Alien astronomers and Benford’s law

In 1881, astronomer Simon Newcomb noticed something curious. The first pages in books of logarithms were dirty on the edge, while the pages became progressively cleaner in later pages. He inferred from this that people more often looked up the logarithms of numbers with small leading digits than with large leading digits. Why might this […]

Benford’s law

This page is an index to articles on the site about Benford’s law. Benford’s law and probability distributions Pareto distribution Weibull distribution Cauchy distribution Benford’s law and number theory Leading digits of powers of two Gelfand’s question Leading digits of factorials Benford’s law and computer science Collatz 3n+1 conjecture Benford’s law and statistics Benford’s law […]

The 3n+1 problem and Benford’s law

This is the third, and last, of a series of posts on Benford’s law, this time looking at a famous open problem in computer science, the 3n + 1 problem, also known as the Collatz conjecture. Start with a positive integer n. Compute 3n + 1 and divide by 2 repeatedly until you get an odd […]

Cauchy, Benford, and a problem with NHST

Introduction Samples from a Cauchy distribution nearly follow Benford’s law. I’ll demonstrate this below. The more data you see, the more confident you should be of this. But with a typical statistical approach, crudely applied NHST (null hypothesis significance testing), the more data you see, the less convinced you are. This post assumes you’ve read the […]

Benford’s law and SciPy

Imagine you picked up a dictionary and found that the pages with A’s were dirty and the Z’s were clean. In between there was a gradual transition with the pages becoming cleaner as you progressed through the alphabet. You might conclude that people have been looking up a lot of words that begin with letters […]

Leading digits of primes

How are the first digits of primes distributed? Do some digits appear as first digits of primes more often that others? How should we even frame the problem? There are an infinite number of primes that begin with each digit, so the cardinalities of the sets of primes beginning with each digit are the same. […]