You might have thought that the distribution of the first digit in a set of data (for example the population of countries in the world) would be evenly distributed? Well, it turns out that it’s not, and this is described by Benford’s Law.
So instead of one-ninth chance of a number starting with each digit from 1 through 9, the probably of a value start with a 1 is about 30%, starting with a 2 is 17%, 3 is 12% and so on, until the change of something starting with a 9 is 4.6% (from Wikipedia).
From Wikipedia again:This result has been found to apply to a wide variety of data sets, including electricity bills, street addresses, stock prices, population numbers, death rates, lengths of rivers, physical and mathematical constants, and processes described by power laws (which are very common in nature). It tends to be most accurate when values are distributed across multiple orders of magnitude.
But then when you think about it, it makes some sort of sense. If you have a population which is doubling every year, then if it starts at 1, next ear it will be 2, then 4, 8, 16, 32, 64, 128, 256, etc. The first digit in that series is 1 pretty often.
Or think of the population of countries. As an example, I think of the population of India over time. When I was growing up, in the ’90s, the population was in the 800 millions (thanks Wikipedia again) and growing at about 20% per decade. In 1991 the population was 846,387,888 and by 2001 it had grown to 1,028,737,436 and was growing at about 20% per decade. The first digit of the population had zipped through 3, 4, 5, 6, 7, 8 and 9 in the years from 1950-2001. Now that the first digit is a 1, it’s going to take a long time to increase a 2 by growing at 20% per decade (or possibly creep back to a 9).
I’m aware that I’m looking at the growth of populations rather than a static snapshot but I find it easier to visualise that way.
I heard about Benford’s Law about a year ago and found it quite interesting and unexpected. Looking back though I’m surprised that I had never noticed that before and it sort of seems logical. Reading the history of the law on Wikipedia makes it seem like it was only reasonably recently discovered and fully explained.
Note: It seems that Wolfram has a few good examples of where this law holds on this page: http://mathworld.wolfram.com/BenfordsLaw.html