Thursday, September 13, 2018

Answer: Can you be a Fermi estimator?

Being a good estimator... 

... requires two kinds of knowledge:  (a) facts about the world, and (b) basic math estimation skills.  Those skills let you work from the basic facts, combining the data and doing quick math estimates to get to some numbers that give you the information you seek.  

How tall was Enrico Fermi, the physicist?  How would you estimate?  Assume he doesn't have extraordinarily long legs.  

Fermi Estimation needs both kinds of knowledge--along with a bit of practice in figuring out how to go from point A to point B.  

Let's practice with a true Fermi estimation!  

1.  Can you estimate how tall Enrico Fermi was?  
In this example, I give you a piece of data: Fermi's head is 9.4 in (23.9 cm) tall.  

How would you estimate his height just from this information?  Ask yourself, What other information do I need to know?  

If you're an artist, you probably learned that when you sketch a person, their body proportions are often expressed in terms of head-heights.  Example:  An ordinary person is about 7.5 heads high--a tall, slender, elegant person is about 8 heads high. 

If you don't know that bit of data, you could look the relationship up quickly:  

     [ body proportion in heads ] 

Later edit (Sep 14, 2018):  In the original post I messed up the calculation with a typo... I wrote that 24 * 7 was 164, which is clearly wrong.  Once I had that typo, all kinds of errors followed.  Bottom line, check your math. Didn't your sixth grade teacher say that? 

So know you know Fermi's head is 23.9 cm tall.  To Fermi estimate, let's round that up to 24 cm.  You can probably guess that 24 * 7 = 168 cm.  If you then add in the half-head height (12), you can add 168 + 12 and get 180 cm.  Make sense?  

In this case, I did the mental math in metric because that value (23.9) could be easily rounded up to 24 without causing many estimation problems, whereas the English measurement (9.4 inches) was kind of a pain--if you round up OR down, you lose a lot of accuracy.  

Plus, I know how to convert from cm to inches pretty easily.  

I know (off the top of my head, OTTOMH) that 10 cm ~ 4 inches.  

(Here, the ~ means "close to" in value.)  

SO... to convert 180 cm  into inches, I would divide 180 by 10 and multiply that by 4.  My mental math was 180/10 gives me 18 --I then multiply 18  by 4 to get your Fermi estimate of 72  inches, or 6 feet.  

When I read that, it SOUNDS really complicated... but it's not.  Here are the steps laid out in a diagram.  

1.  I know 10 cm~ 4 inches.  So... I just have to divide by 10, then multiple the result by 4 to do that conversion. 
2.  180 cm /10 = 18
3.  18 * 4 = 72  inches...  

At that point, most English-unit-using people can convert that into feet/inches without any trouble: 6 feet tall. (Yes, I know that's a weird skill; so be it.  We grew up with it. Point is, it's not that hard.  

So yes, you have to know a basic conversion fact (10 cm ~ 4 in), and know how to break up a complex mental multiplication into parts.

(If you want to learn more about how to do this kind of mental math for estimation, there are many online videos that will walk you through the process.  One that's pretty nice is from the Khan Academy:  Estimating values.)  

2.  Can you estimate (without looking up the answer!) how many people in the United States are over 80 years old?  (For extra credit, how many people worldwide are over 80 years old?)   
This is an interesting question:  How would you go about estimating this? 

Again, you need to know a few things to start. 

1.  The population of the US.  (I know this:  It's around 325M people.) 2.  The population age distribution.  

Once upon a time, I remember seeing a chart like this.  They're sometimes called "age pyramids" or "age distribution" charts.  When I look at this chart in my mind's eye, the top couple of slices of that chart constituted around 3% of the whole.  

To check this, I also remembered that the average life expectancy is around 78 (which is lower than in some countries, higher than in some).  So 3% off the top sounds like a reasonable guess.  

So... 3% of 325 is easy.  Divide by 100 (3.25) and multiply that by 3.  That's 9.75 million people who are older than 80 in the US.  

When I look up the actual numbers to check myself, I see that there are 328.3M people in the US, and this is the age distribution: 

And, surprisingly, my estimate of 3% is pretty close to the actual value.  

Even more surprisingly, I also happen to know that the US is slightly above average for ages > 80 years throughout the world.  Since I know that the population of the Earth is 7.5B, 3% of 7.5B is 225M people, which is a lot of elder wisdom.  

3.  To do Fermi Estimates you actually need to know a few basic facts (e.g., about how many people live in the US).  This brings up a great meta-question for Fermi Estimation and sensemaking of data that you see presented in the news... What facts do you need to know to be a good Fermi Estimator? (There's no perfect answer for this; just tell us what facts you've used to do your own Fermi Estimates!)  

