Why does Benford’s Law work? This law says numbers in a data set are more likely to start with low digits (1, 2) than high digits (8, 9), but there are exactly as many numbers in existence beginning with each digit.

Question

Why does Benford’s Law work? This law says numbers in a data set are more likely to start with low digits (1, 2) than high digits (8, 9), but there are exactly as many numbers in existence beginning with each digit.

In: Mathematics

Answer 1

It’s because the values aren’t just random, the ranges are. A random number 1-99 has an equal chance of starting with any digit. A random number 1-90 has a much lower chance of starting with 9. A random number 1-18, over half the possible values start with 1. Average this out over large datasets & the lower numbers are much more common as an initial digit.

Bedford’s law doesn’t apply in situations where values are not wide open in what they can be. Back in 2020, a lot of people were trying to make hay about vote totals by district in Illinois not following Benford’s law, but the districts are all about the same size, so you wouldn’t expect them to. The same would apply to things like people’s heights, pressure levels in tires, average star ratings on IMDB, dice rolls or anything else where the possible values are fairly constrained.