**Frank Benford**, a man whose curiosity about a book inspired a bizarre discovery. Benford's Law, as it is now known, reveals a cosmic preference for certain numbers. Then **Darrell D. Dorrell**, a forensic accountant, describes how he uses Benford's Law to bust crooks.

Paul Hoffman tells us the story of a boy trapped in a world of numbers, who grew into one of math's greatest proselytizers, Paul Erdös. Joel Spencer and Jerry Grossman help bring to life the man behind the numbers. From producer **Ben Calhoun**.

## Comments [39]

Thanks for the explanation of Benford's law. My daughter, who is in the accounting/auditing field, knew immediately what this is as she uses it all the time when auditing accounts. If the numbers come up distributed not in accordance with this law, it doesn't necessarily mean that there is an attempt at fraud, only that further investigation is needed. There is sometimes another explanation as to why the numbers came up the way they did.

Loved this RadoLab show! Thanks! Unaware of Benford's Law; but, was able to follow/understand because my math used to be good and I grew up with a slide rule. Unfortunately, I think I have a minus Erdos number......

I was fortunate to be in my car for the entire show today. Loved both segments on Benford Rule and Erdős. The interviews at the Combinatorial and Additive Number Theory (CANT 2014) conference was a great way to start the segment on Erdős!

If someone hears your show and is attempting to commit accounting fraud, could they easily defeat the use of Benford's Law to detect their fraud by distributing the initial digits of their cooked numbers in accordance with Benford's Law? From the way the issue was discussed on this program, the answer seems to be yes.

Simple proof that the distribution of first digits can't be uniform:

Suppose you had a bunch of physical distances, measured in yards, and their first digits were all equally common. What would happen if you measured the exact same set of distances in feet instead of yards? All the numbers would be three times what they used to be. The ones that started with 1 would now start with 3, 4, or 5 (1000-1333 become 3, 1334-1666 become 4, 1667-1999 become 5); the ones that started with 2 would now start with 6, 7, or 8; the ones that started with 3 would now start with 9, 10, or11 -- note that both of those have first digit 1. The ones that started with 4 would now start with 12, 13, or 14, all three of which have first digit 1. And so on.

That is, if the first digits of things measured in yards were all equally likely, then the first digits of things measured in feet would definitely NOT be equally likely. If the first digits of things measured in feed were all equally likely, then the first digits of things measured in inches or yards or miles would definitely NOT be equally likely. And so on. But yards, feet, inches, and miles are all human inventions; it's inconceivable that a law of nature would behave differently for one unit than for another. So it CANNOT be true that first digits of distance measurements in the real world are all equally likely.

Benford's Law is really a fact about place-value notation: it would still be true in bases other than ten, but it wouldn't be true in Roman numerals, or tally notation, or any non-place-value system of writing numbers. As some commenters have pointed out, it's also true that smaller numbers are more common in the real world than larger numbers, but that's a separate phenomenon, not Benford's Law.

BTW, my Erdos number is 3 :-)

If you want to hear more about Benford's law, Numberphile has a great video about it that does address some of what others were saying- how for some people it is intuitive.

https://www.youtube.com/watch?v=XXjlR2OK1kM

Also, if you liked this podcast, you might want to check out all of Numberphile's videos.

I understand why Benford's Law works. Lower digits (1, 2,& 3) easily appear more often than higher digits (7, 8, & 9). It is easier to own 1 acre than 9 acres. More people have $100 than $900. The lower numbers are more easily achievable than higher numbers in many categories.

I was listening to your radio program while driving and could not respond by phone or email to the discussion. Thanks for having thought provoking and enjoyable material!

Sad that a program like Radiolab with its interest in psychology and cognition apparently chose to censor one the most interesting facts about Erdos. He used amphetamines on a regular basis to enhance his mathematical ability. We know that these are medicines that canfunction as cognitive enhancers (as well as other prescription medications like modafenil). To me that is a very interesting and relevant part of Erdos's life and a topic that I would have expected Radiolab to address. I suspect it was censored in the interest of political correctness. Very disappointing, especially coming from a fascinating show like Radiolab.

