## Friday, April 17, 2009

### Harmonic Mean

My wife is studying for an actuarial exam. She was doing an online seminar last night and I heard the speaker mention a harmonic mean. I hadn't ever heard of it, so I got curious. For three numbers a, b, and c. We know the average is (a+b+c)/3, but the harmonic mean is the reciprocal of the average of the reciprocals i.e. 3/(1/a+1/b+1/c). So for three particular numbers, 4, 7, and 8, the mean is 6 and 1/3. The harmonic mean is 5 and 23/29. I though cool, but when would I ever use this. Wikipedia gives an nice example:
What is the average speed of a car that goes a certain distance d at a speed of 60 kilometres per hour and then the same distance again at a speed of 40 kilometres per hour?
The (arithmetic) mean is of course 50. If we calculate the average speed directly, we wind up with a different number. The average speed is the distance they traveled divided by the time. The car travels 2d, and the time traveling at 60 kph is d/60. The time at 40 kph is d/40. So we have the average speed as 2d/(d/60+d/40). We simplify 2d/(5d/120) = 240d/5d = 48. This is what we would have go if we'd calculated the harmonic mean of the rates: 2/(1/60+1/40). Cool, huh? I found out there are other means, geometric and quadractic. I'd like to get examples for each of the others.

pic by mince

#### 1 comment:

Tinyc Tim said...

John Derbyshire's "Prime Obsession" opens with "Like many other performances, this one begins with a deck of cards." Pages 3 thru 7
follow.

3 4 5 6 7Your post on the "harmonic mean" is marginally related to this story about cards. I thought readers unfamiliar with this physical interpretation of the "harmonic series" would enjoy it.

If you make one "minor" change to the harmonic series by raising each term to the s power, where s is complex, you get the Riemann zeta function. The formerly "divergent" series suddenly becomes the most fascinating function ever imagined.