Is This Normal?

I saw this article on Hacker News -- it's mainly about pay scales and performance and, you know, tedious real-world stuff¹. It got me wondering about whether I could wave my hands at the real world with a not-entirely-outlandish assumptions and end up with a Pareto distribution.

So: I assumed that there's a firm with 1000 employees, and every person in the universe has a fixed performance rate drawn from ( N(100,10^2))², and this number is known for everyone. Every year, the firm fires its 100 worst-performing employees, and runs a hiring process: they get 1000 applicants, and hire the 100 best of them.

Starting with 1000 normally-distributed employees and running the routine for 100 years (the employees live until they're fired, of course), we get this as the mean performance of the 900 surviving workers:

... which is roughly what I'd expect, the mean rises very sharply at first, but with diminishing returns over time.

What do the surviving employees look like after 100 years, apart from grizzled and slightly terrified of what happens when they do eventually leave? Here's a (slightly funky) CDF plot with a log-scale on the y-axis³:

If the resulting distribution is Pareto, this line should be approximately straight. I'd call that plausible, at least! Incidentally, 20% of the employees are below the blue line. 80% of the performance scale is to the right of the orange line, which also fits with the folk-wisdom Pareto distribution.

To finish: I note that I haven't found (or looked for) any evidence that the distribution isn't lognormal -- I think I'd expect a curve in that case.

The code

... should run like:

python main.py

This is hacked-together code for my own interest. You are welcome to play with it. Let me know if it breaks, but I can't promise to fix it.

Footnotes

1. Saying that, if you want to pay me to do real-world stuff, I'm available for consulting work. Email colin@colinbeveridge.co.uk. ↩

2. No special reason to pick those parameters other than to keep the numbers nice. ↩

3. You would usually expect a CDF to start from the lowest values and move to the highest. Doing it backwards is what gives a good log fit. ↩