It's one thing to say "This tax return violates Benford"; it's quite another to say "These two tax returns out of twenty violate Benford". The entire argument is statistical, and in the second case, you are selecting outliers. That puts you in the realm of multiple comparison problems, which are poorly understood - even by statisticians.

@Chris from MN,

I don't think the theory says that low numbers are more common than large numbers, I think its trying to say that numbers of any size are more common if they begin with small digits than numbers of the same magnitude starting with large digits. So, yes, 10,000 should be more common than 90,000, but 100,000 should also be more common than 90,000.

Yes, the show was fascinating yet simple in its premise, given that it is usually the simplest answers which are the best...Occam's Razor. What is most obvious is not always the simplest to 'see' or understand. Our human tendency is to 'look' far deeper into something than is required, relying too much on academic intelligence than emotional intelligence.

To me, it makes 'sense' that the counting up of numbers takes precedence, from lower to higher, and ties into what the 'Numbers' show told us about how infants learn in their development stage. Going back further, it is something we acquired that is imbedded in our primal brain stem. Everything

'stems' from there...pun intended!! Thanks for another great show!

I agree with the comment below me that this law sort of makes sense (however, I had never heard of the law, and found it fascinating that investigators have used it to discover criminal activity). It especially makes sense to me when counting normal things, including money. It's far more likely that a person has a $10 bill in his wallet than $90, just as it's far more likely that a person has a $100 bill in his wallet than $900. The same in a bank accounts. It's far easier for a person to save $10,000 than it is for a person to save $90,000. Seems like simple math to me, although I wouldn't have noticed it if it wasn't point out.

Absolutely loved this show, and most of the rest!

WHAT??? HOW CAN YOU BE CONFUSED??!

This was my reaction to this segment. Benford's Law seemed to be the most simple, OBVIOUS (like, DUH!) thing I've ever heard. Numbers start with lower numbers more of the time? Uh, yeah, of COURSE they do. Because we count up!

Consider the phrase "the numbers 1 through 10". Ten numbers. And ALREADY you have TWO of them starting with a 1.

How about 1 through 100? You have all the 1's, all the 10's (evenly distributed), and now 100. You ALREADY have more 1's at the start -- and you will for 99 more numbers.

And since we count UP, whenever we ARRIVE at a number (i.e., stop counting something), a "big" number will "cross over" into AN EXTRA DIGIT -- which resets the likelihood that it will start with a low number.

I'm listening to you guys express shock, surprise, and baffled confusion at this, and I'm thinking, "I'm shocked, surprised and baffled THAT you're confused." Honestly, this was the silliest segment I've heard yet.

Talk about self-evident... eesh.

Wonderful episode, as usual. A question just struck me today, while doing some photoshop work: does Benford's apply to something like visual images? As in, is there some way to assign meaningful quantitative values to the visual data that might result in a set which conforms to benford's? I imagine there might be some way to examine the data underlying a computer image, but a way to quantify something like a painting might also be interesting.

The underlying thought is, would Benford's be a way to detect fraudulent images (e.g., edited digital images, counterfeit paintings, etc.)? Or is there some other statistical tool that might apply? Or any?

Cheers!

I'm unsure why Benford's law should seem counter-intuitive. It seems VERY intuitive to me. Data sets in the world measure finite phenomena. Smaller counts are more common in the world than larger counts; smaller AMOUNTS are more common by necessity than larger amounts, for the simple reason that all natural data sets are on their way to infinity but only approaching. This means counts will more commonly end in the lower first-digit range regardless of scale. Besides that, think of a natural group of 100 objects. There are many ways you can count this set. You can have 100 ones. This means you have 100 data that begin with 1 and zero data that begin with anything else. You can have 50 twos. This means you have 50 data that begin with 2 and zero data that begin with anything else. If you have 2 fifties, you have only two data that begin with 5. Compare these two spare 5's with the many 1's or 2's. The 1's or 2's will always be more common. Anyway, it makes complete sense to me.

This episode was excellent for making me aware of the law (which I had not heard of before) but the treatment and accuracy of the story was unfortunate.

It was actually Simon Newcomb (a Canadian-American Astronomer and mathematician) that realised earlier pages in his logarithm book were more worn. Frank Benford was however the man that analysed a vast number of data sets. It's also quite possible to describe scale invariance in a simple way that most people would grasp.

