Does Your Answer Seem Right?

April 1, 2009 3:41 AM

NYT Guesstimation Quiz. Enrico Fermi estimated the yield of the Trinity A-bomb test by dropping some shredded paper. He also asked his students to estimate unusual quantities like the number of piano tuners in Chicago - to show that just about anything can be estimated without detailed knowledge.
posted by Electric Dragon (54 comments total)

In related news, today's XKCD comic is about Fermi as well.
posted by bjrn at 3:50 AM on April 1, 2009

I did badly. 2/8. But I wasn't sure how much "effort" I should have made. I could have pulled out my calculator and worked out the golf balls, but I figured that wasn't guesstimation, that would be calculation... so I just guessed. And got it wrong. Only by one order of magnitude, though - I must have missed counting a zero in my addled brain.
posted by Jimbob at 3:50 AM on April 1, 2009

When I was at university some students came back from interviews at a large bank. They'd been asked, amongst other things, to estimate how much ink and paper would be saved if you dropped the "W." from "George W. Bush" in a year of newspapers. They were rather loudly and arrogantly showing off how detailed their estimates were.
They didn't much like being quietly told that they were all quite wrong as nothing would be saved by having to append "Jr." to the name to make up for the loss of the "W." qualifier, something none of them had recognised.
posted by edd at 3:54 AM on April 1, 2009 [1 favorite]

Of course I did...30,000 golf balls per kilometer, 40,000 kilometers for the circumference of the earth, 3 x 4 = 12 which means I needed another digit, damnit!
posted by Jimbob at 3:55 AM on April 1, 2009

Fermi Paradox.
posted by bardic at 3:59 AM on April 1, 2009

They didn't much like being quietly told that they were all quite wrong as nothing would be saved by having to append "Jr." to the name to make up for the loss of the "W." qualifier

G.W. Bush isn't a junior. Jeez, the banking industry not only sucks at its primary competency, it isn't good at pedantry either.
posted by Makoto at 4:19 AM on April 1, 2009 [5 favorites]

It's always fun to mess around with these kinds of questions. At 3QD recently there was an article that discussed estimating the weight of the air inside the Empire State Building. There's no definitive answer given, but the estimated values and calculations are interseting to compare to your own. One of the comments on that article also links to one of MIT's open courses called 'Street-Fighting Mathematics' - the art of guessing results and solving problems without doing a proof or an exact calculation, which seems to be a more advanced approach to these kind of problems.
posted by Jakey at 4:33 AM on April 1, 2009 [4 favorites]

4/8. For some reason 200,000 "like"s seemed extreme to me. But in hindsight: I recently checked how many words I typed in a week (computer only), and it was well in excess of 50,000; so if you add speech and texting to that I guess 200,000 likes might not be totally unrealistic over a year. But it still sounds parodically (parodistically?) high.
posted by Dumsnill at 4:34 AM on April 1, 2009

I'm going to guess that I answered all of the questions correctly.
posted by not_on_display at 4:46 AM on April 1, 2009 [8 favorites]

This will be very handy for AskMe.
posted by TedW at 4:49 AM on April 1, 2009

This is going to be horrible for AskMe.
posted by bigmusic at 5:01 AM on April 1, 2009

just about anything can be estimated without detailed knowledge

Except, of course, the date at which a software development project will be completed. Zing!
posted by drinkcoffee at 5:04 AM on April 1, 2009 [2 favorites]

I got the first three right, then gave up when the possible answers to the fourth were:

a) A city
b) A state
c) A small country
d) Asia

Now, unless it escaped their notice, 'city', 'state', and 'small country' are not standardised measures. I figured the answer to the question to be about 2,500 km2, which could be equivalent to Istanbul, Rhode Island, or Luxembourg. Admittedly, I was way out anyway, but still...
posted by Sova at 5:09 AM on April 1, 2009

Now, unless it escaped their notice, 'city', 'state', and 'small country' are not standardised measures. I figured the answer to the question to be about 2,500 km2, which could be equivalent to Istanbul, Rhode Island, or Luxembourg. Admittedly, I was way out anyway, but still...

