Benford's Law modeler shows why numbers often start with 1

A little Flash application provides a compelling example of Benford's Law, making it possible to see intuitively something that is otherwise quite abstract.

Yesterday, I blogged about Benford's Law, which predicts that in a set of numbers, numbers that begin with the numeral 1 will appear more often than other numbers. This can be used to catch cheats by checking to see if the data they give in their tax-returns, research data-sets or homework have more numbers that start with 1 than other numerals.

William Fawcett, who made the Flash app, sez, "A couple of years ago I attended an excellent lecture by Simon Singh - he writes popular science and maths books like Fermat's Last Theorem and Big Bang - and this was one of the mathematical curiosities he mentioned. I was intrigued but sceptical, so built a quick Flash app to test it." Link (Thanks, William!)

Update: Gary sez, " If you use a close approximation of PI (say 3.14159265) as a factor, you can avoid getting numbers that start with 1 at all. Using a factor of 3.14159265 and a start number of 3, you get no numbers starting with 1. Change the start number to 6, you get only numbers starting with 1, 4, or 5. I don't know how the modeler is coded, so I wonder whether this is a quirk in the modeler or an exception to the rule."

Update 2: Bob solves the mystery (also thanks to Bernardo for his explanation):

This result comes not because he used pi, but because pi is an approximation of the square root of 10. (3.16 is closer than 3.14.) If you use 3.16 as the multiplying factor (31.6 and 316 also work), the signifigant digit will always be one of two numbers (this works best if you use a single-digit number for the start number).

If you use 1 or 10 or 100 as the multiplying factor, of course all numbers will start with the same digit (the starting digit of the start number).

If you use the cube root of a power of 10 such as 2.154 (21.54, 215.4, and 2154 also work), all results will start with one of 3 digits. And so on.

The results using decimal approximations depend on the Flash app stopping when the 25th occurence is reached. Since an exact square or cube root of ten can't be expressed as a decimal, the distribution would eventually comply with Benford's Law if the 25th occurence limitation was removed.