How many piano tuners are there in Chicago? Ask someone to answer without leafing through a phone book or searching the internet, and most people draw a blank. Yet if you chain together a few things you already know, you can arrive at a surprisingly usable number.
This approach is called Fermi estimation. It is a way of thinking that splits a problem whose exact value you cannot know into numbers you do know, roughly estimates each piece, and multiplies them to gauge the order of magnitude. The physicist Enrico Fermi was famous for using it often. Such questions are also called Fermi problems.
In this article, we will look in turn at what kind of mental framework Fermi estimation is, how the classic piano tuner problem is solved, why rough guesses do not stray far, and finally Fermi's own anecdote and everyday uses.
The Art of Back-of-the-Envelope Math for Unknown Numbers
A Mental Framework for Finding Answers by Estimation
The goal of Fermi estimation is not a value correct to the decimal point but a sense of roughly how big something is. It is commonly described as getting the order of magnitude right, that is, the scale in units of ten.
(1) Breaking the problem down
Turn a value you cannot know in one go into multiplications and divisions of a few values you do know. The key is to work down to numbers you have a feel for, such as population, number of households, or daily workload.
(2) Setting rough values
Put a plausible number into each piece. You might treat a family as four people or assume one in every five homes, and it is best to choose round numbers that are easy to calculate.
(3) Multiplying to gauge the size
Chain the pieces together to get a result, and take that value not as the correct answer but as a baseline meaning “about this much.”
(4) Exposing the assumptions
A big advantage is that every assumption used in the calculation is visible. If the result looks odd, you can immediately point to which piece is suspect.
This method is less a knack for finishing in your head than practice in turning a vague question into one you can calculate. Because you can explain the reasoning in words, it is also good for discussion, since you can compare assumptions with other people.
How Many Piano Tuners Are There in Chicago?
This problem, said to have been posed by Fermi to his students at the University of Chicago, is often presented as the classic example of Fermi estimation. The numbers differ a little depending on who explains it, but the flow is as follows.
- Population: Take Chicago’s population as about 3 million.
- Households: Counting four people per family gives about 750,000 households.
- Pianos: If one home in five has a piano, that is about 150,000 pianos.
- A tuner’s workload: Working four pianos a day, five days a week, 50 weeks a year, one tuner handles about 1,000 pianos a year.
If we assume each piano is tuned once a year, dividing 150,000 pianos by the 1,000 one person handles gives an answer of about 150 people. Depending on the explanation, the value varies from a little over 100 to around 150, but one thing is clear: it is not 15 or 1,500.
What matters here is less the number 150 itself than the fact that all five assumptions behind it are laid out in the open. For example, if you change it so that a piano is tuned only once every two years, the answer immediately drops by half.
Why Rough Guesses Do Not Stray Far
Even when each rough value is wrong, the result is often still usable. Some pieces come out larger than reality and some smaller, and their errors cancel each other out.
(1) Errors cancelling out
As long as you do not have a habit of leaning one way, overestimates and underestimates mix, and the error in the product ends up smaller than you would expect.
(2) The effect of the number of steps
If you assume each step may be off by a factor of two and go through nine steps, the expected error is roughly within a factor of eight. That is very different from the worst case of 512, where every step is off by a factor of two in the same direction.
(3) Value when you are wrong
If the estimate and the real value differ greatly, that gap becomes a clue. You can ask whether it is a calculation mistake, whether you left out a factor, or, for example, whether it is a city with many music schools and so unusually many pianos.
(4) Limits
If the assumptions used are absurd or a single piece is badly wrong, the result will be wrong too. That is why this method is used not for precise answers but as a tool for forming a first judgment.
From a Scrap of Paper in an Explosion to Everyday Questions
The Trinity nuclear test, held in New Mexico, USA, on July 16, 1945, helped make Fermi estimation famous. Just before the blast wave arrived, Fermi dropped small scraps of paper from a height of about 6 feet (about 1.8 meters). There was almost no wind, and from seeing the paper pushed about 2.5 meters away, he estimated the blast’s yield at about 10,000 tons of TNT.
Later measurements put the figure at roughly 18.6 to 21 kilotons, so his estimate was about half the actual value. In effect, he got the order of magnitude right with a few sheets of paper and no instruments.
- Business Check: Estimating the number of customers per day before opening a shop gives you a rough sense of whether the business can work.
- Engineering and Science: Fixing the scale of the answer before moving on to precise calculation is a good way to catch calculation mistakes.
- Interview and Education: It is used as a problem that shows how a person structures their thinking when faced with a question that has no set answer.
- Daily Life: You can practice with questions as simple as how many cafés there might be in your neighborhood.
Not giving up in front of an unknown number, and starting instead from what you do know and chaining it together, helps in every moment when you must make a judgment without accurate information. Rather than waiting for a perfect answer, setting your course first with a usable estimate makes the later checking and refining far easier.
When you meet an unknown number, don’t freeze; start from the values you know and link them one by one. That is the first step of Fermi estimation.
