Fermi Estimations

Video coming soon3 worked examples

Theory

A Fermi Estimation is a specific type of mathematical model used to produce a fast, "ballpark" approximation for a quantity that is incredibly difficult—or completely impossible—to measure directly.

1. Where does the name come from?

They are named after the Italian physicist Enrico Fermi, who was famous for his ability to make surprisingly accurate estimates using very little actual data. (He used these skills while working on the Manhattan Project during World War II).

2. The Core Concept

The primary goal of a Fermi estimation is not to find the exact, correct answer. The goal is to reach a logically sound and reasonable estimate.
To do this, we must make sensible, educated assumptions based on our general knowledge of the world (e.g., assuming a typical human lifespan is 80 years, or a school year has 190 days).
When calculating a Fermi estimation, it is standard practice to ignore minor details and special cases that would overcomplicate the maths.

🚨 Exam Rule 🚨

In an exam, you will rarely be penalised for the specific numbers you choose for your assumptions (as long as they are physically realistic). However, you must explicitly state your assumptions in writing before you do the calculation to be awarded the marks.

Worked examples

Example 1

Example 1: The Lifetime Blinks

Estimate the total number of times a typical person will blink during their lifetime. State any assumptions you make.

Assumptions:

A typical person blinks roughly 15 times per minute.
A person is awake for 16 hours a day.
The average lifespan is 80 years.

Calculation:

Blinks per hour: $15 \times 60 = 900 \text{ blinks}$
Blinks per day: $900 \times 16 \text{ hours} = 14{,}400 \text{ blinks}$
Blinks per year: $14{,}400 \times 365 \text{ days} = 5{,}256{,}000 \text{ blinks}$
Lifetime blinks: $5{,}256{,}000 \times 80 \text{ years} = 420{,}480{,}000 \text{ blinks}$ . (A reasonable Fermi estimate would be approximately 420 million blinks).

Example 2

Example 2: The Coffee Commute

David buys a takeaway coffee every morning on his way to the office. At the end of the year, he reviews his bank statements and estimates he has spent a total of £840 on these morning coffees. Estimate the cost of a single cup of coffee. State any assumptions you make.

Assumptions:

David works 5 days a week.
He takes 4 weeks of annual leave a year (meaning he works 48 weeks a year).
He only buys exactly one coffee on his working days, and ignores bank holidays.

Calculation:

Total coffees bought: $48 \text{ weeks} \times 5 \text{ days} = 240 \text{ coffees}$ .
Cost per coffee: $£840 \div 240 = £3.50$ . (The estimated cost of a single coffee is £3.50).

Example 3

Example 3: The Teacher's Voice

Estimate the total number of words spoken by a secondary school teacher during their lessons in a typical academic year. State any assumptions you make.

Assumptions:

A teacher teaches for 5 hours a day, for the 190 days in a Scottish school year.
The teacher spends roughly 50% of the lesson time speaking.
The average speaking rate is 130 words per minute.

Calculation:

Total teaching hours per year: $5 \text{ hours} \times 190 \text{ days} = 950 \text{ hours}$ .
Total speaking hours: $50\% \text{ of } 950 = 475 \text{ hours}$ .
Total speaking minutes: $475 \times 60 = 28{,}500 \text{ minutes}$ .
Total words: $28{,}500 \times 130 = 3{,}705{,}000 \text{ words}$ . (A reasonable Fermi estimate would be approximately 3.7 million words).

Ready to practise?

9 Fermi Estimations questions with answers and video solutions