Benford’s Law: A curious statistical phenomenon that keeps getting curiouser

Ted Hill, one of the contributors to The Princeton Companion to Applied Mathematics, as well as the coauthor, with Arno Berger, of An Introduction to Benford’s Law, has written a post on this fascinating statistical phenomenon. You’ll be surprised at the rather unexpected places it pops up, from an analysis of Donald Trump’s finances, to earthquake detection.

Benford’s Law

The acclaimed business and technology news website Business Insider proudly offers this advice to its readers, in capital letters:


The curious statistical phenomenon known as Benford’s Law, first discovered by Newcomb in 1881 and later rediscovered and popularized by Benford in 1938, is currently experiencing an explosion of research activity, especially in fraud detection ranging from tax data and digital images to clinical trial statistics, and from voting returns to macroeconomic data. Complementing these new forensic Benford tools, recent applications also include earthquake detection, analysis of Big Data and of errors in scientific computations, and diagnostic tests for mathematical models. As is common in developing fields, the quality of this research is all over the map, from scholarly and insightful to amusing and outlandish. The most recent Benford article I have seen is an analysis of Donald Trump’s finances, and I will let interested readers have fun judging these Benford articles for themselves. Most may be found on the open access and fully searchable Benford Online Bibliography, which currently references more than 800 articles on Benford’s Law, as well as other resources (books, websites, lectures, etc.).

The First-digit Law

In its most common formulation, the special case of the first significant (i.e., first non-zero) decimal digit, Benford’s Law says that the leading decimal digit is not equally likely to be any one of the nine possible digits 1, 2, …, 9, but rather follows the logarithmic distribution

equationwhere D1 denotes the first significant decimal digit. Many numerical datasets follow this distribution, from mathematical tables like the Fibonacci numbers and powers of 2 to real-life data like the numbers appearing in newspapers, in tax returns, in eBay auctions, and in the meta-dataset of all numbers on the World Wide Web (see Figure 1).

For datasets like these that are close to being Benford, about 30% of the leading (nonzero) decimal digits are 1, about 18% are 2, and the other leading digit proportions decrease exponentially to about 5% that begin with 9.

fig 1
Figure 1. Empirical Evidence of Benford’s Law

The complete form of Benford’s Law also specifies the probabilities of occurrence of the second and higher significant digits, and more generally, the joint distribution of all the significant digits. For instance, the probability that a number has the same first three significant digits as π = 3.141… is

eqn 2(For non-decimal bases b, the analogous law simply replaces decimal logarithms with logarithms base b.)

Robustness of Benford’s Law

Benford’s Law is remarkably robust, which may help explain its ubiquity in both theory and applications. For example, it is the only distribution on significant digits that is scale invariant (e.g., converting from dollars to euros or feet to meters preserves Benford’s Law), and is the only continuous distribution on significant digits that is base-invariant.

As an example of stochastic robustness, if a random variable X satisfies Benford’s Law, then so does XY for all positive Y independent of X; thus in multiplying independent positive random variables, say to model stock prices, if you ever encounter a single Benford’s Law entry, the whole product will obey Benford’s Law. Moreover, if X follows Benford’s Law, then so do 1/X and X2, (and all other non-zero integral powers of X).

Benford’s Law is also robust under both additive and multiplicative errors: If an increasing unbounded sequence of values X obeys Benford’s Law, then so does X + E for every bounded “error” sequence E, and if X is Benford and E is any independent error with |E| < 1, then (1 + E)X is also exactly Benford.

Applications of Benford’s Law

The most widespread application of Benford’s Law currently is its use in detection of fraud. The idea here is simple: if true data of a certain type is known to be close to Benford’s Law, then chi-squared goodness-of-fit tests can be used as a simple “red flag” test for data fabrication or falsification. Whether the tested data are close to Benford’s Law or are not close proves nothing, but a poor fit raises the level of suspicion, at which time independent (non-Benford) tests or monitoring may be applied.

A similar application is being employed to detect changes in natural processes. If the significant digits are close to Benford’s Law when the process is in one particular state, but not when the process is in a different state, then comparison to Benford can help identify when changes in the state of the process occur. Recent studies have reported successful Benford’s Law tests to detect earthquakes, phase transitions in quantum many-body problems, different states of anesthesia, signal modulations in electrophysiological recordings, and output changes in interventional radiology.

