Estimation: decompose, bound, and defend the number

Estimation questions are not testing arithmetic. They are testing whether you can build a structure out of things you do know, say how confident you are in each piece, and then hold your ground when someone pushes back. The number matters far less than the decomposition.

Proposition 2.1

The method

Break the quantity into a product of factors, estimate each one separately, multiply, then sanity-check the order of magnitude against something you know independently. Errors in the factors are partly independent, so they cancel rather than compound — which is why a four-factor decomposition is usually more accurate than a single guess at the answer.

Holds when

Prefer factors you can anchor: populations, areas, times, rates. Avoid factors that are themselves the mystery.
State each factor aloud as you pick it, so the interviewer can redirect you before the arithmetic.
Round aggressively. Two significant figures is generous for a Fermi estimate.

Quantity	Rough value
World population	8 billion
United States population	340 million
United Kingdom population	68 million
Seconds in a year	$3.2 \times 10^{7}$
Minutes in a day	1,440
US GDP	$\$29$ trillion
S&P 500 market capitalisation	roughly $\$50$ trillion
Trading days in a year	252

Table 2.2 · Anchors worth carrying. The seconds-in-a-year figure is the one people re-derive under pressure.

\pi \times 10^7

is accurate to half a per cent, and it is easier to remember than the digits.

Example 2.3

How many petrol stations are there in the United States?

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Worked solution

Formula $N = \frac{\text{cars} \times \text{fills per car per year}}{\text{fills per station per year}}$
Substitute $\text{cars} \approx 250\text{m}, \quad \text{fills} \approx 40/\text{year}, \quad \text{station} \approx 8\text{ pumps} \times 20\text{ fills/day} \times 350$
Solve $\text{demand} = 250 \times 10^{6} \times 40 = 10^{10} \text{ fills/year}$
$\text{supply per station} = 8 \times 20 \times 350 = 5.6 \times 10^{4}$
$N = \frac{10^{10}}{5.6 \times 10^{4}} \approx 1.8 \times 10^{5}$
Answer $\text{about } 180{,}000$

Sanity check. The published figure is around 145,000, so this is high by about a quarter — well inside the factor of two a Fermi estimate aims for. The pumps-per-station figure is the weakest link, and saying so is part of the answer.

Estimate it twice. The strongest thing you can do with a Fermi estimate is reach the same number a second way. Here, the United States has about 330 million people and roughly 20,000 towns and cities; two or three stations per few thousand people lands in the same six-figure range. When two independent decompositions agree you have genuine evidence; when they disagree by a factor of ten you have found the factor you got wrong, which is more useful than either answer.

Proposition 2.4

Quote an interval, and pick its width

A point estimate invites the question "how sure are you?", so answer it first. A Fermi decomposition with four factors each good to a factor of two gives an answer good to roughly a factor of three or four — not sixteen, because the errors are partly independent. Say something like "about 180,000, and I would be surprised outside 80,000 to 350,000".

Holds when

Width follows from the weakest factor, so name it: "the pumps-per-station number is the one I am least sure of".
At a trading firm the interval *is* the answer — it is a market on your own estimate, and quoting it absurdly wide defeats the purpose.

Estimation: decompose, bound, and defend the number

After this lesson you should be able to

The method

Quote an interval, and pick its width

The rest of this lesson is in Premium