The real problem I have with those inaccuracies is that now I'll feel the need to double check any fact before telling anyone else about what I heard on Radio Lab.

I was so intrigued by this episode that I wanted to try out Benford's Law myself.

http://lukasvermeer.wordpress.com/2010/05/24/benfords-law/

I've also made a simple website where you can try this on your own data. Instant results, no install required.

http://www.lukasvermeer.nl/projects/benford/

Radio Lab rocks! Keep up the good work!

Every morning I sit at Starbucks and have coffee with a group mostly retired guys from the neighborhood. I stand out in the group for the only fact that I'm 28.

This morning RadioLab enabled me to blow all of their minds. When a guy was introduced to the group as being a math professor I asked him what his Erdös Number was. Nobody but the two of us even knew what it was (FYI his number was 3). It was very nice to be able to sit back and retell the story of Erdös.

Over and over RadioLab gives me the chance to tell a great story. Keep up the good work guys. The time my friends wouldn't believe me that there is a parasite that makes you like cats is another great story...

The elliptical swimming pool problem: the show says that the outer shape is never an ellipse. It seems to me that the outer shape will be an ellipse if both shapes are circles.

I am the very opposite of a mathematician. I majored in art.

As i listened to your "Numbers" episode, i was struck by the incredulity and confusion with which you presented Benford's Law. As soon as i heard it, it sounded like the most sensible, plausible thing. Not that i'd noticed the pattern before, but whether talking about baseball stats, bank statements, centimeters, miles or milligrams, are not the lower numbers simply easier to achieve? Of course there are more bank balances and street numbers and populations in the 100s, 200s, 1000s and 2000s than 900s and 9000s. One is most common because we start at the beginning when counting anything and usually stop before the numbers get tremendously high. Even Wikipedia describes Benford's as "counter-intuitive" but to me it sounds incredibly obvious.

(also as a motorcyclist, i too would wager that low numbers at the gas pump are more likely us than teenagers with four wheels.)

Love this Radiolab show on Numbers. Those who enjoyed this will probably also love a series of short programs on BBC Radio called "5 numbers", "Another 5 numbers" and "A Further 5 Numbers". Check them out.

http://www.bbc.co.uk/radio4/science/5numbers.shtml

I love the show, but it's a shame you didn't tie together Benford's law with the previous segment on infant's understanding of numbers. They fit together perfectly. Once you've explained logarithms, you can easily explain Benford's law.

It's exactly because we live in a logarithmic world that Benford's law holds true. If you pick a random number on a logarithmic number line, you'll get lead digits following Benford's law. Some might also argue that it's because the world has logarithmic properties that infant's judgements seem to follow a logarithmic scale. They are understanding something fundamental about the world.

I was a bit disappointed that there was no significant effort to explain benford's law. To me it seems fairly intuitive, so here goes.

If you are counting something that "grows" out of something else, then the "energy" needed to grow a certain amount will be more related to the percent increase, not the absolute amount.

If I make 80,000, it is far more likely that I will get to 90,000, than if I make 110,000 that I will get to 200,000. so I stay in "low 6 figures" for much longer time and I will have more company. If a city street needs to be lengthened (adding more addresses) Same percentage growth in the city will go much quicker from the 700's to the 800's and 900's than doubling the entire length from 1000 to 2000.

I also enjoyed the show. However, I felt that the segment on Benford's law a little confusing so I looked it up. I was confused why proper accounting numbers follow the law but bad accounting does not. It seems a little weird. Both are sets of data. I looked in wikipedia and its links. As for accounting, it is really emperical. Somebody analyzed a lot of data and showed that cooked books do not follow it and non cooked books follow the law. The next question is do mathematician's empirically show a data set to follow the law or can it be proven analytically?

As stated by Benford's law not all lists universally apply to this law. I enjoy that the occurrences of people's Erdős number is inversely proportional to Benford's law (at least if you focus on the first few natural numbers).

