11.1 — Estimating Anything

A method for producing a usable answer to a question you have no data for, in about two minutes, that will normally be within a factor of ten of the truth. It is one of the highest-value thinking skills available and almost nobody is taught it.

The method

What is a Fermi estimate?

Enrico Fermi was a physicist who was known for producing rough numerical answers to apparently unanswerable questions on the spot. At the Trinity nuclear test in 1945 he dropped torn pieces of paper as the shock wave passed, measured how far they were blown, and estimated the yield of the bomb at about ten kilotons — before the instrument data was analysed, and within a factor of two of the eventual figure.

The method has four steps.

1. Break the question into parts you can guess. Not the answer — the pieces. 2. Guess each piece to within a factor of a few. Not accurately. Roughly. 3. Multiply and divide.4. Sanity-check the answer against something you know.

The reason it works is statistical. Each guess is wrong, but the errors are roughly independent and roughly as likely to be high as low, so they partly cancel when multiplied. Five estimates each wrong by a factor of three do not give an answer wrong by a factor of 243; they typically give one wrong by a factor of three or four.

The classic exercise is: how many piano tuners are there in Chicago?

Work an example properly

How many piano tuners are there in Chicago?

Chicago's population is around 3 million. Take an average household as 3 people, so about 1 million households.

What fraction owns a piano that gets tuned? Pianos are not rare and are not common. Say 1 in 50, giving 20,000 pianos in households. Add schools, churches, hotels and venues — say that doubles it, to 40,000.

A piano that is used gets tuned about once a year. So there are 40,000 tunings a year to be done.

How many can one tuner do? A tuning takes about 2 hours including travel. A working day is 8 hours, so 4 per day. Working 250 days a year gives 1,000 tunings per tuner per year.

\frac{40{,}000 \text{ tunings}}{1{,}000 \text{ per tuner}} = 40 \text{ tuners}

The actual number, from trade directories, has historically been somewhere in the region of 50 to 100. The estimate is within a factor of two, and every input to it was a guess.

Why does the accuracy hold up?

Because of how errors behave when multiplied rather than added.

Suppose each of your five guesses is independently off by up to a factor of 3 in either direction. In logarithms, multiplying is adding, and adding several independent random errors produces a distribution that concentrates around the middle rather than piling up at the extremes — the same reason a hundred coin flips gives close to fifty heads rather than a hundred.

The practical rule: with four or five independent estimates each good to a factor of 3, the product is usually good to a factor of 3 to 5. That is enough to answer "is this a big number or a small one", which is the question that usually matters.

Where it fails is when the errors are not independent — if you systematically overestimate everything, or if one quantity dominates and you get that one badly wrong. So the fourth step, the sanity check, is not optional.

Numbers worth memorising

Which anchor numbers make estimation possible?

A small set of numbers, held roughly, makes almost any estimate tractable. These are worth committing to memory.

People. World population about 8 billion. India about 1.4 billion, China similar, the United States about 340 million, the European Union about 450 million. A large city is 10 million, a mid-size city 1 million, a village 1,000.

Time. A year is about 31.5 million seconds — usefully, close to π × 10⁷. A year is 8,760 hours, about 2,000 working hours. A day is 86,400 seconds.

Distance. Earth's circumference 40,000 km — which is not a coincidence, since the metre was originally defined from it, per 4.6. Earth's radius about 6,400 km. Distance to the moon 384,000 km, to the sun 150 million km.

Sizes. A person is 1.7 m and 70 kg. A car is 4 m and 1,500 kg. A room is 3 m high. A sheet of A4 paper is about 5 g.

Money. World GDP is roughly $100 trillion. India's is roughly $4 trillion. A million seconds is about 11 days; a billion seconds is about 32 years — which is the best available illustration of the difference between the two words.

How do you check an answer is sensible?

By finding a second route to it, or by testing it against something you already know.

The best check is a different decomposition. Estimate the piano tuner question a second way — from the number of music shops, or from the fraction of the workforce in unusual trades — and see whether the answers agree to within a factor of a few. If they differ by a hundred, one of them has a bad assumption.

The second check is a bound. Ask what the answer definitely cannot exceed and definitely cannot fall below. If your estimate of the mass of the atmosphere comes out heavier than the oceans, something is wrong.

The third is unit consistency, which catches an astonishing proportion of errors. If you are computing people per year and your units come out as people-years, you have divided where you should have multiplied.

Worked examples

How much does the Earth's atmosphere weigh?

This one has a beautiful shortcut, and it is the best demonstration of why physics people like this method.

