## Benford’s Law

Sunday, November 27th, 2005

According to Benford’s Law, a number is far more likely to start with the digit 1:

Benford’s Law is named for the late Dr. Frank Benford, a physicist at the General Electric Company. In 1938 he noticed that pages of logarithms corresponding to numbers starting with the numeral 1 were much dirtier and more worn than other pages.

[...]

Dr. Benford concluded that it was unlikely that physicists and engineers had some special preference for logarithms starting with 1. He therefore embarked on a mathematical analysis of 20,229 sets of numbers, including such wildly disparate categories as the areas of rivers, baseball statistics, numbers in magazine articles and the street addresses of the first 342 people listed in the book ‘American Men of Science.’ 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, more often than any other.

To illustrate Benford’s Law, Dr. Mark J. Nigrini offered this example:

If we think of the Dow Jones stock average as 1,000, our first digit would be 1.

To get to a Dow Jones average with a first digit of 2, the average must increase to 2,000, and getting from 1,000 to 2,000 is a 100 percent increase.

Let’s say that the Dow goes up at a rate of about 20 percent a year. That means that it would take five years to get from 1 to 2 as a first digit.

But suppose we start with a first digit 5. It only requires a 20 percent increase to get from 5,000 to 6,000, and that is achieved in one year.

When the Dow reaches 9,000, it takes only an 11 percent increase and just seven months to reach the 10,000 mark, which starts with the number 1. At that point you start over with the first digit a 1, once again. Once again, you must double the number — 10,000 — to 20,000 before reaching 2 as the first digit.

As you can see, the number 1 predominates at every step of the progression, as it does in logarithmic sequences.