The leading digits of the heights of the world’s tallest buildings satisfy Benford’s Law.
Lately, I’ve been having fun reading John D Cook’s Blog. Cook is an applied mathematics consultant who blogs and tweets up a storm about all sorts of topics mathematical, statistical, computational, and scientific. He maintains 18 daily tip Twitter feeds giving daily facts about…well, everything…and one personal feed.
A leading digit 1 is expected to appear a whopping 30% of the time.
But what I like most are his mathematical blog posts. He writes short easy to digest posts about reasonably accessible topics in math, often with a computational bent, and I always walk away feeling like I learned something. This past week he revisited the topic of Benford’s law, which is this totally weird and strange thing that I’ve always wanted to understand more about. Benford’s law says that in many naturally occurring data sets the leading digit is more likely to be small. If the leading digit d from the set {1,…,9} were distributed uniformly you would expect each digit to show up about 11.1% of the time. But in reality, the leading digit is more often distributed according to the chart on the right. Cook can fill you in on some of the more precise formulations of Benford’s Law.
In his blog, Cook describes how the leading digits of factorials satisfy Bedford’s Law, and even gives some tips on how you can use Python to compute leading digits up to 500!! (One of those exclamations is a factorial, the other one is for my excitement.) He also show that the collection of SciPy constants follow Benford’s Law, which Cook explains and computes using Python. Cook blogged about how samples from the Weibull distribution satisfy Benford’s Law, and most recently he even showed that the iterates of the Collatz conjecture seem to follow Benford’s Law.
And you know a party is getting good when the Collatz Conjecture shows up.
These posts just give a small flavor of Cook’s writing. I also really enjoyed his recent posts on harmonic numbers and golden angles (largely because it prompted me to check out the work of the visual artist John Edmark), the lesser known cousin of the golden ratio.