I'd quibble with that one too - I knew it was on the smaller side (c.v. we could all stand on Zanzibar), but how big a city? (York? New York?) How small a state? (Rhode Island? Tasmania? West Australia? Texas?)
posted by outlier at 5:30 AM on April 1, 2009 [1 favorite]

Cool, I remember taking similar quizzes in high school math class. They were even called "Fermi questions" and one I remember in particular was "How many drops of water are there in Lake Ontario?"

They were a lot of fun, but also useful to help develop an intuitive sense of the magnitude that answers should be. Because often times when one makes a mistake with the detailed calculations, the result will be way off and it's useful to be able to go "hmm, that doesn't look right at all."

(Oh, and 6/8 so I am not as good at these as I used to be apparently.)
posted by FishBike at 5:38 AM on April 1, 2009

Now, unless it escaped their notice, 'city', 'state', and 'small country' are not standardised measures.
I got this right only b/c of how it was written. I eliminated state and small country b/c they're essentially the same thing (how many states in the US are bigger than small countries? 80%?). The US and all of Asia seemed to big, so I went with city.
posted by ShadowCrash at 6:01 AM on April 1, 2009

Eh. These things are fine as far as they go, but then when additional claims are put on top of them such as:

This method does not guarantee correct results; but it does establish a first estimate which might be off by no more than a factor of 2 or 3--certainly well within a factor of, say, 10. We know, for example, that we should not expect 15 piano tuners, or 1,500 piano tuners.

That's just silly.

If your guesstimates were off by a factor of ten, then your answer will be off by a factor of ten. It's not some magical mechanism. Nor is "ten" some magical number.

In this particular example, the person guesses that 20% of Chicago households have a piano. I seriously doubt that.

He also assumes without even noting that he is assuming that the average piano gets tuned once per year. I seriously doubt that as well.

I'm not saying his resulting guess is off by a factor of ten. But due to the combination of these particular choices of his, I wouldn't be surprised.
posted by Flunkie at 6:06 AM on April 1, 2009

"G.W. Bush isn't a junior. Jeez, the banking industry not only sucks at its primary competency, it isn't good at pedantry either."
Really, the point is that the W is put in there for a reason and it's a bad idea to just arbitrarily remove it. What you actually replace it with is not a significant detail.
posted by edd at 6:14 AM on April 1, 2009

For some reason 200,000 "like"s seemed extreme to me.

Me too. That would be an average of 547 "likes" per day. There have to be some sullen teenagers out there where you can't get 547 words out of them per day, let alone "like like like" ad nauseam.
posted by jonp72 at 6:20 AM on April 1, 2009

"G.W. Bush isn't a junior. Jeez, the banking industry not only sucks at its primary competency, it isn't good at pedantry either."

Really, the point is that the W is put in there for a reason and it's a bad idea to just arbitrarily remove it. What you actually replace it with is not a significant detail.

Or maybe the point is that an industry capable of bringing down the entire frickin' economy is evaluating job candidates on the basis of completely bullshit questions.
posted by jonp72 at 6:22 AM on April 1, 2009 [2 favorites]

My New England prejudices just shined through on question 4:

I estimated a person as 3 feet wide by 1 foot deep, or 3 persons/m².
That's 2x10⁹ m², or 2,000 square km. (OK, I'm with the NY Times so far, mostly.)

Here's where we diverged: 2,000 sq km is 40 km x 50 km, or just half of Rhode Island! By comparison, our "big city" of Boston has a land area of only 125 square km.

In my continuing hatred of LA, I will point out that any "city" that requires a car is in fact not a city, but a series of suburbs that have merely been incorporated into a conglomerated government. Real cities build upward, not outward.
posted by explosion at 6:25 AM on April 1, 2009 [1 favorite]

I don't think the real question included the Jr. answer.
posted by smackfu at 6:27 AM on April 1, 2009

If your guesstimates were off by a factor of ten, then your answer will be off by a factor of ten. It's not some magical mechanism. Nor is "ten" some magical number.

No, it's a very useful tool.

