Math intuitions
Orders of magnitude
Estimate to the nearest power of ten from a few facts you know; being within a factor of three is often enough to decide.
Enrico Fermi, who taught rough estimation to his physics students and used it at the Trinity test in 1945; such problems carry his name.
An order of magnitude is a factor of ten. Thinking in orders of magnitude means asking whether a quantity is nearer a thousand, a million or a billion rather than what it is exactly, and getting there by breaking the question into pieces you can guess. How many households are there in the United States? The population in 2023 was about 338 million; a household is two to three people, say 2.5; so about 135 million. The Census Bureau's count for 2023 was 132.2 million. That is within 3 percent, from two inputs and ten seconds of arithmetic. Fermi did the same at the Trinity test on 16 July 1945, about ten miles from the tower: as the blast wave passed he dropped scraps of paper from about six feet, saw them carried about 2.5 metres, and from that put the yield at about ten thousand tons of TNT.
The method works because errors in a product tend to cancel. If five inputs are each off by up to a factor of two in random directions, the product is typically off by a factor of about 2.5, not 32: in logarithms the errors add like random steps, some high and some low. So the discipline is to guess each piece honestly at the middle of its plausible range and to keep the pieces few. The payoff is that most decisions need only the order of magnitude. Whether a job takes an hour or a week, whether a dataset fits on one machine, whether a market is ten thousand people or ten million: a factor of three rarely changes the answer and a factor of a hundred always does. Fermi's ten kilotons sat under the later estimates of 15 to 20, inside a factor of two, and he had it within a minute of the blast.
The approach fails when the question is sensitive to the exact number, such as a drug dose, a structural margin or a price, and it fails when one input is itself uncertain by more than an order of magnitude, because the estimate inherits that spread and should be reported as a range. It is also easy to choose the decomposition that gives the answer you wanted. The checks are cheap: estimate the same quantity two different ways and see whether they agree within a factor of three, and whenever a real number exists, look it up afterwards and record how far off you were. The piano-tuners problem usually attributed to Fermi has no primary record, which is a reminder to practise on cases you can check.
Example
At 05:29 on 16 July 1945 Enrico Fermi watched the Trinity test from base camp, about ten miles from the tower. In the memo he wrote afterwards he recorded that the air blast reached him about 40 seconds after the explosion, and that he had tried to estimate its strength by dropping small pieces of paper from about six feet before, during and after the wave passed. The shift was about 2.5 metres, which he estimated to correspond to the blast from ten thousand tons of TNT. The Atomic Heritage Foundation's account of the test puts the yield at between 15 and 20 kilotons: Fermi's one-input estimate was inside a factor of two before any instrument had been read.
Where it applies
Before running a job, estimate its size: a billion rows at 100 bytes each is 100 gigabytes, so it is not a laptop task; a model with 7 billion parameters in 16-bit weights needs about 14 gigabytes of memory before activations. Practise on the Fermi estimation game at /games/fermi-estimation, which scores each answer by how many orders of magnitude you were off.
Think about it
What quantity at work have you never estimated because you assumed it needed a precise answer, and what is your guess to the nearest power of ten?
Applied it to something real? Log it.
Test yourselfWhy is a product of five rough guesses, each possibly off by a factor of two, usually off by far less than a factor of 32?
The errors point in random directions and add in log space like random steps, so they partly cancel, leaving a typical error of about a factor of 2.5. Guess each piece at the middle of its range.
Related models
- First principlesBreak a problem down to what you know is true and reason up from there, instead of copying what others have done.
- Probabilistic thinkingThink in odds rather than certainties: estimate how likely each outcome is, and judge a decision by those odds rather than by the single result that happened.
- The rule of 72Divide 72 by the annual growth rate in percent to get the number of years a quantity takes to double.