governing the distribution
of the first
(most-significant, leftmost) digit
arising from an unbounded
distribution (i.e. it works for lengths of rivers, but not for phone numbers). Contrary to initial expectations, the leftmost digit is a `1' in more than 30% of the numbers, and `9' in less than 4.6%! The actual fraction
of numbers starting in digit d is log10(d+1)-log10(d) (note that no numbers start with a `0'). For different base
s, change the base of the logarithm
; bases like 1000 are relevant to the decimal case, too.
What's going on? Well, suppose a distribution actually exists for numbers like the lengths of rivers. Clearly it can't be related to the units used to measure the length, so multiplying by a constant can't change matters. So it has to be logarithmic.
If you know any probability, you realise that the above paragraph is meaningless! There is no unbounded uniform distribution, which would seem to work best with the argument. And you cannot prove anything about the distribution if it doesn't exist! Nonetheless, people have managed to prove some versions of Benford's Law, by making "reasonable" assumptions. And empirically, it works!