People here have pointed out the problem of "what is a town?" Answer: What does the person defining town define it as? First you define those terms, then you select and continue on.

It may seems surprising, but estimation is *critical* to experimental physics. For example, you have a detector that you're going to use to detect, say, Solar X-Ray flux. Because of limitations of the technology you have to detect solar radiation, you can only detect about 5 orders of magnitude of signal. In terms of absolute energy, it can detect between 10^-20 and 10^-15 Watt meters^-2. Below that, your detector can't see the signal, above that, you saturate, and while you can tell you had a certain flux, you can't tell how much more than 10^-15Wm^-2 it is. You can, however, shield the sensor to move that range. If you need to measure 10^-10 to 10^-5 Wm^-2, you can -- if you put a shield in front of that sensor that absorbs 10^-10Wm^-2 of the flux.

So. The $200 million dollar question¹. What, if any, shielding, do you install in front of this sensor. You don't know the Solar X-Ray flux. If you don't install enough shielding, you get a sensor that reports 100% signal all day, every day, and your experiment fails. If you install too much, you get a sensor that reports 0% signal all day, every day, and you fail.

So. The critical point -- is to make a defensible estimate of what the solar X-Ray flux is. This will depend on a bunch of guesses -- how often solar flares occur, what energy they have, what the background flux of the Sun is. You won't know many of these with much confidence. But you must get somewhat close.

So: Estimation. Trick one is being rational about your guesses. If you posit that 50% of the population of Chicago is Female, you're probably safe. If you posit that 75% of the population is female, you're already in the hole. Trick two is to swap bias. My first guess of population of the Chicagoland Area (Defined as Cook/Dupage/Lake/Will/Kane, plus the county that Gary, IN is in.) is 15 million. I think that's high, so when I make my next guess, I want to bias *low*. By flopping your bias, you tend to average out your errors.

So: Fermi has a Piano Tuner Detector that can measure two orders of magnitude. With no shielding, it could tell you if there are 1, 10 or 100 piano tuners in the Chicago Metropolitan area, but any more than 100 piano tuners and you saturate the entire sensor and not know the real number. What Fermi's lecture is trying to do is show you how to answer the question "Do I need to modify this detector?"

In our example Solar X-Ray flux detector? You'd better. Actually background flux ranges from 10^-8 and 10^-6 Wm^-2. Flares have spiked the way up to 10^-1 Wm^-2. So, if you flew the sensor with a 10¹² Wm^-2 shield, and you'll measure 10^-8 to 10^-3 Wm^-2, and get very useful data -- though really, you need a couple more orders of magnitude bandwidth to see the largest flares.


1) That being what it'll cost to build and fly the satellite with your sensor, because Solar X-Rays don't reach the ground.
posted by eriko at 7:02 AM on April 1, 2009 [6 favorites]

I just estimated I got an 8/8 on this quiz. I knew I was smart!
posted by Mach5 at 7:05 AM on April 1, 2009

Certainly anything can be estimated without detailed knowledge. People do that all the time.
The hard part is to accurately estimate something without detailed knowledge.

For example, I don't know how big Central Park is, or how many squirrels typically live in a hectare, but that didn't stop me from making an incorrect estimate of how many live there.
posted by MtDewd at 7:08 AM on April 1, 2009

I stopped at the area you can cram people into. Exact same problem as explosion. To a person from a small town in a small state, a square 15 miles on a side is a state, not a "large city".
posted by DU at 7:17 AM on April 1, 2009

Or maybe the point is that an industry capable of bringing down the entire frickin' economy is evaluating job candidates on the basis of completely bullshit questions.

Not to defend the banking industry or anything, but what questions should they be asking? If you're hiring someone straight out of school, there's not usually much to go on in terms of experience, and you have to hope that you're hiring someone smart who can learn most of what they need to know to do their job.

