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    Thinking Dictionary

    Fermi estimation

    What it is

    A Fermi estimate is a quick, rough calculation that gets you into the right ballpark. You take a question that seems impossible to answer and break it into smaller pieces you can reasonably guess. Then you multiply the pieces together.¹˒²

    The goal is not an exact answer. It is to be right to within a factor of ten. That means the answer is no more than ten times too big or ten times too small. Scientists call this getting the right order of magnitude: the right number of zeros, more or less.

    Where the name comes from

    The method is named after the physicist Enrico Fermi, who won the Nobel Prize in Physics in 1938.³

    On 16 July 1945, Fermi watched the Trinity nuclear test. As the blast wave reached him, he dropped small scraps of paper. They were pushed about 2.5 metres. From that distance alone, he estimated the bomb's power at about 10,000 tons of TNT.⁴ The figure usually given today is about 20,000 tons. Fermi was off by about a factor of two, but he had the right order of magnitude, using nothing but scraps of paper.

    Why it matters

    We are told big numbers all the time: how many people do something, how much something costs, how much of something is wasted. Most of us have no feel for whether these numbers make sense.

    A Fermi estimate gives you that feel. You won't get the exact answer. But you can quickly tell whether a number you've been told is even plausible. If your rough estimate and the claim differ by a factor of a thousand, something is off.¹˒²

    Breaking a question into pieces also tends to beat one wild guess. Some of your small guesses will be too high and some too low, and the errors partly cancel out. That isn't guaranteed, though.¹

    A worked example: a forest fire

    In 2020, the Bighorn Fire burned near Tucson, Arizona. Official records put its size at about 119,978 acres. That is roughly 486 square kilometres, close to 500.⁵

    How many trees might have burned? Nobody counted. But we can show the method. The numbers below are illustrative guesses, not facts.

    1. Area. 500 square kilometres is 500 million square metres.
    2. How much was forest? Guess about half.
    3. How crowded are the trees? Guess one tree for every 20 to 100 square metres of forest.
    4. Multiply. Half of 500 million is 250 million square metres. Divide by 100, and you get about 2.5 million trees. Divide by 20, and you get about 12.5 million.

    So a reasonable guess is somewhere between one million and ten million trees, give or take. That is a ballpark, not a count. Its real value is that every step is written down, so anyone can question a single guess and redo the sum.

    In everyday life

    • Checking a news claim. A headline gives a huge figure about your city. Estimate it yourself from the number of people, households, or cars. If you're off by a factor of a thousand, look more closely at the claim.
    • Planning. How much paint for a room? How long to save for something? A rough estimate tells you whether you're in the right range before you work out the details.
    • Your own habits. How much do you spend on coffee in a year? Cups per day, times price per cup, times 365. The answer is often a surprise.

    The common trap

    The trap is thinking that an estimate is "just guessing." It is structured guessing. Each step can be checked and improved.

    The opposite trap is trusting an estimate too much. One badly wrong piece can throw off the whole answer. So always write down your assumptions. That way you, or someone else, can spot the weak one.

    Some ways out:

    • Break the question into pieces you have some feel for.
    • Write down each guess and where it came from.
    • Check the final answer against anything real you can find.

    Try it

    How many litres of water does your household use for showers in a year? Estimate it by multiplying these together:

    • the number of people in your household
    • showers per person each week
    • minutes per shower
    • litres per minute (a typical shower head uses very roughly 8 to 12 litres a minute)
    • 52 weeks

    Then check. Look at your water bill, or time one shower while catching the water in a bucket for 30 seconds. Were you within a factor of ten?

    Related concepts

    Scientific models · Probabilistic reasoning · Accuracy vs precision · Credible sources

    Sources

    1. Weinstein, L., & Adam, J. A. (2008). Guesstimation: Solving the world's problems on the back of a cocktail napkin. Princeton University Press. https://doi.org/10.1515/9781400824441

    2. Mahajan, S. (2010). Street-fighting mathematics: The art of educated guessing and opportunistic problem solving. MIT Press. https://doi.org/10.7551/mitpress/7728.001.0001

    3. NobelPrize.org. (n.d.). The Nobel Prize in Physics 1938: Enrico Fermi. https://www.nobelprize.org/prizes/physics/1938/summary/

    4. Fermi, E. (1945). My observations during the explosion at Trinity on July 16, 1945 [Memorandum]. Reproduced at https://www.atomicarchive.com/resources/documents/trinity/fermi.html

    5. U.S. Forest Service, InciWeb. (2020, July 23). Bighorn Fire 100% contained. (Also NOAA NCEI Storm Events Database, event 907564: 119,978 acres.) https://www.ncei.noaa.gov/stormevents/eventdetails.jsp?id=907564