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Fermi Estimates Preliminary Design

Fermi Estimates Preliminary Design
As a process, I expect most people who have ever done preliminary design recognise the method; even if you don’t give Fermi the credit: Idealise, simplify and state your assumptions.

This week I was randomly introduced to the wonderful world of Fermi problems. If, like me, you have no idea what I’m going on about: Enrico Fermi was a physicist who was renowned for making good approximations with no actual data, and a Fermi estimate is a method for making a ball-park guess that might actually be right (for a given value of right). I know what you’re thinking: but that’s just engineering design!

Actually, it gets even eerier. The basis of the method is to divide a problem into several factors that you are better positioned to estimate, and combine them to provide a figure with little or no technical calculation. It’s starting to sound more and more like preliminary design- no? For example:

If everyone in London had helped build the Shard, how long would it have taken?

  • There are 8 million people living in London
  • The Shard took 3 years to build
  • The magnitude of people working, given any day, on the Shard must have been in the hundreds… at an estimate I’d say between 50 – 250…
  • … in the case of a range, take the geometric mean (the square root of the product)…
  • … but because using a calculator is against the spirit of the method, estimate the geometric mean at the lower third: 50 + (250 – 50) / 3 is approximately 120.
  • There are 360 days in a simple year, so it took 1080 days for 120 people to build the Shard…
  • … or somewhere close to 1100 days for 100 people to build it for easy multiplication: 110’000 people days …
  • … which, to the nearest magnitude, is a tenth of a million.
  • So if all 8 million people in London stopped being so damn lazy and chipped in to that tenth of a million people days …
  • … it would have taken a (1 / [8 / 0.1]) 80th of a day …
  • … and as 24 hours fits almost trice into 80; that’s about a third of an hour
  • Which is about 20 minutes; but for the degree of accuracy:
  • Let’s say a day.

Enrico Fermi

Like most prominent physicists of that era, Fermi was involved with the Manhattan Project; which for better or for worse gave rise to his most famous estimate. Without the apparatus to measure it, Fermi used the distance travelled by pieces of paper dropped from his hand during the Trinity test to estimate the power of the atomic detonation. He approximated 10 kilotons of TNT, which (all things considered), was amazingly close to the back-calculated value of 20.

As an example, this one is a bit more deliberate than it first seems. As a process, I expect most people who have ever done preliminary design recognise the method; even if you don’t give Fermi the credit: Idealise, simplify and state your assumptions. Similarly, the accuracy of the calculation aims for degrees of magnitude; whilst you might not know how many years the Shard took to build, you could guess it is a single figure.

Something I hadn’t been introduced to, before I started looking at the method, was use of a geometric mean to reduce a range to a representative value. I think there is a tendency in engineering to conservatism, which can cause a significant over-estimate. The geometric mean therefore helps us on two levels, at once it removes the temptation to play it safe with the highest value, and, because it tends towards the smaller number, allows us to acknowledge that our lower bound would have been conservatively picked, and the upper bound is most definitely exaggerated.

My final step of moving from 20 minutes to a day is a reflection of this. Because of the way the assumptions are multiplied, working always with conservative estimates compounds the conservatism of the design until your estimate is so far away from likely that you might as well have just guessed at the start. As the use of the method is most helpful prior to any detailed design, however, the end is the best place to apply a safety factor.

There is a limitation of the method; it requires linearity, or at least a scalar between the basis upon which the assumptions were made and the answer.

An advantage of using a formal Fermi method is that this estimate can provide a basis of sanity for further calculations. When a detailed analysis is undertaken, answers not within the magnitude of the estimate require the identification of the assumptions that led it out of kilter; justifying the sanity of the detailed calculation.

Finally, I’m sure anyone with any site experience might argue that, even if you could handle the logistics, eight million people could not build the Shard in a day, because actions such as curing, lifting, boring, etc. all impose time constraints. This highlights a limitation of the method; it requires linearity, or at least a scalar between the basis upon which the assumptions were made and the answer. Whilst there will be a range that throwing more people at the construction would have got it built faster, somewhere between that and a day new, non-modelled, effects begin to dominate.

Despite all this, an awareness between the similarities of the Fermi method and preliminary design calculations is useful. Appreciating the analogy helps both to formalise the process, prevent our tendency from conservatism snow-balling into a significant over-design, and leaves us with a set of assumptions to hone and evaluate when it comes to the details.

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  1. Fermi was a real genius, who is undervalued even by most physicists. As you can see from the photo, he had a strong visual sense & could quickly reduce problems to simple geometric approximations. Visualization is the key to creativity, as it draws on the visual sub-system in the human brain; the one with the largest number of neurones. analytic (or mathematical) thinking is deductive & only works for well-defined problems; it runs the risk that one error in the logic chain can produce a totally erroneous result, which will remain undiscovered without strong intuition.

    • I completely agree; there’s a bit of an adage in civil engineering, at least: If you can’t draw it, you don’t understand it.

      • Hi Tom:
        Great comment about ‘drawing’ that I will steal for physics that has a similar deep problem with quantum mechanics; where there has been a mathematical theory since 1925/26 which very accurately matches the experimental results (only for the hydrogen atom though). But since it was just a math model there is NO explanation – in fact, if you check Wiki you will find 12 competing theories (or explanations). As the great Ernest Rutherford (discover of the the proton-nuclear model) also said: “if you can’t explain your theory to a barmaid, then you don’t understand it; accordingly, he dismissed Einstein’s theory of relativity (as I do) because no physicist could explain it to him. These stories illustrate the importance of developing good image-based theories BEFORE the math is introduce. otherwise, we end up with a Ptolemaic model of the world: accurate fit to the measurement numbers but still “dead wrong”. This is the current state of QED (quantum electro Dynamics – “the best theory in physics”).
        Best wishes,
        Herb Spencer
        (Vancouver, BC)


  1. […] you will want to normalise your conservatisms by taking geometric means between ranges (see Fermi Estimates for a […]