Tests for goodness-of-fit to Benford are also useful as a diagnostic tool for assessing the appropriateness of mathematical models. If current and past data obey Benford’s Law, it is reasonable to expect that future data will also obey Benford’s Law. For example, the 1990, 2000, and 2010 census statistics of populations of the some three thousand counties in the United States follow Benford’s Law very closely (see Figure 1), so to evaluate a proposed mathematical model’s prediction of future populations, simply enter current values as input, and then check to see how closely the output of that model agrees with Benford’s Law (see Figure 2).

fig 2
Figure 2. Benford-in-Benford-out Diagnostic Test

The appearance of Benford’s law in real-life scientific computations is now widely accepted, both as an empirical fact (as reported in Knuth’s classic text), and as a mathematical fact (e.g., Newton’s method and related numerical algorithms have recently been shown to follow Benford’s Law). Thus, in those scientific calculations where Benford’s Law is expected to occur, knowledge of the distribution of the output of the algorithm permits better estimates of both round-off and overflow/underflow errors.

Recent Theoretical Developments

Complementing these applications are new theoretical advancements, which are useful in explaining and predicting when Benford analysis is appropriate, and which are also of independent mathematical interest. Recent results include:

  • The outputs of many numerical algorithms, including Newton’s method, obey Benford’s Law.
  • Iterations of most linear functions follow Benford’s Law exactly, and iterations of most functions close to linear, such as f(x) = 2x + ex, also follow Benford’s Law exactly.
  • Continuous functions with exponential or super-exponential growth or decay typically exhibit Benford’s Law behavior, and thus wide classes of initial value problems obey Benford’s Law exactly.
  • Powers and products of very general classes of random variables, including all random variables with densities, approach Benford’s law in the limit (see Figure 3 for the standard uniform case).
  • Many multidimensional systems such as powers of large classes of square matrices and Markov chains, obey Benford’s Law.
  • Large classes of stochastic processes, including geometric Brownian motion and many Levy processes, obey Benford’s Law.
  • If random samples from different randomly selected probability distributions are combined, the resulting meta-sample also typically converges to Benford’s Law. (This may help explain why numbers in the WWW and newspapers and combined financial data have been found to follow Benford’s Law.)


Fig 3
Figure 3. Powers of a Uniform Random Variable

The study of Benford’s Law has also at times been entertaining. I’ve been contacted about its use to support various religious philosophies (including evidence of Benford’s Law in the Bible and Quran, and its appearance in tables of the earth’s elements as evidence of Intelligent Design), as well as a website where Eastern European entrepreneurs sold Benford data to people who need it for 25 euros a pop. For me, however, the main attraction has been its wealth of fascinating and challenging mathematical questions.

Ted Hill is Professor Emeritus of Mathematics at the Georgia Institute of Technology, and currently Research Scholar in Residence at the California Polytechnic State University in San Luis Obispo. He is the co-author, with Arno Berger, of An Introduction to Benford’s Law, (Princeton University Press, 2015).

Intro to Benford's Law


Leah Wright Rigueur on Making Sense of Ben Carson

Leah Wright Rigueur

Photo Credit: Chion Wolf WNPR

Harvard Kennedy School of Government Professor of Public Policy, Leah Wright Rigueur, who was extensively quoted in the Washington Post  this week on Ben Carson, has written the first in a series of posts she’ll be contributing to the PUP blog. Today she explains the surge in Carson’s popularity among Republican voters in a race that has, until recently, been dominated by Donald Trump. Leah’s recent book, The Loneliness of the Black Republican, offers further insight into the seemingly incongruous intersection of civil rights and American conservatism. We’re delighted to have her. –PUP Blog Editor

Making Sense of Ben Carson

A modified version of this post appears at The Monkey Cage blog of the Washington Post

According to recent polls, Ben Carson has surged in popularity among likely Republican voters and now finds himself at the top of the GOP presidential primary pack. A recent poll of Iowa Republicans, for example, found Carson tied with Donald Trump for first place, with each candidate garnering 23 percent support. That an African American with zero political experience is now a front-runner in the Republican primaries is shocking. A political campaign that many dubbed a joke now appears to have political legs, and the public and press are scrambling to make sense of it.