In the case of the ink question, it's not really about the answer itself. Nobody takes the estimate and compares it to some official correct estimate to evaluate the candidate. The point is that you're giving the person a problem that (hopefully) they haven't thought about before but that they should be able to work out. If the person punts on it and says "I wouldn't know where to start," it doesn't tell you much, other than that maybe they aren't a very good problem solver. If they point out that taking out the W. wouldn't help much because differentiating between the two presidents is important, you might give them cleverness points or something, but that's not really the object of the exercise. If they start asking you clarifying questions, walking through their thought process, throwing out some ideas, etc. then you'll have a good idea of how they will approach new unknown problems in their job.
posted by burnmp3s at 7:19 AM on April 1, 2009

One square foot per person, so roughly 25 million people per square mile, or 6 billion people in 240 square miles. About the size of Chicago, so...
posted by notswedish at 7:19 AM on April 1, 2009

No, it's a very useful tool.

I didn't say it wasn't a useful tool.

I said that the idea that you know whatever answer you get out of it is correct to within a factor of ten is silly.
posted by Flunkie at 7:34 AM on April 1, 2009

Interesting quiz, but it would really have helped if they had actual answers. After all, you guess an answer, and then it tells you if the guess was "right" or "wrong", but in reality it isn't telling you that, only if your guess was the same as their guess.

For example, the last question, the "teenagers saying like" one. My thought process was:
Kids spend 1/3 of the day sleeping, and 1/3 at school or work. So an average of 8 hours a day free time. Half of that is spent on non-talking activities (driving, video games, homework, shower, YouTube, tv). So 4 hours a day of talking time. Kids are social, and often in groups, so I figured that 1/4 of the talking time was ACTUAL talking time (with another 2/4 spent listening to their 2 friends talk, and 1/4 spent doing random shit like running for shotgun in the car, jumping off mailboxes, etc.). So 1 hour of full talk-time. That's 60 minutes of talk time. Some kids say "like" a lot, some almost never say it (note: calculations would have been far different back in the 80's "valley girl" heyday). So let's guess 3 times a minute, with some going over and some going under. That's 180 "likes" per day, 365 days a year, for a total of 65,700 likes / year. Which is a lot closer to 20,000 than 200,000. Am I right about that number? No idea, we'd have to actually measure kids. But they said I was wrong, not based on actually being wrong, but because my methods and results were different from theirs.
posted by Bugbread at 7:38 AM on April 1, 2009

That's just silly.

Well, using a similar series of estimations, I tried, myself, a couple of years ago to guess the number of piano tuners in Baltimore. Here's how I approached it (I had no idea of the population of Baltimore, as you can see):

Baltimore is a medium sized American city, I'll guess 2,000,000 people. I'd guess one in 200 families owns a piano. If the average family size is 3, that means 2,000,000/3/200=3300 pianos in Baltimore.Round that down to 3,000.

A piano tuner as far as what I've seen on TV takes most of a day to tune a piano. Pianos probably need to be tuned about once a year. That means the tuner can service 365*(5/7) or about 250 customers a year. 3,000 customers in baltimore/250 customers per tuner=12 piano tuners in Baltimore.

Going to the Baltimore area website of the American Piano Technicians guild shows 10 tuners who list their area of practice as Baltimore.
posted by Turtles all the way down at 7:38 AM on April 1, 2009

Plus all the "miles" and "inches" and "foot-pounds" and "slugs" was confusing. It's to be expected, but if you're dealing with Newtons, you may as well start out with meters, kgs, etc., instead of starting out in Imperial and then doing all the conversions.
posted by Bugbread at 7:39 AM on April 1, 2009

I really do think that making estimates like this is one of the most satisfying and useful cognitive exercises. It's good for getting around some common human reasoning hickups. It's good for compartmentalizing big problems into manageable chunks, making reasonable and useful assumptions, searching out information from experience, and maintaining focus.
posted by I Foody at 7:54 AM on April 1, 2009 [1 favorite]

12 piano tuners

Your guess for the number of piano tuners per unit population is less than an eighth of the other guy's, using the same methodology as he did. If anything, your example backs my point.
posted by Flunkie at 8:07 AM on April 1, 2009

a total of 65,700 likes / year. Which is a lot closer to 20,000 than 200,000.

