Estimation: getting the right answer to within a factor of ten
Knowing an answer is about a million is often worth more than calculating it exactly.
Enrico Fermi, working on the first nuclear test in 1945, estimated the blast's strength by dropping scraps of paper as the shockwave passed and watching how far they blew. His figure was within a factor of two of the answer the instruments produced days later.
The skill is called Fermi estimation, and it is the most practically useful piece of mathematics most people never get taught.
The method
- Break the impossible question into parts you can guess at.
- Guess each part to the nearest power of ten - you are aiming for the right number of zeros.
- Multiply through.
- Sanity-check the result against something you know.
Because you are working to the nearest power of ten, the errors are usually small and, crucially, they tend to cancel. Overestimate one factor and underestimate another and the product stays close.
Every input is a guess. None is more than a factor of two or three off, and the answer lands in the right range - the real figure has historically been around 50 to 100. This is the classic example, and the point is that you started knowing nothing and finished with a usable number.
Why it works
Because errors in a product behave better than intuition suggests. If you guess six quantities and each is off by up to a factor of 2, some high and some low, the product is usually within a factor of 3 or so - not 2^6 = 64. Independent errors partly cancel.
So the goal is never precision in any single input. It is avoiding a catastrophic mistake in any of them - a factor of a thousand, or the wrong units.
Numbers worth carrying around
Estimation needs anchors. A handful of memorised figures makes almost any question tractable:
- World population: 8 billion. Your country's, roughly.
- Seconds in a day: 86,400, so about 10^5. In a year: about 3 x 10^7.
- A human lifetime: about 2.5 x 10^9 seconds.
- Distance around the Earth: 40,000 km.
- Speed of light: 3 x 10^8 m/s.
- A sheet of paper: about 0.1 mm thick.
Orders of magnitude
An order of magnitude is a factor of ten. Getting within one order - somewhere between a tenth and ten times the true value - is usually a success.
This matters because people are poor at large numbers. A million seconds is about 11 days. A billion seconds is about 32 years. Those two words sound similar and differ by a factor of a thousand, which is exactly the kind of gap estimation is for.
Use it as an error check. Before any calculation, estimate the answer. If the calculator says 4,700 and your estimate said 'about 5 million', you have made a mistake - probably a units error or a misplaced decimal - and you have caught it in five seconds. This habit alone prevents more wrong answers than being careful does.
In real arguments
Someone claims a policy will save 10 billion pounds a year. Divide by the population - if that is 70 million people, it is about 140 pounds per person per year. Now you can ask whether that is plausible, which you could not do with the original figure.
Large numbers are often quoted precisely because they are hard to feel. Reducing them to something per-person, per-day, or per-household is the fastest way to tell a serious claim from a hollow one, and it takes ten seconds.