The art of the flying guess
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I used to read a lot of all kinds of books when I was a kid. Not anymore – now I just read professional literature. I figure that’s a shame, so I made a new year’s promise for 2025 to read 10 non-physics books. Being impatient, I started already in December – and as I cracked open the first book on the list, I wondered if other people have similarly stopped reading, and what the size of the global book market is now.
Ever wonder about things like that? The approximate size of some number, like the size of a book market or the amount of energy in a gallon of gas. I wonder about things like that all the time.
You can, of course, Google it – but where’s the fun in that? Googling is for dullards. Besides, not everything is easy to google anyway. Wouldn’t it be great if you could just calculate the answer yourself?
I’ll answer for you: yes, it would be. And we’re in luck, since estimating numerical quantities like that is very much possible, as long as we don’t want to be TOO precise. If we’re happy to just get within a factor of 10, we can get calculating – we’ll know the answer is ‘hundreds’, but not whether it’s 352 or 638.
The key to estimating is to break down the problem in to several different quantities for which you can make a guess (like how many people read books and how many books do they read in a year on average). Much like a drunk man who, in aiming for a doorway, first stumbles to the left and then accidentally course corrects by stumbling to the right, we will often find our mistakes cancelling each other out. This makes it possible to find pretty good guesses for almost any number.
These types of problems are sometimes called "Fermi problems" after the physicist Enrico Fermi, perhaps the greatest master of such problems. As he said,
I can calculate anything in physics within a factor 2 on a few sheets; to get the numerical factor in front of the formula right may well take a physicist a year to calculate, but I am not interested in that.
Here, we won’t do any physics, though – just numerical estimates. Before we get going, a digression to powers of ten.
“Scientific” notation
You need to know how to write numbers in powers of ten to get guessing. For some unconscionable reason, that’s called "scientific notation". If you already know how it works, just skip this – although at the end of this section I show you how to quickly calculate geometric averages using powers of ten, so you might want to check that.
Writing in powers of 10 is very easy. It means writing a number as follows:
\[\begin{aligned} 12 = 1.2\cdot 10^1\end{aligned}\]It’s obvious how that works: 12 is 1.2 times 10. Multiplying by 10 simply means moving the dot one spot to the right. Since $10^2 = 10 \cdot 10$, multiplying by $10^2$ means moving two places to the right, and so on. Dividing by 10 works the opposite: moving the dot to the left.
Take the number $143$. You can read it as $143.0$. To get the dot between the first and second number, we need to move the dot two places. Hence:
\[\begin{aligned} 143.0 = 1.43\cdot 10^2\end{aligned}\]Check for yourself – if you move the dot two places to the right, you get 143 back.
Once you get used to it, it’s simple to do this conversion. To aid your memory, I’ve tabulated the names of early powers of ten.
| Power | Meaning |
|---|---|
| $10^0$ | 1 (one) |
| $10^1$ | 10 (ten) |
| $10^2$ | 100 (a hundred) |
| $10^3$ | 1000 (a thousand) |
| $10^4$ | 10 000 (ten thousand) |
| $10^5$ | 100 000 (a hundred thousand) |
| $10^6$ | 1 000 000 (a million) |
| $10^7$ | 10 000 000 (ten million) |
| $10^8$ | 100 000 000 (hundred million) |
| $10^9$ | 1 000 000 000 (a billion) |
| $10^{10}$ | 10 000 000 000 (ten billion) |
So, when I say things like $3\cdot 10^2$, you can use that table to read “three hundred”. But once you get the hang of it – or remember these from your school days – you won’t need the table anyway.
The final thing we need is the geometric average. The arithmetic average – the “normal” average – of two values $a$ and $b$ is $(a+b)/2$. The geometric average is instead $\sqrt{a\cdot b}$. This has an advantage for the sort of thing we’re about to do: if we take the numbers 1 and 100, the geometric average is 10, which is 10 times the lower limit and a tenth of the upper limit. If we took the arithmetic average instead, we would get $\approx 50$, which is 50 times larger than the lower limit but only half the upper limit.
