How do numbers begin?

In many data series a surprising number of entries begin with the number 1, and the number 2 is also more common than a random distribution might suggest.  This is called Benford’s Law.  For instance about one third of all house numbers start with one.  That may be a quirk of bureaucratic numbering psychology, but the principle also applies to the Dow Jones index history, size of files stored on a PC, the length
of the world’s rivers, and the numbers in newspapers’ front page headlines.  It does not apply to lottery-winning numbers, see the graph at the above link.  Here is an exact statement of the law:

Besides the number 1 consistently appearing
about 1/3 of the time, number 2 appears with a frequency of 17.6%,
number 3 at 12.5%, on down to number 9 at 4.6%.  In mathematical terms,
this logarithmic law is written as F(d) = log[1 + (1/d)], where F is
the frequency and d is the digit in question.

I feel as if someone is pulling my leg.  And I keep thinking of nominal interest rates being bounded from below at zero.  Yes this has practical implications:

…because a year’s accounting data of a company
should fulfill the law, economists can detect falsified data, which is
very hard to manipulate to follow the law. (Interestingly, scientists
found that numbers 5 and 6, rather than 1, are the most prevalent,
suggesting that forgers try to “hide” data in the middle.)

The law was first discovered by an economist (and astronomer), Simon Newcomb.  Here is Wikipedia on the law.  Here is more startling data on where the law applies.  From a completely orthogonal but I suspect not totally irrelevant direction, here is Tim Harford on price stickiness.

This whole topic makes me feel like an idiot for even bringing it up, with apologies to Pythagoras. 


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