Benford lays down the Law

A few months ago I received in the mail a book called An Introduction to Benford’s Law by Arno Berger and Theodore Hill. I eagerly opened it but I lost interest once I realized it was essentially a pure math book. Not that there’s anything wrong with math, it just wasn’t what I wanted to read.

But, hey, the book got reviewed by Frank Benford himself (well, actually, Frank Benford the grandson of Frank Benford himself), and the review has some math too. Should be of interest to some of you.

P.S. On the applied end, Ethan Rouen sends along a recent paper, Financial statement errors: evidence from the distributional properties of financial statement numbers, where Dan Amiram, and Zahn Bozanic, and Rouen “use Benford’s Law to measure the amount of error contained in a firm’s financial statements and show how the law can be used to help predict fraud.”

Here’s an example:

Which somehow reminds me of this classic plot:

1. zbicyclist says:

This post inspired me to post some data I’ve been sitting on since 2008 on grocery store packages and Benford’s Law.

On an individual category basis (e.g. beer) the package sizes don’t — for example, there are 6 packs, 12 packs, and 24 packs, but no 7 packs. But when we pool across categories there’s a decent fit.

http://www.truncatedthoughts.com/2015/11/benfords-law-and-package-sizes.html

2. Rahul says:

Someone ought to plot a Benford for the financial statements of Enron, Tyco, etc.

• Jonathan (another one) says:

Benford’s law violations only affect particular types of accounting fraud. If items are increased by percentage factors you won’t see violations. In Tyco and Worldcom’s cases, entries were misclassified, not made up, so Benford’s law won’t be implicated. In Enron’s case, there wasn’t any real accounting fraud of this sort, simply exploitation of gaps in GAAP.

• Rahul says:

Do fraudsters increase *ALL* items by same percentage factors? Because if they don’t, and selectively change some line items, wouldn’t there be a chance of detection via Benford?

• James C. Whanger says:

You would simply have to know the thresholds for noticeable change.

3. MNL says:

I’ve got to ask an embarrassingly simple question: Isn’t this obviously true whenever one numbers or counts something–provided the thing being numbered is finite, the number scheme has a set number of digits (i.e., you can’t go to infinite decimal places), and the numbering starts at 1? Take house numbers from the image above for example. Of course addresses starting with low numbers are going to appear more often if one starts the numbering at 1 (or 2 on the other side of the street), moves sequentially, and city streets or blocks don’t go to infinite length. After all, there are more streets with at least one block than streets with at least nine blocks. The graph title calls this a “striking pattern,” but it almost seems too trivial to call a “law.” (It’s simply called “counting”). What am I missing here?

• James C. Whanger says:

I think you are right about the naming convention — I mean you could call it the Benford Pattern and it would still be important. But, I don’t think you are right in minimizing the fact that Benford identified it and saw the implications of it for identifying and solving other problems. It is easy in hindsight to say, “Of course that makes sense, who wouldn’t see it” but is far less common to be the one who sees past the blind spots of almost everyone else and says, “Hey look at this, this might be useful.”

4. gdp says:

You are missing the fact that Benford’s Law is the distribution for the _first_ digit in a number with an _arbitrarily large_ number of digits. E.g., 0.0004, 4, 42, 437, and 4395271 each contribute a single count to the “4” bin. So the “absolute smallness” of numbers has nothing to do with Benford’s Law; Benford’s Law is a statement about the distribution of “Mantissas,” not “Characteristics.”