Benford's Law


I’ve been listening to the Radiolab podcast (which is excellent).

I recently listened to their issue about numbers. Some cool stuff in there, like how and when babies start to understand quantities (logarithmically and very young).

The thing that interested me the most was Benford’s law. Here’s my succinct definition.


The world prefers to start numbers with a 1. Its second choice is 2. Its third choice is 3, and so on.

Longer description:

When compiling lists of numbers (like, say, bank account balances of different people over a certain number of years), the numbers that come up will start with a 1 much more often than any other number.

This phenomenon works for lots of different things. Areas of rivers; populations; death rates; stock prices; heights of buildings, and a lot more.

Numbers that start with 1 appear about 30% of the time. 2s appear less frequently (17%-18%), 3s less again, 4s less again, until we get to 9s, which appear only about 5% of the time.

This is crazy, but has been shown again and again. In fact, it’s so accepted that its absence is used as an indicator of foul play!

When the numbers don’t follow Benford’s law, then there’s a deep suspicion that someone fudged something. This is apparently acceptable in a court of law as part of a case to show fraud.

Like, if you want to know whether you’re being cheated by the bank, check your balances over the last few years. If there are too many 7s and not enough 1s leading the numbers, then call a lawyer!

Math is awesome.

3 Responses to Benford's Law

  1. BruceS March 18, 2010 at 3:02 pm #

    Being a C programmer, I start all my numbers with 0 (much more logical), but the banks keep trimming them off.
    This sounds more like voodoo than math. My bank *balances* should really start with 1 much more often than with other numbers?! I’ve *got* to check that out. Thanks for another fascinating one.

  2. BruceS March 22, 2010 at 3:30 pm #

    Update: I checked our monthly closing balance on our primary bank account for 18 statements, and got the following frequency of 1st digit:
    1 9
    2 2
    4 1
    7 2
    8 3
    9 1
    While this does show a surprisingly high frequency of ‘1’ (50%!), it doesn’t follow the predicted pattern. I wonder if it would work better with a larger sample. While it all makes perfect sense for systems that grow exponentially, I still find it dubious for things like my bank balance, which has an unfortunate failure to grow exponentially. A base of 2 and a period of 6 months would be just fine by me.

  3. weeklyrob March 22, 2010 at 3:36 pm #

    I think you’d need to do it for a few years.

    It’s definitely not limited to things that grow exponentially. I mean, death rates, for example, or heights of buildings.

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