Making Estimates

An estimate is a deliberate, defensible approximation, made when an exact answer is unavailable or unnecessary. The skill is knowing what to ignore.

Order of magnitude — the nearest power of ten. Two quantities are of the same order of magnitude if their ratio is less than about ten. Estimating to an order of magnitude is often enough to answer the real question: is this possible at all?

How to estimate — break the problem into pieces you can each guess to within a factor of two or three, then combine them. Errors in the pieces partly cancel, so the combined estimate is usually better than any individual guess.

Example: how many breaths do you take in a lifetime? About 15 breaths a minute, about 5 × 10⁵ minutes in a year, about 80 years. That gives 15 × 5 × 10⁵ × 80 ≈ 6 × 10⁸ — a few hundred million. Every input is rough, and the answer is still trustworthy to an order of magnitude.

Useful quantities worth knowing — an adult is about 1.7 m tall and 70 kg; a sheet of paper is about 0.1 mm thick; g is about 10 m s⁻² for estimates; the density of water is 1000 kg m⁻³; atmospheric pressure is about 10⁵ Pa; a year is about 3 × 10⁷ s.

Modelling shapes — treating a person as a cylinder or a raindrop as a sphere is not laziness, it is what makes the estimate possible. Choose the simplest shape that keeps the feature you care about.

Example: estimate the volume of an adult by treating them as a cylinder 1.7 m tall of radius 0.15 m. V = πr²h = π × 0.15² × 1.7 ≈ 0.12 m³. Check it against something you know: multiplied by the density of water, 1000 kg m⁻³, this gives about 120 kg — a bit high, which correctly suggests a person is slightly thinner than a 0.15 m cylinder and slightly less dense than water. Every estimate should end with a sanity check like this.

Why estimation is worth the effort — it tells you whether a calculated answer is plausible. If a calculation gives a car's kinetic energy as 3 × 10⁹ J, an estimate of ½ × 1000 × 20² ≈ 2 × 10⁵ J shows you are four orders of magnitude out, and you have almost certainly slipped a prefix.