Friday, November 6, 2009

Benford's Law: preference for the starting digit of 1

Frequency of first significant digit of physic...Image via Wikipedia

Very nice write-up of Benford's Law: why so many scientific constants and other published numbers start with the digit "1".
I have seen Knuth's write-up, and, I admit, could not make heads-or-tails of it. This is very nicely and simply explained.
Benford's Law: "

Dow Illustrates Benford's Law

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.

One of Knuth’s reward checksImage via Wikipedia

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

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1 comment:

Overglo Music ~ Chris said...

Very clear explanation. I hope you don't mind if I post a link...but I'm part of a group that does "rapid composition songs" where we write and record music on an extremely tight schedule. Our most recent song is an ode to Benford's law...