When I start thinking about how you do estimates in your everyday life, I think about all of the times I do the Fermi estimations on news stories that I read.  Usually, these days, these are stories about income, wealth, distribution, immigration, and science stories. 

To do any kind of Fermi estimation you need to know a few things.  I've been writing down some of what I'd consider core knowledge over the past week, just taking notes when I did my own estimates to see what core knowledge I used.  Here's my list (yours is probably different): 

- population of the US in 2017:  327M 
- population of the world in 2017:  7.5B
- area of California:  ~100M acres
- conversion from inches to cm:  1 inch = 2.54 cm 
- number of stars in our galaxy:  300B stars

And so on... 

The reality is that nobody can take the time to spend 5 minutes doing online research for everything.  But it's pretty simple to do Fermi estimates to see if what's being said actually makes sense.  

It's a really good skill to have.  I highly recommend you practice this whenever you see something that sounds a little off.  It just might be.  But with your Fermi estimator skills, you can see through the mistakes.  

Here's a short video by my friend Jevin West talking about Fermi estimation in his class at the University of Washington.  It's well worth watching--his examples are great!    (Video link.)  

Search Lessons 

A few lessons spring out at me... 

1.  You still have to know basic facts about the world.  Dates, places, quantities, names, sizes, durations...  For instance, how important are plastic straws as a source of plastic pollution?  (Can you estimate what fraction of the total plastic trash they are?)  Once you know how to do Fermi estimates, your reading and understanding of news stories  / current events changes.  Beyond just knowing facts--e.g., the population of Japan is around 127M--once you know things like this, you know what kinds of information can be derived from them. Example:  While 3% is a good estimate of the number of people world-wide who are older than 80, you might also know that Japan is an outlier in the age pyramid--they have many more older residents than most countries, so you should Ferminate a higher fraction for folks > 80 years of age.  

2.  Realize that you CAN do estimates that are pretty good, and use them to do basic fact-checking of a story.  If it doesn't add up, you have more research to do.  This more of an attitude about research than anything--but it's important.  Many people don't know how to do estimates, and so they don't know they can guesstimate what they're reading or hearing about. And that makes them more susceptible to incorrect data and news stories.   

3.  Of course, online search is handy to get to the basic facts and formulas.  This is really true for breaking stories where there simply isn't any good information yet.  Even if you can't determine a particular value for something you've read, there's a good chance that you can estimate it by looking up other information and combining the data together into a fuller picture.  

Keep estimating.  It's a great personal skill to have! 

Search on! 


  1. Well, Professor Russel, we have a problem :

    - You can probably guess that 24 * 7 = 164 cm, well NO. 168 I guess

    17*4 easy to see it's NOT 64 but 68 cm

    Are you using some kind of alternate arithmetic, Dan ?

  2. There's a problem with your first example. The method works, but your math is off.
    17 * 4 equals 68, not 64. 176 centimeters is ~ 69.25". Quite a bit taller than your result. Also, 66.4 inches is 5 feet, 6.4 inches, not 5 feet 8 inches.

  3. Oh... you're both correct. That's a typo on my part. I'll go edit the post now (and fess up to my mistakes)!

    1. Thanks, Dr. Russell for sharing the answer and method. It is as always, very interesting.

      Also, want to share this new source of data. Our World in Data
      "...The project is produced by the Oxford Martin Programme on Global Development at the University of Oxford..." Have someone of you tried?

    2. I was reading this link and think this could be very helpful for others too. So, I am sharing here. I wonder how many other tricks and tips Google Sheets has

      Translate Cells in Google Spreadsheets and ImportFeed Function

  4. A bit late to this conversation but I think all these estimates are way off. As Jon said in an earlier post, the average size Italian was 5 feet 6, but I would say he is below average height. My estimates would make him between 5 and 5"3"
    Using this image and assuming an bench height of about 30 inches, this would make him only 60 inches tall (5 foot). Using this image with Robert Oppenheimer (left) I estimate him at about 5'2" (according to the NY Times Obituary Oppenheimer was 6 ft and according to IMDB he was 5'10"). Likewise in this photo he is a good foot shorter than Chancellor Hutchins (right) who was 6'3" according to

    1. This is a GREAT comment, Chris. I'll write an entire post about it (maybe next week?). Estimates are only as good as the initial data and the operations involved. What you describe is an excellent validation step, which I should have described in the article.. but it was getting long as it was. SO... I'll write up Part 2. Thanks for pointing this out!