no matter fourth, etc. significant decimal digit, so that, for example, what positive number was chosen in the beginning. Thus, ffiffiffi p even though many common sequences such as the natural D ð 2Þ¼4; D ðpÀ1Þ¼8; D ðepÞ¼4: 2 3 4 numbers and the primes do not follow BL, those that do are The ‘‘laws’’ (BL1) and (BL2) were apparently first dis- so ubiquitous that many authors have assumed that a simple covered by polymath S. Newcomb in the 1880s [N]. They explanation must exist. were rediscovered by physicist F. Benford [Ben]; New- Raimi [R1, R2] reviews many of the attempts at such comb’s article having been forgotten at the time, they came explanations: Some authors simply labeled BL self-evident; to be known as Benford’s Law. Today, BL appears in a thus Benford himself wrote that ‘‘the logarithmic law broad spectrum of mathematics, ranging from differential applies particularly to those outlaw numbers that are equations to number theory to statistics. Simultaneously, without known relationship’’, Goudsmit and Furry opined the applications of BL are mushrooming—from diagnostic that it ‘‘is merely the result of our way of writing numbers’’, tests for mathematical models in biology and finance to and likewise Weaver claimed that BL ‘‘is a built-in charac- fraud detection. For instance, the U.S. Internal Revenue teristic of our number system’’. For the more mathematical Service uses BL to ferret out suspicious tax returns, political ones among the back-of-the-envelope derivations, Raimi Figure 1. Different interpretations of (BL1) for sequences, datasets, and random variables, respectively, and scenarios that may lead to exact conformance with BL. carefully points out their shortcomings, e.g., Flehinger’s the discovery that the error is in Pinkham’s implicit assumption Cesa`ro-summation method, Herzel’s urn model, and the that there exists a scale-invariant probability distribution on the two different fallacies in Logan and Goudsmit’s urn model positive real numbers, when clearly there is no such distribu- derivation. tion. To see this, simply note that for instance multiplying any In [R1, sec.7], Raimi also explains the basic flaw in Pinkham’s positive random variable X by 2 doubles the value of its widely cited scale-invariance argument. That is the argument median, and hence X cannot be scale-invariant. To the best of that BL is the only distribution on significant digits that is the authors’ knowledge, the first correct proof that BL indeed is invariant under changes of scale, meaning that (BL1) and (BL2) the unique scale-invariant probability distribution (and also the remain unchanged if a sequence, dataset, or random variable is unique continuous base-invariant distribution) on the signifi- multiplied by any positive constant. Raimi credits Knuth [K]for cant digits is in [H2]. A closer look at sequences of numbers reveals some now include a brief introduction to BL, it is not surprising surprises that may help explain why correct and quick that Feller’s derivation is still in frequent use to ‘‘provide derivations of BL in a purely mathematical context may some insight’’ about this phenomenon. For example, a class be hard to come by. For instance, iterating the func- project report for a 2009 upper-division course in statistics tion x 7! x2 þ 1 results in a sequence following BL for at UC Berkeley [AP1, p.3] said, (Lebesgue) almost all starting points, but not for all starting ...like the birthday paradox, an explanation [of BL] points. Thus, if the initial value x1 is chosen randomly from occurs quickly to those with appropriate mathematical any positive distribution with a density, such as, say, the background ... To a mathematical statistician, Feller’s uniform distribution on (0,1) or an exponential distribution, paragraph says all there is to say ...Feller’s derivation has 2 then the resulting sequence (xn) with xnþ1 ¼ xn þ 1 will been common knowledge in the academic community follow BL exactly with probability 1. But there are excep- throughout the last 40 years. tional points also. For example, choosing x1 ¼ 9:9496230... The online database [BH] lists about twenty published implies D1ðxnÞ¼9 for all n, that is, every number xn begins references since 2000 alone to Feller’s argument (e.g., [AP1, with a decimal digit 9. (See [BBH, exp.4.3] to find out what is Few]) the crux of which is Feller’s claim (trivially edited) special about this remarkable value for x1.) Whether or not that the sequence (xn) follows BL when x1 = 0 is still an open problem. (2) If the spread of a random variable X is very large, then All in all, even though it would be highly desirable to log10 X will be approximately u.d. mod 1. have both a rigorous formal proof and a reasonably sound heuristic explanation, it seems unlikely that any quick The implication of (1) and (2) is that all random variables derivation has much hope of explaining BL mathematically.