Benford’s Law And A Theory of Everything
In 1938, the physicist Frank Benford made an extraordinary discovery about numbers. He found that in many lists of numbers drawn from real data, the leading digit is far more likely to be a 1 than a 9. In fact, the distribution of first digits follows a logarithmic law. So the first digit is likely to be 1 about 30 per cent of time while the number 9 appears only five per cent of the time.
That’s an unsettling and counterintuitive discovery. Why aren’t numbers evenly distributed in such lists? One answer is that if numbers have this type of distribution then it must be scale invariant. So switching a data set measured in inches to one measured in centimetres should not change the distribution. If that’s the case, then the only form such a distribution can take is logarithmic.
But while this is a powerful argument, it does nothing to explan the existence of the distribution in the first place.
Then there is the fact that Benford Law seems to apply only to certain types of data. Physicists have found that it crops up in an amazing variety of data sets. Here are just a few: the areas of lakes, the lengths of rivers, the physical constants, stock market indices, file sizes in a personal computer and so on.
However, there are many data sets that do not follow Benford’s law, such as lottery and telephone numbers.
(Note from NT: I don’t post this because I pretend to understand it.)