The triathlon training blog of Phil Barnes

The (evil) Harmonic Mean

Quick question:

You're out for a bike ride, on the way out you average 10 mph (uphill, into the wind); on the way back, you average 30 mph (downhill, with the wind). What was your average speed?

If you said 20 mph ... you are wrong. In actuality it is 15.


Let's tighten up the wording. If you go for 6 mi and average 10 mph and then return 6 mi and average 30 mph -- your average speed for the 12 mi distance would be 15 mph.


Ok, how much time did it take you to complete 6 mi at 10 mph? 6/10ths of an hour right? Yes. 0.6 hours.

Next, how much time did it take you to complete 6 mi at 30 mph? 6/30ths of an hour? Yes.
0.2 hours (one third the time).

So... in 0.6 hours + 0.2 hours, you completed 6 + 6 mi? Yes. So, in other words you did 12 mi in 0.8 hours? Yes. So in other words your speed was 12 mi/0.8 hr. In other words, you did 15 mph.

The (evil) Harmonic mean is at work here. You can use an equation to figure out the harmonic mean of 2 numbers: 2 / ( (1/n) + (1/m)). Where n and m are the 2 numbers. If you have more than 2 numbes: n / ((1/x1) + (1/x2) + (1/x3) + ... + (1/xn))

Why do I bring this up?
1. It's late and I can't sleep, so I feel like writing something.
2. I got fooled by this over the weekend. I brain cramped and thought I got jipped by Garmin on my average speed for a trip.
3. "Harmonic means" are often used in stream flow statistics (sorry, forgot to tell you I was a water resources engineer).

Want to know something crazy?
If you go in a straight line from point A to point B at 10 km/hr, you would have to return in the same straight line from point B to point A at over 100,000 km/hr to average 20.0.

It's true.. I wouldn't make that up. You can check it here.

Good night.