I loved this episode too. I was absolutely captivated by the computerized music that followed this episode. There was a piece that ended the episode that was a series of chimes the pulsated and moved laterally outward through the stereo. It was like a musical mantra. Does anyone know the composer?

to David in DC:

The Erdős number distribution seems to be more of a bell curve. Go about half way down this page:

http://www.oakland.edu/enp/trivia/

under 'The distribution of Erdös numbers'

Lots of data sets do conform to Benford, but many don't. To see the battle between Benford and the bell curve, we'll have to hope for a Radiolab short.

Fascinating show. I was really engrossed by the Benford's Law stuff...tested it out on a few of my recent expense reports (sample size of 34 and 49 numbers) and it roughly approximated Benford.

Then I found this really great link to a video that does a great job of illustrating Benford's law with some large data sets, and shows numbers (such as rankings) that do not follow Benford: http://www.kirix.com/blog/2008/07/22/fun-and-fraud-detection-with-benfords-law/

Jad and Robert, please do keep up the great work!

The Erdös number is fun, but what starts making it really interesting is when you look at the Erdös-Bacon number.

http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Bacon_number

Great episode. I really enjoyed it. But I don't really agree with the notion that everyone with an Erdös number was "influenced" by the man. Just because I collaborate with someone who collaborated with him means that now I'm influenced by him, or connected to him? Why? Working with someone who worked with someone else doesn't make one influenced by the original person in any way, and surely not as the chain is expanded further.

Referring to the use of Benford's law to discover fraud. In order for this to work, numbers generated through purposeful human deception (i.e. me making up a fake salary), and those generated through more genuine means (i.e. the real salary that my boss decides to give me) must be fundamentally different. I am aware that consciously forged numbers probably differ from Benford's law, and possibly approach actual randomness. However, wouldn't all the permutations that occur on any tax form (subtract paid taxes, consider dependents, adding in other income, etc) eventually make my "fake" salary follow Benford's law?

David, I think there's every reason to belief that the distribution of Erdős numbers follows Benford's Law. However, the trouble in verifying this is that people with Erdős numbers greater than 6 either can't calculate it, don't wish to report it, are embarrassed it isn't lower, or are in a profession so distant from mathematics they have no reason to care! Hence in practice only a non-random subset of Erdős numbers are ever reported.

By the way Radio Lab, it's Erdős, not Erdös!! I enjoyed the show!

I like the Wiki quote which notes that Alfréd Rényi said, "a mathematician is a machine for turning coffee into theorems"

I'm still listening to the show here in D.C., at 6:47 p.m. and I thought you were going to relate the Bendford Rule and the Erdos number, but I don't think you are.

However, it seems to my VERY untrained ear as if the Erdos numbers do sort of follow the Benford Rule.

I.E., at first the numbers are big, though not in order (one is not bigger than two, etc.). By the five and six numbers, the amount of these is very large, i.e., robust.

And yet the numbers after six (I think) go down, which goes against the "robustness" of the lower Erdos numbers, yet seems to follow the Benford Rule.

Will someone comment to set me straight?

This was a fantastic show. My father is a mathematician, and during the Erdos segment, it hit me, I have no idea what this man does! No idea of what the heat kernels or eigenvalues of 3 spheres he speaks of are. As a first step towards that understanding, I think I am going to try to find his Erdos number....

I've used Benfords Law as a bundled application in some powerful data mining software to discover payroll fraud.

I really enjoyed this program. I find math fascinating and scary. I am currently enrolled in college (online)and have a remedial math class so naturally, your talk caught my attention. I agree that you cannot live without numbers I have focused on getting jobs based on how much math I needed to know. I worked in some aspect of bookeeping/accounting for the first four years of my professional life. At age 49 I am trying to beat my fear of math which reduces my mind to chaotic confusion until I just give up and just guess the answers. But I have managed to pass some math classes post high school and even understand your basic accounting. Paul Erdos and Benfords law will probably find their way into my class at some point.

Hello Mark Migrini, don't forget about this possibility: If you see a low amount on another gas pump, it *could* be from a cash-strapped teenager, but it could also be a motorcycle that needs only a fraction of what an automobile needs.

Oh, and be careful for motorcycles and bicycles. I ride both...

This episode was totally engrossing. I only wish I'd ever turned any of my random, maddening numbers obsessions into a "law."

Thanks for this show - loved it.

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