The idea of a black conservative and/or a black Republican often feels a little like an oxymoron. Black people are partisan voters, overwhelmingly affiliating with the Democratic Party since 1948. In 2012, more than 90 percent of black voters cast ballots for Barack Obama. Their partisanship was, and continues to be, strategic – after all, many of the post-World War II advances in racial and social justice have come by way of Democratic liberalism. Coupled this with the modern – as in, post 1960s – GOP’s move to the extreme right and hostility toward racial justice and race-conscious solutions, and it would appear that politically, black voters have nothing in common with the Republican Party or modern conservatism.

But clearly we know that there are exceptions to this rule. According to the Pew Research Center, between five and eleven percent of the black public either identify as Republican or Lean Republican. Carson is not the first black Republican to run for president, and he won’t be the last. He’s also not the first black Republican to be discussed as part of a potential Republican presidential ticket. Speculation has long surrounded moderate Republican Colin Powell, who has declined to run, time and time again. In 1968 Richard Nixon toyed with tapping liberal Republican Edward Brooke as his vice-presidential running mate, while Gerald Ford placed the black senator high on a private list of potential vice-presidential appointees, in 1974.

Carson, Powell and Brooke, of course, all exhibit different forms of black Republicanism, ranging from liberal to the extreme right. But to some degree, their belief in Republicanism is undergirded by a kind of general black conservatism. African Americans are no strangers to conservatism. We see strong strands of it crop up in nineteenth and early twentieth century religious thought, especially among black churches, despite their political radicalism. Despite their beliefs in racial equality and justice, we see conservative thought in the behaviors of even some of the most progressive of civil rights leaders of the 1960s and 1970s. Conservatism also underscores the respectability politics of the black middle and working class, historically and in the present-day. Even today, studies have shown that about a third of black people self identify as conservative, although their conservatism rarely translates into support for the Republican Party.

And this is how we make sense of Ben Carson. He comes from a long conservative tradition, one that is rooted in a belief in religious morality, personal responsibility, self-help, individualism and free-market enterprise, and one that sometimes exists outside the boundaries of partisanship. Some have attributed Carson’s switch from ardent Democrat to conservative Republican as a matter of opportunism. That may very well be true, but one read of Gifted Hands, indicates that Carson has long exhibited the kind of “everyday black conservatism” that defines a portion of black communities.

Carson also comes from a partisan tradition that has given us figures like Clarence Thomas, Mia Love, Tim Scott, and many others. Organizations like the Black Silent Majority Committee in the 1970s, and the Lincoln Institute for Research and Education in the 1980s and 1990s, built on the black conservatism of past, cultivating a harsher kind of partisan Republican activism and rhetoric that Ben Carson currently articulates.

It’s the kind of position that conservative audiences, almost exclusively white, embrace. Carson is unique no doubt in attracting so much popularity. Historically, black Republicans have been unsuccessful at commanding this kind of attention, at this level of politics. And there are many, many reasons why Carson is surging in the polls: his religious roots, his “niceness” especially when contrasted with Donald Trump, his image as a brilliant surgeon, his position as a political outsider at a moment when people universally distrust politicians, his plain-spoken ability to “tell it like it is,” and his willingness to criticize, unapologetically, Barack Obama, and more broadly, black social justice movements like Black Lives Matter (BLM).

For white conservative audiences, Carson is “safe.” His words on racism, for instance, while profoundly critical of racist acts, are striking when compared to criticisms employed by black liberals. For Carson, racism is something to be changed through individual acts rather than something to be eradicated through structural change. In an era when explicit acts of racism are taboo, Carson’s rhetoric is both palatable to white audiences and comforting. In fact, Carson’s race lends a certain level of legitimacy to his remarks. In other words, conservative voters can look at Carson and have their personal beliefs on race validated, because a black man is articulating their exact same ideas. Historically, we consistently see this, as the GOP moved further to the right. Republicans in 1975, for example, used the Black Silent Majority Committee’s various conservative platforms to validate their views on various racial issues – even as African American voters routinely rejected the organization’s positions. This fraught relationship even led Clarence Thomas to once quip that black converts to the GOP’s acceptance hinged on becoming – in Thomas’ own words – a “caricature of sorts, providing sideshows of anti-black quips and attacks.”

