Uncertainty estimates for Fermi estimates

In a final year course that I co-teach, I teach Fermi estimation* (my notes are here). Fermi estimates are simple back-of-the-envelope calculations. Let’s say you want a Fermi estimate of how many people in the UK take a train journey on a normal week day. You start by saying “Well the population of the UK is about 60 million people”, then you say “I guess about 10% take a train journey on a given day, as 1% of people taking a train looks too low, while it is clearly not 100%”. The Fermi estimate is then that about 6 million people take the train in one day. To check this estimate, I did a little Googling, and there are about 6 million journeys per day in the UK, so assuming that people who travel in a day take two trips (eg to and from work), it looks like I am about a factor of two, too large. Not bad for a simple estimate.

I think Fermi estimation is a useful skill, both for physicists, and for others, and it is fun to teach. But I still have work to do when I teach it. One problem that comes up quite frequently, is when a student estimates a value as, say, 30, and the estimate in the model solutions I provide is, say, 10. Then the student is often uncertain as to whether their answer is right or wrong. This is a fair point, so I think I need to work on that.

To determine whether or not estimates of 10 and 30 are the same or different, you need uncertainty estimates for the Fermi estimates. Only with uncertainty estimates, can you determine if 10 and 30 are genuinely different, or if the uncertainty estimates overlap and they are effectively the same. The Wikipedia page on Fermi estimates goes through a standard-ish uncertainty analysis. Typically Fermi estimates are obtained by multiplying factors together, so the central-limit-theorem of statistics says that on a log scale if the Fermi estimate is say a product of = 4 factors, each of which has the same uncertainty on a log scale, for example, say a factor of σ = 3, then on a log scale the uncertainty is √n*σ = 2*3 = 6. So then if our a Fermi estimate was 10, we have 10 multiplied or divided by a factor of 6, i.e., we predict that our Fermi estimate is (with some high probability) in the range 2 to 60. Note that as errors here are on a log scale we multiply or divide, instead of adding or subtracting.

The Brilliant website considers the other limit, where instead of the uncertainties of all the factors being the same, one is much larger than the rest. Then the largest uncertainty will dominate, and the other much smaller uncertainties can just be neglected. Say for example, we again are multiplying four factors together, but the uncertainty in one is a factor of 10, while the uncertainties in the other three are much lower. The uncertainty in the Fermi estimate is a factor of 10.

Lesswrong points out that often it is easier to work with intervals not central estimates for the parameters and so get uncertainty estimates that way. To go back to my Fermi estimate above of how many travel by train in a day, there the uncertainty is dominated by how uncertain I am about what fraction of people travel by train on a week day. I am confident that 60 million is a pretty accurate estimate of the UK’s population, i.e., will be a lot more accurate that my estimate for the fraction. So I could also have said that I think at least 2% of people travel by train, while I don’t think more than 15% do. Then my Fermi estimate would be that between 1.2 and 10 million travel by train on a typical week day.

So, when I next teach this, I need to beef up my notes and questions on Fermi estimates, to include uncertainty estimates. I think that will be a useful thing to do. Being able to estimate uncertainty is a very useful skill, not only in science but also in finance, insurance, and in many other fields that employ physics graduates.

* Fermi estimation is often called doing Fermi problems, or answering Fermi questions.

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