In simplest terms Fermi estimation involves using available data or assigning a best guess to a series of related estimates to arrive at an estimate of a particular outcome or quantity. So for example you were evaluating a business case that rested on an estimate of the number of new cars sold in Tampa every year and you wanted to get a rough guess at that number to see if the business case justified further investigation you might start with the population of Tampa (~400k in 2020) divide that by the number of people per household (~2.5) to arrive at an estimate of households in Tampla (160k). You then multiply this result by the number of cars per household (~1.9) to arrive at the number of cars in Tampa (304k). Finally you could then use the average lifespan of a car (~12 years) to conclude 8.2% of cars must be replaced in a typical year to arrive at a new car sales estimate for Tampa of (~25k). Now if you were looking at a business case that required 120k new car sales in Tampa every year this very quick estimate would allow you to put your pencil down and go do something more productive with your time because we can quite confidently say there is no way that business case is viable.
Sometimes it is really that simple and you can walk away from a problem that another analyst might have spent hours or even days digging into in less than a few minutes but you will rarely get that lucky. The main weakness in Fermi estimation is that because it cascades estimates and available data which may also be inaccurate the end result can be significantly off base. As a simple shorthand people assume that a reasonably constructed Fermi estimate is order of magnitude correct so in the example above any business case from 10k to 99k sales would be at least plausible but the fact is with just a bit more work we can tighten that range considerably. To achieve this we need to get a little bit more formal and talk about error bars.
Error bars are actually a fantastically complex topic and to really understand how to put them together you need a fair bit of statistics and an understanding of normal distributions so if you are a Physics student do not use the following technique in your Statistical Mechanics final. For the typical product manager doing a quick napkin calculation however all you really need to understand is that any estimate will have some amount of error and that we can often put a relatively narrow range around an estimate and have a very high degree of certainty that the actual number is within that range. For example in the example above we used 400k as the population of Tampa. This number is a projection based on a US census in 2020 so while it is certainly accurate enough it is not 100% correct. The question is how far off might it be? Could we say with almost absolute certainty that the population of Tampa is between 340-460k? Absolutely we can, we could actually with even a little bit of research narrow the potential range much further but we won’t need to. Let’s re evaluate our data and estimates above and see what we get:
Population of Tampa - 340-460k as previously discussed.
Number of people per household - 2-3. Remember this is an average number over the entire population so we are really being comically conservative with our estimate here - in reality averages of large populations do not vary much at all.
Number of households in Tampa. - This is a calculated value based on the above so to arrive at the possible range we divide the lowest plausible population by the highest plausible household size and the highest plausible population by the lowest plausible household size. So 340k/3 - 460k/2 gives a range of 113k - 230k
Number of cars per household - Our guess of 1.9 came from the US department of transportation via a Google search. Let’s put a range of 1.5-2.4 on this value. Putting this together with the previously arrived at estimated range of the number of households in Tampa we arrive at a possible range of the number of cars in Tampa of 170k to 552k. Remember we are trying to find the broadest possible estimate so you need to multiply the smallest possible number of cars per household by the smallest possible number of households.
Average Lifespan of a Car - In this case let's take the lowest possible average to be 10 years and the highest to be 14 years. As previously mentioned averages over large populations just do not vary that much so we are being wildly overcareful in selecting such a large range. Multiplying the largest possible population of cars in Tampa obtained previously by the smallest plausible lifespan of the average car gives us 55k new car sales per year as an upper limit.
Being a bit more formal about it we arrived at an estimated range of between 12k and 55k new car sales each year in Tampa. This result is much more useful than the rule of thumb 10k to 99k result that you would get by just taking the Fermi estimate as a magnitude indicator. Further we were very over cautious in the ranges we used in this example and if needed we could collapse the plausible range even further by researching some of our data and estimates for even a few minutes.
The real value of the Fermi estimation method comes when you begin to understand a bit more about normal distributions which is a topic I will need to cover another time. Suffice to say that the likelihood of the actual number of new cars sold in Tampa is vastly more likely to be between 22-27k per year then 50-55k per year. This means if you see a business case that relies on 25k sales per year you can be far more comfortable with it then one that relies on 50k sales per year. Knowing this you can quickly identify the assumptions where you should focus your time and energy and those that you can accept with a bit of faith that they are either correct or unlikely to be wrong enough to be significant.
The key thing you need to remember is that this is an estimation technique and the result will not be perfect and may have serious flaws if your logic is flawed so it is always worth doing a couple quick sanity checks to see if your estimate holds up. In the Tampa car example we might find the number of new cars sold in the United States, ~15 million from Google, and estimate the fraction sold in Tampa by using easily available population numbers. This method nets us an estimate of 20k a little lower than our other method but very much in the same ball park. The obvious weakness here is we are relying on a similar logical path going from population to car ownership and this is likely to underestimate the number of cars in a car dependant area like Tampa, if we want to be particularly robust we should double check via a very different estimation route. We could use the autotrader.com website to discover that there are ~40 new car dealerships listed in Tampa and a Google search indicates the average new car dealership sells ~1000 cars per year. This nets us a higher estimate of ~40k new car sales in Tampa which as a car dependant area is likely closer to the truth. Again no amount of estimating will get you an exact answer but at this point if someone put a business case in front of you that required even 50k+ new car sales a year in Tampa you could with a great degree of confidence push back on the number.
I hope this brief intro was useful as always if you have questions do not hesitate to reach out.