To calculate the geometric average quickly (but not precisely) in powers of 10, we have the following trick:
\[\begin{aligned} \text{geom. avg }(a\cdot 10^b, c\cdot 10^d) \approx \frac{a+c}{2} \cdot 10^{\frac{b+d}{2}}\end{aligned}\]In other words, average the numbers in front of the 10 and the exponents.
Here’s an example. Take the numbers $3\cdot 10^2$ and $5 \cdot 10^4$. Then we get for the geometric average:
\(\begin{aligned} \text{geom. avg}(3\cdot 10^2,5 \cdot 10^4) = \frac{3+5}{2}\cdot 10^{\frac{2+4}{2}}=4\cdot 10^{3}\end{aligned}\) What if we end up with a half in the power of ten? For example, the numbers $3\cdot 10^2$ and $5 \cdot 10^3$. Observe:
\[\begin{aligned} &\text{geom. avg}(3\cdot 10^2,5 \cdot 10^3) = \frac{3+5}{2}\cdot 10^{\frac{2+3}{2}} = 4\cdot 10^{2+\frac{1}{2}} \\ &= 4\cdot 10^{2}10^{1/2}\approx 4\cdot 10^2\cdot 3 = 12 \cdot 10^2 = 1.2\cdot 10^3\end{aligned}\]Confused? No worries: if you have a calculator at hand, you can always use $\text{geom. avg}(a,b) = \sqrt{a\cdot b}$ – multiply the numbers and take the square root. This approximate method is just easier to do in your head. Even a pretty cheap calculator will often be able to do scientific notation, too.
Guessing by example
The book market
The best way to learn is to just get guessing. So, what IS the size of the book market in the world?
Well, there’s about 8 billion people in the planet. In other words, $8\cdot 10^9$ people. However, hundreds of millions of those people are entirely illiterate, and many more are functionally illiterate. Many others don’t have the financial resources to buy books, and some just don’t care to do so.
Let’s asume about half of that 8 billion occasionally buys books. That’s $4\cdot 10^9$ people.
Well, how many books do they buy on average per year? Some people buy many books a year, some only buy a book once in a few years. It’s hard to say. To get going, we need to estimate lower and upper limits.
First of all, the AVERAGE can’t possibly be more than
$~$<!-- -->{=html}26. If someone buys 26 books a year, that’s a book
every 2 weeks. Very few people read that much – so certainly the
average number of books per person can’t be that high.
On the other hand, surely the average reader buys more than one a year. So the lowest estimate is 1 and highest estimate 26. We now employ our geometric average trick:
\[\begin{aligned} \text{geom. avg} (1\cdot 10^0,2.6\cdot 10^1) \approx \frac{1+2.6}{2}\cdot 10^{\frac{0+1}{2}} \approx 6\ \frac{\text{books per year}}{\text{reader}}\end{aligned}\]So, let’s say that the average reader buys 6 books a year.
How much does the average book cost? Well, here we have a difficulty: different countries price things differently – the pricing depends on the average income in the country, the average price of labor, and so on. So we don’t know. But let’s assume the average is no less than 5 dollars (convert to your currency if you wish), and probably not higher than about 30 dollars. In western countries, books may regularly cost more than that, but most people don’t live in western countries – so this seems like a reasonable upper limit.
Geometric average again:
\[\begin{aligned} \text{geom. avg} = (5\cdot 10^0,30\cdot 10^1) \approx 12\ \text{dollars/book}\end{aligned}\]So, we have 4 billion people reading 6 books costing 12 dollars. The revenue of the book market in a year:
\[\begin{aligned} &\text{Book market revenue} \\ &= 12\ \frac{\text{dollars}}{\text{book}}\cdot 6\ \frac{\text{books per year}}{\text{reader}}\cdot 4\cdot 10^9 \text{readers} \\ &= 288\cdot 10^9 \text{dollars per year} = 288\ \text{billion dollars per year}\end{aligned}\]According to google (I know – dullard move), the actual size of the market is 140 billion. What did Fermi say about factors of 2 again?
