How many hair salons are there in Seoul? Could you find out without searching?
Fermi estimation breaks an unknown big number into known pieces to nail its order of magnitude. It aims for the scale, not the exact answer.
Experiment
Hands-on experiment
🔮 Predict first — how many hair salons does Seoul have? Trust your gut.
✂️ Compute it from three pieces
Nobody knows the answer, but each piece can be estimated. Move the sliders.
Salon visits per month (population ÷ cycle)
5,000,000
Salons (visits ÷ customers per salon)
≈ 10,000
📖 Read more — why it exists · insights · common mistakes · formulasExpand ▾
Why
Why does this exist?
Questions nobody knows the answer to — 'salons in Seoul', 'cups of coffee a country drinks per day' — are everywhere. Yet business, policy, and interviews demand answers.
Physicist Enrico Fermi's trick was decomposition: turn one unknown into a product of a few known pieces. You know Seoul's population; you know how often people get haircuts. Multiply the pieces and the scale of the answer appears.
The rounding lab cut a single number at a place value; Fermi estimation cuts the problem itself into pieces. Like Fermi estimating a bomb's yield from scraps of falling paper, nailing the order of magnitude is judgment itself.
Insight
Insights from the video
“Even without an answer, there is an order of magnitude.”
Ten thousand versus a hundred thousand changes a business plan completely. You may not know the exact number, but telling the scale apart is where judgment starts. Fermi estimation bridges 'unknown' and 'known'.
“The more you split, the more the errors erase each other.”
A one-shot guess is easily off by 10×, but a product of five pieces mixes overshoots and undershoots that cancel. Splitting a big problem finely is itself a strategy for accuracy.
Misconception
Common misconceptions
If you don't know the answer, you can't say anything.
You can say the order of magnitude. Just telling apart 'a thousand, ten thousand, or a hundred thousand' is enough to make decisions. Fermi estimation targets scale, not the exact answer.
If every piece is a little off, the final answer is way off.
In a chain of multiplications, overshoots and undershoots tend to cancel. Even if each piece is off by 2×, the final order of magnitude usually survives. Splitting makes you safer.
Formula
Writing it as math
What you did in the salon estimate, written in the language of math.
Decomposition
One unknown becomes a product and quotient of known pieces. Each piece is small enough to estimate from experience.
The language of magnitude
A Fermi answer speaks in powers of ten. The place-value sense from the rounding lab works again here.
Error cancellation
Set one piece 2× too big and another 2× too small, and the product erases both. The more pieces, the more likely this cancellation.
In Real Life
Where you meet it in real life
Startups and market sizing
'If I open a laundromat here, how many customers?' Split into population, wash frequency, and rival shops, and a first judgment forms without searching.
The classic interview question
'How many piano tuners in Chicago?' Google and consulting firms ask these to watch the decomposition, not to hear the answer.
Fact-checking news numbers
See 'ten thousand tons of food waste a day'? Estimate population × waste per person. The order of magnitude checks out (or not) in three seconds.
Science's first calculation
At the Trinity test, Fermi estimated the blast's yield from how far scraps of paper flew. Sizing the scale before precise measurement is a scientific habit.
Try Yourself
Test yourself
Q1How many milk cartons does your whole school drink in a year? Split it into pieces.Show answer ▾
Students × school days × cartons per day. 500 × 190 × 1 ≈ 100,000 — order 10⁵. Every piece was a number you already knew.
Q2How many cups of coffee are sold in your country per day?Show answer ▾
Take 50 million people, half of them drinking one cup a day: about 25 million cups — order 10⁷. Real statistics land in the same range.
Q3You split a problem into five pieces, each possibly off by 2×. How far off could the worst case be?Show answer ▾
If all five miss in the same direction, 2⁵ = 32×. But mixed directions cancel. That's why the knack is picking pieces without a one-sided bias.
💡 Try answering yourself before revealing it — getting it wrong is where learning starts.
Watch
Related video
Connection
Concepts connect
Previous concept
Rounding
Cutting at a place value comes first. Fermi scales that skill up to entire problems.
← Rounding labLeads to next
Sampling & Estimation
You estimated with pieces; next, estimate with data — inferring 50 million minds from a sample of 1,000.
Go to the Sampling & Estimation lab →Related
Labs worth exploring together
Related lab
Rounding
Learn to cut a single number first; Fermi extends that cut to the whole problem.
Go to the Rounding lab →Related lab
Counting Cases
The multiplication principle — the grammar of building a whole from pieces — comes from here.
Go to the Counting Cases lab →