The Fermi Rule: Better be Approximately Right than Precisely Wrong

What’s the size of the market for razors in China? How many golf balls does it take to fill a Boeing 747 aircraft? How many piano tuners are there in the world?

Non-standard problems such as these are called “Fermi problems” after the distinguished Italian-American nuclear physicist Enrico Fermi (1901–54.) Fermi delighted not only in creating and solving them, but also in challenging his fellow scientists with similar problems.

Physicist Enrico Fermi Was a Master of Guesstimation

Fermi was celebrated for his ability to make fast, excellent approximate calculations with little or no concrete data. In one well-known example, when the first atomic bomb was detonated during the Manhattan Project, Fermi dropped a few scraps of paper as the shock wave from the detonation passed. After some coarse calculation, Fermi estimated the power of the blast from the motion of the scraps as they fell. Fermi’s guesstimate of 10 kilotons of TNT was remarkably close to the now-established value of 20 kilotons. Even though Fermi’s estimate appears 50% off, it was a reasonable order-of-magnitude estimate.

Fermi believed that the ability to guesstimate was an essential skill for physicists. A good way to solve physics problems—and complex problems in any line of work—is by coming up with simple shortcuts to make approximate, but meaningful, calculations.

Teaching Physics Students the Fermi Way of Contemplating Open, Non-Standard Problems

Based on Fermi’s technique, at the beginning of many physics courses, professors pose problems such as “how many piano tuners are there in Chicago?” Such questions require students to employ quick reasoning and unsophisticated numerical methods to attack problems without the knowledge of any core physics concepts.

The historical emphasis on the order-of-magnitude calculation was propelled by the lack of computing power available to solve complex problems. Such approximate calculations were considered necessary to decide if an onerous and lengthy full-blown calculation was required.

Classic Fermi Problem: Number of Piano Tuners in the City of Chicago

Fermi problems are typically restructured by breaking them up into smaller problems that are easier for the students to approach than the original problem.

The challenge of estimating the number of piano tuners in the city of Chicago is the classical example of a Fermi Problem. A problem-solver guesstimates the total population of Chicago, then the fraction of families in Chicago that may own a piano, and the frequency of piano-tuning, the time it takes to tune a piano, and so on. This sequence of thinking, accompanied by a few conversion factors, can lead to an adequate assessment of the number of piano tuners in Chicago.

Back-of-Envelope Calculations for Fermi Problems

The Fermi technique is so popular that math buffs organize competitions in Fermi’s honor. Contestants are asked to estimate unusual assessments (the fraction of the surface area of the United States that’s covered by automobiles, the number of cells in the human body, the number of pizzas ordered this year in the state of California, for example) as closely as they can.

One distinctive feature of Fermi problems is that precision is impossible to achieve quickly, but it’s easier to arrive at a fast estimate of the range for the right answer. Before investing a big effort to measure something with precision, problem-solvers can estimate the answer approximately—and only then determine if it’s sensible to do the extra steps to calculate the accurate answers.

The Ability to Guesstimate: A Key Problem-Solving Aptitude

The ability to reach first-order estimations is an important skill in daily life. In a world where we are continuously bombarded with qualitative and quantitative information (and disinformation,) acquiring a solid grounding in numeric literacy has almost become an important intellectual obligation.

Many problems are too complicated for you to come up with an accurate answer immediate. In analyzing such problems, precision may be impossible, but you can quickly estimate a range for the right answer. Guesstimation enables anyone with basic math and science skills to estimate virtually anything quickly using realistic assumptions and elementary mathematics.

Microsoft, McKinsey Consulting, Google, Goldman Sachs, and many leading businesses use guesstimate questions in job interviews to judge the ability of the applicants’ intelligence, their flexibility to think on their feet, and to apply their numerical skills to real-world problems.

Idea for Impact: Use Effective Guesstimation Techniques Before Undertaking a More Complete and Formal Investigation

Learn to do a first approximation of value and then, if the problem merits, refine your estimate further for much nuanced decision-making. Before putting much effort into calculating anything with precision, make a rough estimate of the answer, then decide whether it’s worth investigating further.

In my line of work as an investor, for example, I use fund manager Eddy Elfenbein’s “simple stock valuation measure”:

Growth Rate/2 + 8 = PE Ratio

Let me emphasize that this is simply a quick-and-dirty valuation tool and it shouldn’t be used as a precise measure of a stock’s value. But when I’m first looking at a stock and want to see roughly how it’s priced, this is what I’ll use.

For example, let’s look at Pfizer ($PFE). Wall Street expects the company to earn $2.34 per share next year. They also see the company’s 5-year growth rate at 2.79%. If we take half the growth rate and add 8, that gives us a fair value P/E Ratio of 9.40. Multiplying that by the $2.34 estimate gives us a fair price for Pfizer of $21.98. The current price for Pfizer is $22.98, so it’s about fairly priced.

Let’s look at IBM ($IBM) which has a higher growth rate. Wall Street sees IBM earning $16.61 next year. They peg the five-year growth rate at 10.58%. Our formula gives us a fair value multiple of 13.29, and that multiplied by $16.61 works out to a value of $220.75. IBM is currently at $201.71.

Recommended Resources for Guesstimation

If you’re interviewing with one of those companies that use guesstimate questions in job interviews, or if you’re interested in developing your ability to make rough, common-sense estimates starting from just a few basic facts, I recommend the following learning resources:

• Guesstimation: Solving the World’s Problems on the Back of a Cocktail Napkin by Lawrence Weinstein and John A. Adam is a fun introduction to guesstimation.
• Sanjoy Mahajan teaches a course on “down-and-dirty, opportunistic problem solving” at MIT. His Art of Approximation in Science and Engineering course is accessible free of charge on OpenCourseWare. Mahajan has also written the resourceful textbook Street-Fighting Mathematics: The Art of Educated Guessing and Opportunistic Problem Solving.