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
ReplyDeleteJohn 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.