Science a GoGo's Home Page Forums General Discussion General Science Discussion Forum Benford's Strange Math Law

Data Sets of Numbers follow a surprising law of digits, and scientists can't explain why.

Does your house address start with a 1? According to a strange mathematical law, about 1/3 of house numbers have 1 as their first digit. The same holds true for many other areas that have almost nothing in common: the Dow Jones index history, size of files stored on a PC, the length of the worlds rivers, the numbers in newspapers front page headlines, and many more.

The law is called Benford’s law after its (second) founder, Frank Benford, who discovered it in 1935 as a physicist at General Electric. The law tells how often each number (from 1 to 9) appears as the first significant digit in a very diverse range of data sets.

Intuitively, most people assume that in a string of numbers sampled randomly from some body of data, the first non-zero digit could be any number from 1 through 9. All nine numbers would be regarded as equally probable. But you would be wrong.

Given a string of at least four numbers sampled from one or more sets of data, the chance that the first digit will be 1 is not one in nine, as many people would imagine; according to Benford's Law, it is 30.1 percent, or nearly one in three. The chance that the first number in the string will be 2 is only 17.6 percent, and the probabilities that successive numbers will be the first digit decline smoothly up to 9, which has only a 4.6 percent chance.

Benford's Law is a powerful and relatively simple tool for pointing suspicion at frauds, embezzlers, tax evaders, sloppy accountants and even computer bugs.
The income tax agencies of several nations and several states, including California, are using detection software based on Benford's Law, as are a score of large companies and accounting businesses.
Benfords law has also been recently applied to electoral fraud in order to detect voting anomalies. Scientists found that the 2004 US presidential election showed anomalies in the state of Florida, as well as fraud in Venezuela in 2004 and Mexico in 2006.

However you can't use it to improve your chances in a lottery. "In a lottery someone simply pulls a series of balls out of a jar, or something like that. The balls are not really numbers; they are labeled with numbers, but they could just as easily be labeled with the names of animals. The numbers they represent are uniformly distributed, every number has an equal chance, and Benford's Law does not apply to uniform distributions."
http://plus.maths.org/issue9/features/benford/
and
http://mathworld.wolfram.com/BenfordsLaw.html
and
http://www.rexswain.com/benford.html

That's interesting, and it makes sense. After all, number ranges most often start at '1' and, before you know it, they reach the teens, so it's not surprising that it occurs more frequently - especially since there will be many ranges that don't go very far beyond the teens. By the same token, the least likely occurrence should be '9' since the '90s' (and '900s') are farther up the numerical scale. There are probably more streets with 100 - 199 houses than with 200 - 299, less in the 300 - 399 range, and so on. Even so, it is surprising that as many as 1/3 of house numbers start with a '1'.

Very interesting, Mike. If I ever have to file income taxes again I'll be sure and check them over with this in mind.

Thanks for sharing.

Amaranth