Benford's Law is virtually as interesting as Pi () [3.14159,26535,89793. ...] Instead it is a mathematical law used to catch Tax Cheaters. It is based on certain mathematical properties of the universe.
If, for instance, we were given a compilation of a million bank account balances from any particular bank, respecting the leading (first) digit, the number might be any from 1-9. We might assume and suppose that each of the nine digits would have a one-ninth (1/9th) chance to be the leading digit. But that is not true.
As it turns out, the leading digit is a “1" 30.1% of the time; a “2" 17.6% of the time; a 3 approximately 13% of the time and a 9 only about 4-1/2% of the time.
A professor teaching Benford's law to his class, gave a homework assignment of tossing a coin 200 times and writing down the result. He suggested it was okay to cheat by making up results. He could tell who cheated mathematically.
The neat thing about this law is that it applies to all sorts of phenomena be it lengths of rivers, populations of cities, numbers on pages #1 and 2 of the Wall Street Journal, etc. The more numbers in the sample, the more closely the percentages follow the mantissas.
So how does it work? We take the mantissas namely the Base 10 Logs of numbers and subtract them from each other. For instance, to find results for #1, we take the log of 2 which is .30103 and we obtain 30.1%. To obtain the results for #2, we subtract the log of 2 (.30103) from the log of 3 (.47712) and we obtain .176 or 17.6% and so on. It is an extremely elegant formula and with its occurrences in nature is essentially as real and as interesting as Pi (T).