Every Third Number Starts with 1

An interesting fact to think about; there is 30% chance for any given number in big databases to start with 1. Whether it's population, sales, or areas the theorem Benford Law comes true. It says; "In great data of numbers, 30% of them have 1 as the first digit."

17% have 2 as the first digit, and the percentages decrease all the way down to 9 that is the first digit in only 5% of numbers.

Simon Newcomb was the first person to discover this theorem, as he noticed that certain pages of his table of logarithms were more worn out then others. He then noticed that on those pages, the logarithms started with the number 1. Seeing as he doubted he had an overly fascination with the number 1 he studied other scientist's tables, and found the same thing. A small article was published, but then the idea was forgotten about for 60 years.

Dr. Frank Benford, who at the time was working for General Electric found the forgotten article from Newcomb, and was immediately interested. Huge amounts of work was put in to analyze over 20 000 sets of numbers to see if it was actually true, and in that case, what cases it could be applied to.

He dealt with a wide range of numbers, including known rivers, base ball statistics, and numbers in newspaper and magazine articles, bills, and street addresses. All the tests proved the same conclusion: The number 1 was the first digit in 30% of the cases, and was always the most frequent number. Even the house Benford lived at had a street number that started with the number 1. The theorem is now called the Benford Law.

In an article in the New York Times the mathematician Mark Nigrini explains the number magic with an example from the stock Dow Jones:

["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."]

Mark Nigirini was one of the first to suggest this law as a way to control tax and number fraud. Early in the 1980's he cooperated with the RIS in New York, and received access to both fraudulent and true tax data. The result came back and further proved the theory, in the true tax reports the number 1 was in 30% of the cases the first digit. In the fraudulent however, the number 1 appeared far less, and the most prominent numbers were 5 and 6.

This law has already uncovered several fraud attempts, and in the future it will probably be used more and more as a way to solve crime.

A quick check on Wikipedia of a list of countries and outlying areas confirms that the theorem holds true. I counted 65 areas with 1 as first digits, and out of 232 countries total that would equal 28%. Very impressive indeed!