Fermi estimation: how to estimate anything

Where the name comes from

Enrico Fermi, one of the physicists behind the first nuclear reactor, liked to ask his students questions such as how many piano tuners work in Chicago. Nobody knows that number offhand, yet a chain of reasonable guesses lands close to it.

At the first atomic bomb test in 1945, Fermi dropped scraps of paper as the blast wave passed and estimated the yield from how far they were blown. His figure was within a factor of two of the measured one.

Why rough estimates work

When you multiply several guesses together, some are too high and some are too low, and the errors partly cancel. As long as no single step is wildly wrong, the product stays within an order of magnitude of the truth.

This is also why the goal is a power of ten, not a precise figure. Knowing whether something costs a million or a billion is usually what a decision needs.

A four step method

• Restate the question as a product of quantities you can guess.
• Anchor each quantity on something you know: a population, a distance, a duration, a price.
• Round every number to one significant figure and multiply the powers of ten.
• Sanity check the result against a different route to the same answer.

Worked example: heartbeats in a lifetime

A heart beats roughly once per second, a little faster in practice. A day has about 100,000 seconds (86,400). A year has about 400 days at this precision, so 40 million beats a year. Over 80 years that is around 3 billion heartbeats. The accepted figure is between 2.5 and 3 billion.

Anchors worth memorizing

• World population: about 8 billion.
• Seconds in a year: about 30 million.
• Circumference of the Earth: 40,000 km.
• Distance to the Moon: about 400,000 km.
• Mass of a liter of water: 1 kg.

Practice daily

The method only becomes intuition with repetition. The daily question on Magnitudle is a Fermi problem with a sourced answer, and the score tells you immediately how many powers of ten you were off.