[Excerpt] Fermi Estimates

Excerpt From: How to Measure Anything book
Part Of: Bayesianism sequence
Content Summary: 1200 words, 6 min read

Eratosthenes

Our first mentor of measurement did something that was probably thought by many in his day to be impossible. An ancient Greek named Eratosthenes (ca 276-194 BCE) made the first recorded measurement of the circumference of the Earth. If he sounds familiar, it might be because he is mentioned in many high school trigonometry and geometry textbooks. 

Eratosthenes didn’t use accurate survey equipment and he certainly didn’t have lasers and satellites. He didn’t even embark on a risky and potentially lifelong attempt at circumnavigating the Earth. Instead, while in the Library of Alexandria, he read that a certain deep well in Syene (a city in southern Egypt) would have its bottom entirely lit by the noon sun one day a year. This meant the sun must be directly overhead at that point in time. He also observed that at the same time, vertical objects in Alexandria (almost directly north of Syene) cast a shadow. This meant Alkexandria received sunlight at a slightly different angle at the same time. Eratosthenes recognized that he could use this information to assess the curvature of Earth.

He observed that the shadows in Alexandria at noon at that time of year made an angle that was equal to an angle of 7.2 degrees. Using geometry, he could then prove that this meant that the circumference of Earth must be 50 times the distance between Alexandria and Syene. Modern attempts to replicate Eratosthenes’ calculations put his answer within 3% of the actual value. Eratosthenes’s calculation was a huge improvement on previous knowledge, and his error was much less than the error modern scientists had just a few decades ago for the size and age of the univers. Even 1700 year later, Columbus was apparently unaware of Eratosthenes’s result; his estimate was fully 25% shorrt. (This is one of the reasons Columbus thought he might be in India, not another large, intervening landmass where I reside). In fact, a more accurate measurement than Eratosthenes’s would not be available for another 300 years after Columbus. By then, two Frenchmen, armed with the finest survey equipment available in eighteenth-century France, numerous staff, and a significant grant, finally were able to do better than Eratosthenes. 

Here is the lesson: Eratosthenes made what might seem like an impossible measurement by making a clever calculation on some simple observations. When I ask participants in my seminars how they would make this estimate without modern tools, they usually identify one of the “hard ways” to do it (e.g., circumnavigation). But Eratosthenes, in fact, need not have even left the vicinity of the library to make this calculation. He wrung more information out of the few facts he could confirm instead of assuming the hard way was the only way. 

Enrico Fermi

Consider Enrico Fermi (1901-1954 CE), a physicist who won the Nobel Prize in Physics in 1938. 

One renowned example of his measurement skills was demonstrated at the first detonation of the atom bomb on July 16, 1945, where he was one of the atomic scientists observing the blast from base camp. While other scientists were making final adjustments to instruments used to measure the yield of the blast, Fermi was making confetti out of a page of notebook paper. As the wind from the initial blast began to blow through the camp, he slowly dribbled the confetti into the air, observing how far back it was scattered by the blast (taking the farthest scattered pieces as being the peak of the pressure wave). Simply put, Fermi knew that how far the confetti scattered in the time it would flutter down from a known height (his outstretched arm) gave him a rough approximation of wind speed which, together with knowing the distance from the point of detonation, provided an approximation of the energy of the blast. 

Fermi concluded that the yield must be greater than 10 kilotons. This would have been news, since other initial observers of the blast did not know that lower limit. Could the observed blast be less than 5 kilotons? Less than 2? These answers were not obvious at first. (As it was the first atomic blast on the planet, nobody had much of an eye for these things. After much analysis of the instrument readings, the final yield estimate was determined to be 18.6 kilotons. Like Eratosthenes, Fermi was aware of a rule relating one simple observation – the scattering of confetti in the wind – to a quantity he wanted to measure. The point of the story is not to teach you enough physics to estimate like Fermi, but that, rather, you should start thinking about measurements as a multistep chain of thought. Inferences can be made from highly indirect observations.

The value of quick estimates was something Fermi was known for throughout his career. He was famous for teaching his students skills to approxximate fanciful-sounding quantities that, at first glance, they might presume they knew nothing about. The best-known example of such a “Fermi question” was Fermi asking his students to estimate the number of piano tuners in Chicago. His students – science and engineering majors – would begin by saying that they could not possibly know anything about such a quantity. What Fermi was trying to teach his students was, to figure out that they already knew something about the quantity in question. 

Fermi would start by asking them to estimate other things about pianos and piano tuners that, while still uncertain, might seem easier to estimate. These included the current population of Chicago (a little over 3 million in the 1930s), the average number of people per household (two or three), the share of households with regularly tuned pianos (not more than 1 in 10 but not less than 1 in 30), the required frequency of tuning (perhaps once a year, on average), how many pianos a tuner could tune in a day (four or five, including travel time), and how many days a year the tuner works (say, 250 or so). The result would be computed:

Tuners in Chicago = population / people per household
* percentage of households with tuned pianos
* tunings per year per piano / (tunings per tuner per day * workdays per year)

Depending on which specific values you chose, you would probably get answers in the range of 30 to 150, with something like 50 being fairly common. When this number was compared to the actual number (which Fermi would already have acquired from the phone directory of a guild list), it was always closer to the true value than the students would have guessed. This may seem like a very wide range, but consider the improvement this was from the “How could we possibly even guess?” attitude his students often started with. 

Implications

Taken together, these examples show us something very different from what we are typically exposed to in business. Executives often say “We can’t even begin to guess at something like that.” They dwell ad infinitum on the overwhelming uncertainties. Instead of making any attempt at measurement, they sometimes prefer to be stunned into inactivity by the apparent difficulty in dealing with these uncertainties. Fermi might say, “Yes, there are a lot of things you don’t know, but what do you know?”

Viewing the world as these individuals do- through calibrated eyes that see things in a quantitative light – has been a historical force propelling both science and economic productivity. If you are prepared to rethink some assumptions and put in the time, you will see through calibrated eyes as well. 

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