Basic Concepts: Fermi Questions

Closely related to the idea of order-of-magnitude estimates is the idea of Fermi Questions, a type of problem that demonstrates the power of estimation techniques. The idea is that you can come up with a reasonable guess at an answer for a difficult question by using some really basic reasoning, and a few facts here and there.

The classic example of a Fermi Question is “How many piano tuners are there in Chicago?” This has never really worked for me, though, because I don’t know anything about pianos, and I have no real way of knowing how often they’re tuned, or any of the other estimates. So, to demonstrate the basic idea, I’ll make up a different question:

How many shopping malls are there in the United States?

It’s not immediately obvious how to attack this, but a reasonable assumption might be to say that a mall needs a certain number of customers within reasonable driving distance in order to remain open. If the population is too low, you won’t get enough shoppers to keep a mall open.

So, how many shoppers does it take to keep a mall open? Well, within easy driving distance of my house (~30 minutes), I can think of five malls, so this area must contain at least the minimum number for five malls.

The population of this general area is probably between 100,000 and 200,000 people– Schenectady and Troy are probably 20-30,000 each, Albany proper is probably 40-50,000, and the various suburbs account for a good number more. Let’s call it 150,000, because the math works out nicely that way.

Five malls for 150,000 people works out to 30,000 customers per mall, if we assume they’re just barely keeping the malls open (a reasonable assumption for a couple of those). The total population of the United States is something like 300 million people, so we would expect 300,000,000/30,000 = 10,000 malls in the United States.

This happens to be a question that’s susceptible to Googling (which is why I picked it, though I didn’t look at the answer in advance), and a few seconds of searching turns up a Census Department pagefrom 2005 claiming that the number of malls and shopping centers was 47,835 in 2004. So, at the very worst, the Fermi question reasoning gets within a factor of five of the correct answer, using very little concrete information. And that statistic includes “shopping centers,” however you define those, so the number of malls meeting my mental definition is probably closer to the estimate of 10,000.

This is somewhere between a party trick and an essential physics technique. If you’re thinking about doing an experiment in physics, you almost always start off with this sort of reasoning about what sort of signal you can expect. It’s a good way to avoid hours and days of wasted effort, not to mention wasted money. And with a little knowledge of basic physics, you can get remarkably close to a lot of answers.

Of course, it’s also a fun way to kill time, in a nerdy sort of way…

If you know of a particularly entertaining Fermi Question, leave it in the comments.