If you examine all the lengths of all of the rivers in the world, the sizes of all of the files stored on your computer, or all of the numbers on your tax return, what is the probability that the first digit and any number is a 1? It should be 1 in 9, right?

Well, it’s not. The answer is 31% (almost 1 in 3)

The probably of a 9 is only 4.6% (less than 1 in 20). How can that be?

Distributions of numbers follow a power law. In 1881 Simon Newcomb noticed that the first few pages of his logarithm tables were more worn that the last pages. In the 1920’s Frank Benford noticed the same thing with his logarithm tables. Benford then looked at lots of other data sets, and found the same distribution occurring in lots of measured data.

Random number sets and assigned numbers like lottery numbers, dates, and telephone numbers typically do not follow Benford’s Law, but data taken from observations of the real world does tend to follow it.

To make it even freakier, it’s scale invariant. From Physorg.com:

In 1961, Pinkham discovered the first general relevant result, demonstrating that Benford’s law is scale invariant and is also the only law referring to digits which can have this scale invariance,” the scientists wrote in their letter. “That is to say, as the length of the rivers of the world in kilometers fulfill Benford’s law, it is certain that these same data expressed in miles, light years, microns or in any other length units will also fulfill it.

The best explanation I found is at the Journal of Accountancy:

Mutual fund math. An intuitive explanation of Benford’s law is to consider the total assets of a mutual fund that is growing at 10% per year. When the total assets are $100 million, the first digit of total assets is 1 . The first digit will continue to be 1 until total assets reach $200 million. This will require a 100% increase (from 100 to 200 ), which, at a growth rate of 10% per year, will take about 7.3 years (with compounding). At $500 million the first digit will be 5 .

Growing at 10% per year, the total assets will rise from $500 million to $600 million in about 1.9 years, significantly less time than assets took to grow from $100 million to $200 million. At $900 million, the first digit will be 9 until total assets reach $1 billion, or about 1.1 years at 10%. Once total assets are $1 billion the first digit will again be 1 , until total assets again grow by another 100%. The persistence of a 1 as a first digit will occur with any phenomenon that has a constant (or even an erratic) growth rate.

The numbers 5 and 6 show up most when people try to fake a random distribution. The 2004 election results show anomolies in the State of Florida (surprise). Tax agencies use Benford’s Law to look for irregularities, so if you’re going to cheat make sure you use lots of 1s and 2s.