Review of “An Introduction to Benford's Law” by Berger and Hill

By J.D. Meiss
An Introduction to Benford's Law
by Arno Berger and Theodore P. Hill
Princeton University Press
ISBN: 978-0-691-16306-2
Reviewed by: James Meiss
Department of Applied Mathematics
University of Colorado at Boulder

Benford's law corresponds to a probability distribution that decreases logarithmically with leading digit $d$, specifically ${\rm prob}(d) = \log(1+1/d)$ for $d = 1,2,\ldots, 9$ (for numbers in base-10, $\log$ is the base-10 logarithm). Thus in a table of numbers with the Benford distribution, about 30.1% will have the leading digit $1$ while only 17.6% will have leading digit $2$. In 1938, Benford [1] observed this law in data obtained from fields as diverse as the specific heats of chemical compounds, numbers appearing on the first page of newspapers, and baseball statistics. Some sequences also follow this distribution; for example, 30 of the first 100 numbers in the Fibonacci sequence, $1,1,2,3,5,8,13, \ldots$, begin with the digit $1$, while 18 begin with the digit $2$. For the first 1000 numbers of the Fibonacci sequence, 301 begin with the digit $1$ and 177 begin with the digit $2$. However, not all mathematical sequences have a Benford distribution: the prime numbers do not.

More precisely a real-valued random variable is Benford if its “significands” have a logarithmic probability distribution. In base-10, the significand $t = S(x)$ of $x$ is the unique $t \in [1,10)$ such that $|x| = 10^k t$ for some $k \in \mathbb{Z}$. For example, if $x = -0.00137$, then $t = S(x) = 1.37$ with $k = -3$. Note that $S(10^n x) = S(x)$ for any integer $n$. The cumulative Benford distribution is $$ {\rm Prob}(S(x) < t) = \log(t) . $$

The book “An Introduction to Benford's Law” by Berger and Hill, is a delightful introduction to this area and also a careful exposition of the underlying mathematics. Its introductory chapter discusses the history---as is often the case, the eponym is not always the discoverer---and presents some of the empirical evidence for this logarithmic distribution of significant digits, and some of the contradictions inherent in some applications. The second chapter sets out some of the mathematical tools, especially a particular $\sigma$-algebra, the significand algebra, that is scale invariant, which is used to make probabilistic sense of the law. The third chapter formalizes the statement of the Benford property. The focus of most of the book is on sequences that may or many not have the Benford property, and later chapters give a number of examples and theorems are given. For example it is shown that sequences that with exponential asymptotics, $x_n \sim \gamma^n$ are Benford whenever $\log(\gamma)$ is irrational. This is the reason that the Fibonnaci sequence is Benford, since for this case $\gamma$ is the golden mean.

Why am I reviewing this for the dynamical systems magazine? The question “Do dynamical systems follow Benford's law?” was raised in [2], and positive examples were given in [3], leading to a sequence of papers by Berger, that are explained and extended in the book under review. For example, if $f: \mathbb{R} \to \mathbb{R}$ is a one-dimensional map with a stable fixed point $x^*$, then the sequence of deviations $f^t(x)-x^*$ is base-10 Benford whenever the multiplier $\mu = f'(x^*)$ has an irrational base-10 logarithm.

The Benford property appears to be less common in chaotic dynamics. For example there is a zero-measure set of orbits of the tent map with slope 2, $f(x) = 1-|2x-1|$, that have the Benford property. Indeed, almost all bounded orbits, in the sense of Lebesgue measure, have uniform measure in $[0,1]$, and are not Benford. However, there is a dense set of initial points with the Benford property. For example orbits that spend increasing amounts of time near the fixed point $0$ do have the Benford property, since $\log(f'(0)) = \log 2$ is irrational. Such orbits can be constructed using the conjugacy of the dynamics to symbolic dynamics with the partition $L = [0,\frac12]$ and $R = (\frac12,1]$. One class of appropriate initial conditions have symbol sequences that interleave an arbitrary sequence $s_t \in \{L,R\}$, with increasingly long strings of $L$'s, e.g., $\{R\,L\,R\,LL\,R\,LLL\,R\,LLLL\ldots\}$. The implication is that an uncountable, dense set of initial conditions have orbits with a Benford distribution. A similar symbolic argument gives the same result for the logistic map $f(x) = 4x(1-x)$.

Later chapters of the book consider multidimensional linear maps, Markov chains, and random processes. The book ends with a short discussion of some of the many applications of Benford's law. We close with a quote from this chapter:

From natural science to medicine, from social science to economics, from computer science to theology, Benford's law, even in its most basic form, provides a simple analytical tool that invites anyone with numerical tables to look for this easily recognizable pattern in their own data.


  1. F. Benford (1938). “The Law of Anomalous Numbers.” Proceedings of the American Philosophical Society 78(4): 551-572.
  2. C.R. Tolle, J.L. Budzien, and R.A. LaViolette (2000). “Do Dynamical Systems follow Benford's Law?” Chaos 10(2):331-336.
  3. M.A. Snyder, J.H. Curry, and A.M. Dougherty (2001). “Stochastic Aspects of One-Dimensional Discrete Dynamical Systems: Benford's Law.” Phys. Rev. E 64:026222.
Categories: Magazine, Book Reviews

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