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Monday, January 28, 2013

We're Number One: Benford's Law

Did you know? The digit 1 occurs much  more frequently as a first digit in our lives than the other digits. Just take a look at the first digit of bank account numbers, income statements, grocery receipts, restaurant tabs, tax forms and any other area of life where  lists of numbers occur.

Statistically, we would predict the digit 1 to occur as the first digit in a list of numbers one out of nine times or 11.1%. But instead, the digit 1 tends to occur in tables, listings, and statistics with probability ~30%, much greater than we would expect. This phenomenological law is called Benford's law, the first digit law, first digit phenomenon, or leading digit phenomenon.

Not every student will become a mathematician or develop an interest in Benford's law, but all can learn to view math as something useful to everyday life. Schools across the nation that use Excel Math consistently report improved test scores. Many graduates of Excel Math tell us math is now their favorite subject. See how Excel Math can help your students at
Benford's law was first published in 1881 by the astronomer Simon Newcomb. It is named for the late Dr. Frank Benford, a physicist at the General Electric company. In 1938 he discovered, after examining tables of logarithms, that the first pages were much more worn and smudged than later pages.

A logarithm is an exponent. Any number can be expressed as the fractional exponent—the logarithm—of some base number, such as 10. Published tables let us look up logarithms corresponding to numbers, or numbers corresponding to logarithms. But logarithm tables (remember the old slide rules derived from them?) are no longer used much for calculating. Calculators and computers are easier and faster. But logarithms remain important in many scientific and technical applications, and they were a key part of Dr. Benford's discovery.

Dr. Benford concluded that it was unlikely that physicists and engineers had some special preference for logarithms starting with 1. He therefore began a mathematical analysis of 20,229 sets of numbers, including the areas of rivers, baseball statistics, numbers in magazine articles, random samples from a day's stock quotations, a tournament's tennis scores, the numbers on the front page of The New York Times, the populations of towns, and electricity bills in the Solomon Islands, to name a few. All these seemingly unrelated sets of numbers followed the same first-digit probability pattern as the worn pages of logarithm tables suggested. In all cases, the number 1 turned up as the first digit about 30 percent of the time. Read more at

It doesn't seem to matter whether the numbers are based on the dollar prices of stocks or their prices in yen or marks, nor does it matter if the numbers are in terms of stocks per dollar. As long as there are enough numbers in the sample, the first digit of the sequence is more likely to be 1 than any other. Read more at

In general, this first digit law says that the probability of the first digit being a "d" is

This formula suggests that the number in a table of physical constants is more likely to begin with a smaller digit than a larger digit. Read more at

Since most people aren't aware of this phenomenon, it has been used to help detect fraud. Software companies have developed detection software based on Benford's Law. The income tax agencies of several nations and several states, including California, are using this software, as are a score of large companies and accounting businesses. Benford's law is even used in "Statistics forensics" to help bring criminals (such as embezzlers, tax evaders, frauds and even sloppy accountants) to justice. Learn more about fraud detection at

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