It’s fun to think about which quantity we overestimated. Perhaps, once taxes are taken in to account, the averge price of a book is even lower. Or maybe a larger number of humanity basically never buys books (libraries exist, after all). Or perhaps the average reader only reads 3 books a year! Who can say, but we got the right ballpark (hundreds of billions). More likely, we made both under- and overestimations, and some of them cancelled out, just not all of them.
How many words in a book?
Speaking of books, how many words in a novel?
This one is much easier. First, we could estimate how many words are on the average page. That’s not difficult – if you have a book around, you can eyeball the number pretty quickly. Let’s say it’s around 250 words.
Then how many pages are in the average book? Don’t know. Let’s get an upper and lower estimate again. I’ll assume that the average book has no fewer than 100 pages and no more than 1000. Geometric average: 300, or thereabouts. Thus, the average novel has
\[\begin{aligned} \text{words in a novel} = 250\ \frac{\text{words}}{\text{page}}\cdot 300\ \text{pages} = 75000\ \text{words}\end{aligned}\]According to google, that’s about right for an adult novel. This time, we’re not even off by a factor of two! Perhaps we should modify our guesses to be worse lest Fermi’s ghost believe we spent a year on this.
How many meters of toilet paper used per year per family?
A domestic question of some relevance – budget your toilet paper! The first step is to guess how large the average family is. Let’s say we’re talking about families in western countries. Many people live alone or in dormitories – how to count a family there? In any case, we know that a household by definition has 1 or more people. Presumably, the average family living together has less than 10 people. Geometric average: 3.
Well, how much toilet paper does a person use per day? No clue – it’s hard to guess at daily use – I’ve never counted how many pieces of toilet paper I use per day!
I do know this, though: there’s no way people use less than one roll per 10 weeks. I doubt anyone uses more than 10 a week (or if they do, something is seriously wrong). The geometric average of those two: 1 roll per week.
So, we have 3 people using a roll per week on average, which makes for
\[\begin{aligned} 52\ \text{weeks per year}\cdot 3\ \frac{\text{rolls}}{\text{week}} \approx 150\ \text{rolls per year}\end{aligned}\]How many meters of toilet paper in a roll? Again, hard to say. You could calculate by unrolling the one from your toilet. Easier, though, to guesstimate by hand: unrolling a meter doesn’t make a big dent. It’s pretty clear it has to be longer than 5 meters (15 feet). On the other hand, it surely can’t be more than 50 meters (150 feet). Geometric average to the rescue: we’ll guess 15 meters (45ish feet).
So, we have 150 rolls of 15 meters each, which makes for 2250 meters. Since our family had 3 people, that’s about 750 meters per person.
Incidentally, Statista estimated that the average PERSON uses 150 rolls a year! Can our estimate be that badly wrong? It seems doubtful; the page says they calculated the estimate assuming the average roll is 90 grams, but I think it’s heavier than that by quite a margin. You can do that experiment yourself.
The book Guesstimation 2 also contains an estimate on this; they get about 700 meters per person as well by estimating the number of sheets used per person (each sheet being about 10 centimeters in length).
How off can you be?
In this article, I’ve constantly given upper and lower estimates for various numbers. The worst case: the number is not between my upper and lower estimate. I’ve been pretty generous, though – seems unlikely that, for example, the average toiler paper roll would really be longer than 50 meters or less than 5 – there’s hardly a way I could be wrong.
The worst thing, then, is that you make several estimates and in each case, the true number happens to be near either the lower or upper boundary.
Take, for example, the book market calculation. Suppose the upper guess is true every time: the average reader buys 26 books a year, the average price is 30 dollars a book. Then the market is over 3 trillion dollars! That would be way off from our estimate. It also intuitively seems way too big.
Always keep your wits about you. If the end result of your estimate is that the average person uses 500 rolls of toilet paper per day, you’ve probably gone wrong somewhere.
Finally, I want to mention some books that deal with this topic: Guesstimation 1 and 2 by L. Weinstein and A. Adam. They contain many more examples of this kind – work through them and you’ll be calculating almost like Fermi in no time.