Benford’s law, also called the first-digit law, states that in lists of numbers from many (but not all) real-life sources of data, the leading digit is distributed in a specific, non-uniform way. According to this law, the first digit is 1 almost one third of the time, and larger digits occur as the leading digit with lower and lower frequency, to the point where 9 as a first digit occurs less than one time in twenty. This distribution of first digits arises logically whenever a set of values is distributed logarithmically. For reasons described below, real-world measurements are often distributed logarithmically (or equivalently, the logarithm of the measurements is distributed uniformly).

This counter-intuitive 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). The result holds regardless of the base in which the numbers are expressed, although the exact proportions change.

It is named after physicist Frank Benford, who stated it in 1938,^[1] although it had been previously stated by Simon Newcomb in 1881.^[2]

^{Benford’s law – Wikipedia, the free encyclopedia.}

I heard about Benford’s Law this weekend via a fascinating Radio Lab podcast on numbers. I came home and tested a few series of numbers and naturally enough they matched Benford’s Law. Below is the distribution of the first two hundred Fibonacci Numbers.

The podcast covered a lot of interesting ground on numbers. It seems we’re born with logarithmic programming and are re-trained by our parents and almost everyone around us to think linearly, 1 2 3 4 5 6 and so on.

What is the mid point of 1 and 9? If you think 5 then you’ve been reprogrammed, if 3 then you’re you’ve still got the programming nature intended, if both then well done you’ve reprogrammed yourself. 1+4=5+4=9 vs 1×3=3×3=9.

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1. Some good numbers
2. Safety in Numbers – Q4 2009 Results
3. Satety Insurance: Safety In Numbers
4. Inch Worm – Enjoy the Journey while Planning for the Future