John Cook checks out fat tails:
Someone with no exposure to probability or statistics likely has an intuitive sense that averaging random variables reduces variance, though they wouldn’t state it in those terms. They might, for example, agree that the average of several test grades gives a better assessment of a student than a single test grade. But data from a Cauchy distribution doesn’t behave this way.
This is one of my favorite distributions to mess with, though I’m glad few things naturally follow a Lorentzian. I had a colleague of mine build an Excel spreadsheet to describe the behavior, showing that results never converge and you could easily pull values 50+ “standard deviations” from the “mean.” I use scare quotes there because neither concept actually exists for the distribution, but if you mistake it for an awkwardly thin normal distribution, it can lead to some interesting results.