Quoting a Range Instead of a Single Number

Say an interviewer asks how many gas stations there are in your city, and after a decomposition tree you land on 214. If you say "214," you've just handed them a single number that is almost certainly wrong — not because your reasoning was bad, but because a Fermi estimate is built from several rounded inputs multiplied together, and each rounding compounds. The interviewer isn't testing whether you can recite 214. They're testing whether you know how wrong 214 might be, and whether you can say so out loud without flinching.

The problem with a bare point estimate

A point estimate hides its own uncertainty. If you say "214" and the true number is 340, you look like you got it wrong — even though your method might have been sound and the miss came entirely from one shaky assumption three steps back (average stations per neighborhood, say). A range communicates the opposite: "somewhere between 150 and 400, most likely around 200–250" tells the interviewer you understand where your estimate is shaky, not just what your best guess is. That's the actual skill being tested — the same skill a trader uses when quoting a market rather than declaring a single fair value.

Building the range, not just picking one

The range should come from the same decomposition you used to get the point estimate, not be bolted on afterward as a vague "give or take." Walk back through your chain of multiplications and ask which input you're least sure about — that's usually where the range should be widest. If your estimate was

$$N = P \times f \times \frac{1}{c},$$

where $P$ is population, $f$ is the fraction of the population that drives, and $c$ is customers served per station per day, you're probably confident in $P$ to within 10%, less confident in $f$, and least confident in $c$ — station capacity assumptions are usually the roughest guess in the chain. Widen your final range mostly on the back of that weakest link, not evenly across all three.

Worked example: gas stations in a city of 2 million

Suppose your chain was: population 2,000,000; 60% of adults drive regularly, so roughly 1,000,000 drivers; each driver fills up roughly once a week, so about 143,000 fill-ups a day; a station can handle roughly 500 fill-ups a day. That gives

$$N = \frac{143{,}000}{500} \approx 286 .$$

Now stress-test the shakiest input: "fill-ups per station per day" could reasonably be anywhere from 300 (a small, slow station) to 800 (a busy highway station), not just 500. Swapping in those bounds gives $143{,}000/800 \approx 179$ and $143{,}000/300 \approx 477$. So instead of reporting "286," you report: "my point estimate is around 280–290, but because I'm least sure about throughput per station, I'd put the real range at roughly 180 to 480, with 280 as my best single guess." That's a far more defensible answer, and it shows the interviewer exactly which assumption you'd go check first if you had real data.