Given all of this, what are we to make of Ben Carson? Are we to take him seriously? Well, in short, yes – we absolutely should take his candidacy seriously. Regardless of whether or not his campaign fizzles or he ends up at the top of the Republican presidential ticket, the general public, scholars, and journalists need to grapple with what, exactly, Carson represents. At this point, the scenarios are nuanced – at the extreme end of the spectrum, Carson could end up as the Republican nominee. He could also end up as vice-presidential nominee, a future presidential cabinet member, a member of Congress, or as a consultant to various public and private Republican factions. There’s a very real chance that Carson may ultimately end up influencing public policies, either directly or indirectly. After all, over the course of the last 80 years, the GOP has implemented black Republicans policies and programs, often favoring those ideas that are firmly couched within right-wing thought. And as we have seen in the past, previous Republican contenders rarely fade from the limelight, choosing instead to use their popularity to influence popular opinion. Ultimately then, we must pay attention to how Carson uses this platform, no matter how precarious, to influence American politics and life.

Leah Wright Rigueur is an Assistant Professor of Public Policy at the Harvard Kennedy School. She is the author of The Loneliness of the Black Republican: Pragmatic Politics and the Pursuit of Power (Princeton University Press, 2015).

PUP Celebrates Mothers — Making a Splash

This Mother’s Day, Princeton University Press is trading in the perfumed soap and jewelry for a different type of celebration for moms. We’ve gathered a group of experts on a range of interesting subjects and compiled a group of mom-related shorts. Zumba class instructor or Pinterest lover – we have a special story for your mom. We hope that this series will provide you with some interesting conversation topics to get family members thinking (and chuckling) during that Mother’s Day brunch.

We’re thankful for our moms, who shuttled us to soccer and ballet practice and made sure that we stuck to our bedtime (even when we wanted to stay up reading). Moms are always a phone call away when college homesickness sets in or after a bad day. For the mothers who keep us afloat, we’re diving deep to explore the world of under-the-sea families. It turns out that clownfish love their moms just as much as we do!


Family of ocellaris clownfish, Miyako Island, Nukuluna false ocellaris

Family of ocellaris clownfish, Miyako Island, Nukuluna false ocellaris *

Marine Mothers

Stephen R. Palumbi & Anthony R. Palumbi


Mothers are always precious, nowhere more so than in the sea. Marine creatures don’t do much parenting, usually preferring to dump their eggs in the water and promptly forget about them. But where they do appear, mothers work diligently to support their young. They are anchors of their communities, often assuming matriarchal leadership roles. Clownfish cluster into intimate family units rooted physically to their anemone homes and emotionally to their mothers—an ever-shifting role that challenges our own conceptions of family.

A gaggle of small, immature males coexists peacefully under the watchful eyes of a single mature male and an even larger mature female. Mom rules the roost, but the family’s got a plan should something ever happen to her. No, not the plan from “Finding Nemo,” where the grieving father raises his young alone.  Actual clownfish react to death with the detachment of Westerosi nobles, simply shuffling along the line of succession.

The mature male begins a rapid hormonal change—within a week he’s producing eggs, acting in every sense as the new matriarch. The biggest and halest juvenile male matures to fill the father role, while his brothers patiently wait for their turns at reproduction. There must always be a mother. Without her, they’re lost.

Without mothers, the whole ocean is lost. Not just in the literal sense; mothers turn out to be especially crucial for maintaining healthy ecosystems. Unlike humans, many marine mothers see their fertility only increasing with age. The oldest mothers don’t just produce the most offspring, they produce the best offspring: the biggest, the healthiest, the most likely to reach adulthood. As a consequence, fisheries become massively easier to manage when mothers—particularly older, highly successful mothers—are protected.

We celebrate Mother’s Day not just for the wonderful women in our own lives, but also for the animals who’ll provide us a healthy future.



* Copyleft. Multi-license with GFDL and Creative Commons CC-BY-SA-2.5 and older versions (2.0 and 1.0)

Launch of The Princeton Encyclopedia of Poetry and Poetics— “004”!

We were delighted to host the launch of the Fourth Edition of The Princeton Encyclopedia of Poetry and Poetics at the London Review Bookshop last Thursday evening. Contributors, well-wishers and lifelong fans gathered together to celebrate this magnificent book. Among them was the contributor on Poetry of Russia, Andrew Kahn, who was kind enough to share his admiration for this much-loved work in a speech:

“Like the appearance of a new James Bond film, the appearance of the fourth edition of The Princeton Encyclopedia of Poetry and Poetics—004!—is cause for jubilation.

