What "mean" means

Dear Dr. Math,
My parents live about 200 miles away from me, so I make the drive back and forth a lot, with no stops. Almost exactly halfway in between the speed limit changes, so instead of driving 55 mph I drive 80 mph. Since my average speed is 67.5 mph, shouldn't it take me 200/67.5 = 2.96 hours to get there? I've noticed it always takes a little longer, but I don't get it. I've even set the cruise control and kept the speeds exactly constant.
Chuck

Dear Chuck,

I'm going to go ahead and assume that you live in one of those places in Utah or west Texas where the speed limit actually is 80 mph. Otherwise, you've been speeding, and I can't endorse that kind of behavior. OK? OK. Don't make me write a post about the correlation between speeding and traffic fatalities. I swear I will turn this blog around.

Here's why your numbers didn't add up: while it's true that the average, in the sense of arithmetic mean, of 55 and 80 is mph, that's actually the wrong kind of average to be using in this circumstance. "Kinds of averages?" Oh yes. Allow me to explain:

In the course of your trip, you drive half the distance, 100 miles, at 55 mph. So that leg takes you hours. On the second half, you're going the legal speed limit of 80 mph, so that half should take you hours. Altogether, then, your driving time is 1.81 + 1.25 = 3.06 hours, a little more than you expected.

Rather than the arithmetic mean here, you should have been calculating your harmonic mean, which for two numbers A and B is defined as . To see why that's the right quantity, let's denote by S your real average speed for the trip, that is, the total distance you traveled divided by your total time. If T is the total time you spent driving, then ; equivalently, . If A is the speed you went for the first half and B is the speed for the second half, then another way you could calculate the total time is as , just like we did previously. As usual, in math when we compute the same thing two different ways we end up with an interesting equation. In this case, since the times are equal, we get:
.
Dividing through by 100 on both sides gives us

which, if you take reciprocals of both sides and multiply by 2, yields the formula for the harmonic mean. In this particular example, mph, so your guess of 67.5 mph was only off by a little bit.

So, when is the arithmetic mean the right one? If you had gone on a trip and spent an equal amount of time driving 55 mph and 80 mph, then your average speed would be the arithmetic mean of the two. To see that, let's just assume you drove 1 hour at each speed. Thus, your total distance traveled would be miles, and your total time is 2 hours, so the average speed is mph. Voilà! If you look at that calculation closely, you can pretty clearly see why it should always give you the arithmetic mean--you're just adding the two speeds together and dividing by 2. Similarly, another way to see that the arithmetic mean is inappropriate for the equal distance problem is to notice that by driving the same distance at each speed, you spend more time at the slower speed and less time at the faster one.

There's actually yet another kind of mean, called the geometric mean, which shows up when you're computing ratios, percents, interest rates, and other things that are typically multiplied together. For two numbers A and B, it's defined as . For example, let's say you were a rabbit farmer and your population of rabbits grew by 50% one year and only 10% the next. The combined effect at the end of two years would be that the population had increased by a factor of , for an increase of 65%. To achieve that same growth at a constant rate, say a factor of R for each year, you'd need , so . So in a sense the "average" growth rate was 28% per year. Many people in this kind of situation would be tempted to guess that the average was 30%, splitting the difference between 50% and 10%. You can see that it's not far off from the truth, but it's not quite right. And why be almost right when you can be exactly right?

The point of all these means is to replace the net effect of two different values with the effect of just a single value repeated. But you have to be careful to consider exactly how those quantities are interacting to produce that combined effect. When they simply add together, the relevant type of mean is the arithmetic one, when they multiply, the correct mean is geometric, and when they do that weird thing of combining via their reciprocals, you use the harmonic mean. Interestingly enough, for any two numbers, if M is their arithmetic mean, G is the geometric mean,and H is the harmonic mean, it's always the case that . In fact, there are other means, too, but these three are the major players.

Other situations where the harmonic mean might come up include: calculating average fuel economy of a car given an equal amount of city and highway driving, computing the total length of time it takes two people working together to complete a task, figuring out the net resistance of two electrical resistors in parallel, finding a pleasant harmonic note (hence the name) between two other musical notes, calculating the height of the intersection between two crossed wires, and answering questions about the uses of the harmonic mean!

-DrM