At Booz Allen in 2012, I needed to find the average time of day that something happened as part of my code. Taking the normal (linear) average of times of day doesn’t work because there is always a discontinuity somewhere, usually midnight. Since times of day are periodic, 11:59 pm and 1:01 am should average to midnight, not noon. But they’ll average to noon (1200) if you average something like 2359 (military time) and 0001.

I thought we had stumbled on to an area of mathematics that needed further study: finding the average and variance of periodic variables. Because all of my internet searches were things like “how to average periodic variables” and “average time of day” and the like, I could not find anything on the internet about how to solve this problem. So I started to derive an answer and ended up with something very similar to what I now learn is called Directional Statistics. Wiki article on directional statistics. I never thought to search for anything like “Directional Statistics” or “Circular Statistics” or anything similar, so I didn’t find Directional Statistics until Dave Anderson showed them to me this week.

The basic idea (that AJ Mobley, Dave, and I derived at Booz Allen) is that you treat times of day as points on the unit circle, and then average the points in two-dimensions. Invert the “average point” to get the average time of day. Additionally, the farther the point is from the origin, the lower the variance of the points.