How many piano tuners are there in Chicago? That question is usually attributed to the physicist Enrico Fermi, who liked to ask students questions that nobody could look up and which could still be answered with surprising accuracy. The method that came from it, called Fermi estimation or a back-of-the-envelope calculation, is among the most useful thinking tools there is. It does not require precise knowledge. It requires a habit: break the unknown into pieces that you can guess.
Fermi at Trinity
The best-known story involves the first nuclear test, at Trinity in New Mexico in July 1945. As the blast wave reached the observers, Fermi dropped small pieces of paper from his hand and watched how far they were pushed. From the distance he estimated the strength of the explosion at about 10 kilotons of TNT. The measured value was about 21 kilotons. He was off by a factor of two, from a handful of confetti, within seconds. That is a typical Fermi result: not exact, but in the right range, which for most purposes is what matters.
The method
- State the question as a number. "How many piano tuners are there in Chicago?" is a good question, since it has a numerical answer.
- Break it into factors. Number of pianos in the city, divided by how many pianos a tuner can service in a year, gives the number of tuners.
- Estimate each factor. The city has about 3 million people, about 1 in 20 households has a piano, and households have about 2.5 people. That gives 3,000,000 / 2.5 / 20, about 60,000 pianos.
- Estimate the work rate. A piano is tuned about once a year. A tuner does perhaps 4 tunings a day for 220 working days, which is about 900 a year.
- Combine. 60,000 / 900 is about 70 tuners. The actual number is of the order of 100, which is in the same range.
Each step above is a rough guess. Nobody knows precisely how many households have a piano. The remarkable thing is that the answer is still close.
Why errors cancel
When you multiply several guesses, some are too high and some are too low. Because the errors are multiplicative, they combine like the logarithms of the factors: positive and negative log-errors tend to cancel. If each of five factors is within a factor of 2 of the truth, in random directions, the product is typically within a factor of about 2 to 3, and not 2 to the power of 5. This is the same effect as in the wisdom of crowds, and it is the reason why splitting the question into parts is better than guessing the total directly. Studies of estimation have found that breaking an estimate into components, called decomposition, improves accuracy particularly for large and unfamiliar quantities.
Tools for estimating
- Anchor facts. Learn a few reference numbers: the world has about 8 billion people, the United States about 335 million, the European Union about 450 million, a year has about 31.5 million seconds, a day 86,400 seconds, and light travels about 300,000 km each second. With some of these in your head, many questions become multiplications.
- Powers of ten. Do not worry about the digits. Work with rounded numbers, count zeros, and only at the end refine.
- Bounds. Ask what the answer certainly is not. It is more than 10 and fewer than 10,000: that already narrows it, and the geometric mean of 10 and 10,000 is 300.
- Sanity checks. Compare the answer with something you do know. If your tuner estimate was 70,000, you would know that it is wrong, because that would be as many tuners as pianos.
Examples to try
- How many heartbeats does a person have in a lifetime? About 70 per minute times 60 times 24 times 365 times 80 years: roughly 3 billion.
- How many litres of water does a household use each day? A shower is about 50 litres, flushing a toilet 6 litres, with cooking and washing, perhaps 100 to 150 litres per person.
- How many words are there in a thick novel? 500 pages times 300 words: around 150,000.
How the games help
The games on our site train the parts that Fermi estimation depends on: a feel for how big a number is, a reference point for populations and distances, and the ability to compare quantities at a glance. In Geo Population, for example, you build the anchor facts you need for questions about countries. In Gut Total and Number Sense you practise estimating a quantity without counting it. The best way to build a habit of decomposition is to ask, after each reveal, which assumption was the largest source of your error.