This new edition is a magnificent book and achievement. Was there ever a work that taught us more about the ideal and the practical, the historical and the theoretical? Was there ever a work that in a single volume ranged across so many forms of the imagination? Perhaps the Bible, but then for many of its readers, and I include myself, the Princeton Encyclopedia is something of a Bible, containing revelations, divine writings, miracles of concision and lightly worn authority, the precepts of wisdom literature and abundant storytelling. Except that the God of Poetics wears her learning lightly. While deeply serious, and executed with great technical finish, this Good Book is a lovable and playful work. One would want to praise it in terms commensurate it with its contents and achievement. One would therefore want to be a ‘Meistersinger’ (p. 860) gripped by a ‘furor poeticus’ (p.531), ‘inspired’ (p.709) with ‘intensity’ (p. 710) to dithyrambic flights (p.371), to new heights of ‘agudeza’ (p.26), to praise Princeton Press ‘phonesthemically’ (p.1038) in rhyme, near rhyme or even ottava rima, to lavish ‘hovering accents’ (p.640) or devise hypograms (p.649), to roar with leonine rhymes or fire a cybertext, and then to repeat the pythiambic ode, a paplindrome of rispetto or, if you all joined in, to stage a ‘poetry slam’ (p.1070)—a Zulu izibongo (p. 1553) or an epinikion in the Pindaric mode.

It’s not news that the art of poetry has many rules and forms from ‘agudeza’ to ‘Zulu’. But the Princeton Encyclopedia always manages to make it new. This indispensable manual has a history of being savoured and cherished, and the fourth edition will instruct and inspire faithful users and new readers alike. Its reach is global–the expanded selection of national chapters bears witness to the universality and vitality of poetry. It’s worth its considerable weight in gold (but well priced so have no fear). But there’s a further aspect to the Princeton Encyclopedia that I find profoundly wonderful. Poetry as we see it assembled, explored, taxonomized, appreciated and renewed here is a mirror of civilizations and hearts and minds. It turns out that poetry is nothing less than the sum total of virtually everything that goes into thinking and writing about life. In fact, one has only to glance at topical chapters to see that poetry IS life because poetry goes hand in hand with anthropology, belief, culture, dance, gender, history, linguistics, music, painting, philosophy, politics, psychology, religion, science, technology and therapy. And if I might strike a personal note, there are many reference works about poetry, but there is only one that commands universal respect. Contributing a chapter on my subject, and writing an essayistic account of the lives and lines of the poets of Russia, was a privilege and uplifting responsibility.

Horace, a grand old man of poet legislators and sometimes a killjoy, says ‘Nil admirari est’—‘It’s better not to admire’. But the learning, style and sheer scale of Princeton Encyclopedia is worthy of Horace’s own famous Poetics, now fitted for our times yet ‘more lasting than bronze’. 007 may only have so many lives, 004 is imperishable! The contributors, editors and publishers deserve all our ungrudging admiration, congratulations and thanks for the latest incarnation of this tremendous work of learning and spirit.”

Andrew Kahn — Contributor, Poetry of Russia

The Ultimate Book of Saturday Science – see the experiments in action!

    Neil Downie, author of the intriguingly titled ‘Vacuum Bazookas, Electric Rainbow Jelly, and 27 Other Saturday Science Projects’, has a new book out from Princeton in June called ‘The Ultimate Book of Saturday Science: The Very Best Backyard Science Experiments You Can Do Yourself’.

    For a taster of the treats in store for you see Neil’s youtube video.

Pterosaurs are almost here!

Are you a big fan of flying reptiles?

Here at the PUP, we certainly are.  Last year we gave you a sneak peak at a book we have in development with author Mark Witton on Pterosaurs.  If you are not quite sure what a Pterosaur is, here’s an image to give you an idea:

(Cool, right?)

That’s why we at the press are very pleased to announce the launch of Mark Witton’s new blog, at which you can find more of the beautiful images and information about Pterosaurs (such as this characterization by Witton himself: “Even the boring ones have natty looking teeth and preposterous bodily proportions, while more extreme variants wouldn’t look out of place in a Guillermo del Toro movie.”)