Atmospheric pressure at sea level is about 101,000 pascals, which is newtons per square metre. Pressure at the surface is simply the weight of all the air above that square metre. So the weight of air per square metre is 101,000 N, and the mass is that divided by gravity:

\frac{101{,}000}{9.8} \approx 10{,}300 \text{ kg per square metre}

Earth's surface area is 4\pi r^2 with r = 6{,}400{,}000 m, which is about 5.1 × 10¹⁴ square metres.

10{,}300 \times 5.1 \times 10^{14} \approx 5.3 \times 10^{18} \text{ kg}

The accepted figure is about 5.15 × 10¹⁸ kg. The estimate is within two per cent, because no guessing was involved — only the recognition that pressure is weight per unit area.

How many words have you spoken in your life?

Take a person of 30. Studies of conversational speech put average output somewhere around 15,000 words a day, with enormous individual variation.

Assume speech starts meaningfully at age 2, so 28 years:

15{,}000 \times 365 \times 28 \approx 1.5 \times 10^{8}

About 150 million words. For comparison, the complete works of Shakespeare are about 900,000 words, so a 30-year-old has spoken roughly 170 times the collected Shakespeare.

How many breaths does a person take in a lifetime?

About 15 breaths a minute at rest, lower asleep, higher active. Call it 15.

15 \times 60 \times 24 \times 365 \times 75 \approx 5.9 \times 10^{8}

Roughly 600 million breaths in 75 years. The heart, at about 70 beats a minute, manages about 2.8 billion.

Is a molecule from Caesar's last breath in the air you just breathed?

Yes, probably several, and the calculation is the standard demonstration of how large Avogadro's number is.

A breath is about 0.5 litres, which at ordinary conditions contains roughly 1.3 × 10²² molecules.

The atmosphere's total mass is 5.15 × 10¹⁸ kg, mostly nitrogen and oxygen with an average molar mass around 0.029 kg. So the number of molecules in the atmosphere is:

\frac{5.15 \times 10^{18}}{0.029} \times 6 \times 10^{23} \approx 1.1 \times 10^{44}

The fraction of the atmosphere that was in Caesar's breath is:

\frac{1.3 \times 10^{22}}{1.1 \times 10^{44}} \approx 1.2 \times 10^{-22}

Multiply that fraction by the number of molecules in your breath:

1.2 \times 10^{-22} \times 1.3 \times 10^{22} \approx 1.5

About one or two molecules. The result depends on assuming two thousand years is enough for complete mixing of the atmosphere, which is roughly true for the well-mixed gases, and on ignoring the fraction that has since been absorbed into oceans and organisms. The order of magnitude survives both objections.

How many cells are in your body?

The commonly repeated figure of 37 trillion comes from a 2013 estimate that summed organ by organ, and it is the best available.

A rough route to the same place: a typical human cell is about 10 micrometres across, so about 10^{-15} cubic metres in volume. A 70 kg person is about 0.07 cubic metres. Dividing gives 7 \times 10^{13} — 70 trillion.

That overestimates by about a factor of two, because it assumes cells fill the body completely and are all average-sized, when in fact red blood cells are far smaller and far more numerous while fat and muscle cells are much larger. Within a factor of two on a first attempt is exactly what this method promises.

The related claim that bacterial cells outnumber human cells ten to one has been revised: a careful 2016 recount put the ratio at close to 1:1, with roughly 38 trillion bacteria, most of them in the colon.

Where this is actually used

Who uses Fermi estimation professionally?

Physicists and engineers, constantly, before committing to a calculation — to check whether an experiment is even possible and whether a result is plausible.

Consultants and investors, for market sizing: how many units of a thing could be sold, which cannot be looked up and must be built from population, adoption rate and replacement cycle.

Public health and disaster planning, where the question is how many hospital beds or how much water, and a precise answer arriving in six weeks is worthless.

And it is the reason interview questions like "how many golf balls fit in a bus" exist. The question is not a puzzle with a right answer. It is a test of whether the candidate decomposes the problem, states assumptions out loud, and checks the result — or freezes because there is no data.

What is the real value of the habit?

Immunity to a number that is wrong by a factor of a thousand, which is the most common form of misinformation in public life.

A claimed figure — a cost, a death toll, a market size, a saving — can usually be checked in thirty seconds against an independent estimate. If somebody claims a policy will save a billion rupees a year and a rough calculation says the whole activity is worth ten million, you know without any research that the claim is wrong.

That check is available to anybody willing to guess in public and multiply. It is the closest thing to a general-purpose defence against being told nonsense.

What comes next

The next page applies the method to a specific question that is genuinely hard to look up: at this exact moment, what is everybody on the planet doing?