The site also features a new excerpt from the upcoming book.

Check it out, and stay on the lookout for more news about the Pterosaurs release!

Happy 5th Anniversary to Our Friends at

Congratulations to our friends at for their 5th year annivesary.  To commemorate the momentous occasion, those creative geniuses went back through their treasure trove of diavlogs to chronicle the past five years, Billy Joel-style!  I count two Princeton University Press authors making appearances (Daniel Drezner and Mark Kleiman).  Check it out below!

Kuran comments on what it means to live under sharia law

Timur Kuran, author of The Long Divergence: How Islamic Law Held Back the Middle East, has been busy posting on his book’s Facebook page! His most recent posting commented on a leader in last week’s Economist about controversies regarding Sharia, the holy law of Islam.

Here’s an excerpt:
An observation of the leader is that few people know what the imposition of the sharia would mean for daily life. Indeed, to people whose knowledge of Islam comes from sensational headlines the sharia involves cutting hands and stoning. To many Islamists, it involves fair adjudication and submission to an inerrant divine code.

To read more, please click here. And if you haven’t already, “Like” The Long Divergence‘s Facebook page so you can receive notice when Kuran posts more fascinating Notes! To get to the book’s Facebook page, click here.

Featured PUP Blogging Author, Raghuram G. Rajan

To learn more about an author, there is no better way than to check out their blog.  Our featured PUP blogging author today is Raghuram G. Rajan, author of Fault Lines: How Hidden Fractures Still Threaten the World Economy.  Get to know the blogger, the blog and his books.

Raghuram G. Rajan is the Eric J. Gleacher Distinguished Service Professor of Finance at the University of Chicago Booth School of Business and former chief economist at the International Monetary Fund.

Introducing Raghuram Rajan’s blog at:

You can also check out this Q&A with Rajan on the NYT’s blog, Freakonomics:


Fault Lines:
How Hidden Fractures Still Threaten the World Economy
By Raghuram G. Rajan

Q&A with Rajan:

Also by Raghuram G. Rajan:

Saving Capitalism from the Capitalists:
Unleashing the Power of Financial Markets to Create Wealth and Spread Opportunity
By Raghuram G. Rajan & Luigi Zingales

The Squam Lake Report:
Fixing the Financial System
By Kenneth R. French, Martin N. Baily, John Y. Campbell, John H. Cochrane, Douglas W. Diamond, Darrell Duffie, Anil K Kashyap, Frederic S. Mishkin, Raghuram G. Rajan, David S. Scharfstein, Robert J. Shiller, Hyun Song Shin, Matthew J. Slaughter, Jeremy C. Stein, and René M. Stulz

(Photo by Dan Dry. © University of Chicago Booth School of Business)

Following PUP Authors on their Blogs – First up, Marisa A. Abrajano & R. Michael Alvarez

Welcome to a new feature on our blog that has the catchy name, “PUP Blogging Authors”.  To learn more about an author, there is no better way than to check out their blog.  Author blogs provide upfront, personal, informative, and entertaining information about their books, their research, and their day-to-day lives.  Our “PUP Blogging Authors” feature consists of three parts: Blogger, Blogs, and Books.

Our featured bloggers are the authors of New Faces, New Voices: The Hispanic Electorate in America.

Marisa A. Abrajano is assistant professor of political science at the University of California, San Diego. She is the author of Campaigning to the New American Electorate. R. Michael Alvarez is professor of political science at the California Institute of Technology. He is the coauthor of Electronic Elections and Hard Choices, Easy Answers (both Princeton).

Blogs: We invite you to learn more about their book and election research on their blog, New Faces, New Voices:

You can also find more blogging from R. Michael Alvarez on the blog, Election Updates.  The blog covers new research, analysis and commentary on election reform, voting technology, and election administration.

Books:  For descriptions, sample chapters, and table of contents, visit:

New Faces, New Voices:
The Hispanic Electorate in America
By Marisa A. Abrajano & R. Michael Alvarez

Also by R. Michael Alvarez:

Electronic Elections:
The Perils and Promises of Digital Democracy
By R. Michael Alvarez & Thad E. Hall

Hard Choices, Easy Answers:
Values, Information, and American Public Opinion
R. Michael Alvarez & John Brehm