Well, not really. It's a factor of 3ish out either way, so your original answer is in fact correct within an order of magnitude. It's difficult to answer these questions as multiple choice, as you quite often end up in this region and have to toss a coin to choose. (I did so 3 times in this quiz and lost twice). In any case, from a recreational point of view, I think the most fun in Fermi questions is comparing assumptions and results with others. e.g, I got the 'like' question 'correct' as follows: I assume people talk at 60wpm, 8 hrs a day = approx 25k words a day. I know this is high, so I multiply by 300 days to keep the arithmetic easy and round down a bit = 7.5million words. How many of these words are like? Probably more than 1% and less than 10%, but I still feel I'm a bit high, so I plump for 1% = 75k. This is in-between territory as it's within a factor of ~2.5-3.5 between the listed answers, so I toss a coin and go for 200k.
posted by Jakey at 8:11 AM on April 1, 2009

burnmp3s - Good point, but you may be giving the average campus recruiter a lot of credit there. Still, it doesn't take away from your point that a good recruiter could use that question as a tool to start a discussion (doesn't sound like they did in this example though). It's certainly better than the (apocryphal?) story that the Morgan Stanley approach was to invite the candidate for lunch and observe if s/he puts salt on their food before tasting it.
posted by Nick Verstayne at 8:14 AM on April 1, 2009

Your guess for the number of piano tuners per unit population is less than an eighth of the other guy's, using the same methodology as he did. If anything, your example backs my point.

Different cities, bud.
posted by Turtles all the way down at 8:16 AM on April 1, 2009

A better approach might be to say, "Hey, I wonder how many grains of salt are in that salt shaker you've got in your hand there, Bub."
posted by Nick Verstayne at 8:18 AM on April 1, 2009

Going to the Baltimore area website of the American Piano Technicians guild shows 10 tuners who list their area of practice as Baltimore.

Which only goes to show that you can sometimes get a (maybe) reasonable estimate by getting lucky and having your ridiculously unfounded assumptions cancel each other out.

For example, your population estimate is off by a factor of about 3, a piano tuner only takes a couple of hours to do one piano, but you helpfully forgot about the fact that every school and church in the area has one too, which happily happens to balance out the previous errors (assuming the 1/200 guess was reasonable to begin with). If your errors didn't happen to cancel, you'd quickly end up two orders of magnitude off.
posted by Dr.Enormous at 8:22 AM on April 1, 2009

Interestingly, I just finished reading William Poundstone's "How Would You Move Mt Fuji", which is nominally about the sort of questions that Microsoft asks in interviews (but actually ranges over IQ, interviewing styles and brain teasers). The piano tuner question is a classic of the the tech interview, but here's the catch: no body knows.

The piano tuners union doesn't know. They can give you their number of members, but not everyone belongs, a lot of piano tuners actually just do it on the side, many of their members are retired ... guesstimation can be useful. And sometimes you have to guesstimate because there is no answer.
posted by outlier at 8:25 AM on April 1, 2009

Which only goes to show that you can sometimes get a (maybe) reasonable estimate by getting lucky and having your ridiculously unfounded assumptions cancel each other out.

And that, my esteemed medical colleague, is the beauty of this method. You expect your assumptions to be wrong, but rely on the probability that some are going to be higher and some lower than the real number, some are going to be overestimates and some underestimates, and in the end they will balance each other out to give you a number within an order or mangnitude or so of the true number either way. And if you have to estimate, being within an order of magnitude plus or minus is much more useful than plucking a number out of thin air.

Nobody is saying that, for example, the architect charged with building the Baltimore Piano Tuners' Retirement Home should base its size on this calculation. But if you arrive in a smallish town and see a sign at the airport welcoming the Baltimore chapter of the Tuners' Guild, your strategy for finding a hotel room will be different depending on whether there are going to be 20 or 2000 drunken, tuning-fork-wielding revelers competing with you.
posted by Turtles all the way down at 8:38 AM on April 1, 2009

I understand the value of the method; I can't tell you the number of times I've had students who never notice that "1.2x10⁴ kJ" isn't a reasonable answer for the energy of one 533 nm photon (unit conversions kill them). But that value only exists if you have some clue what you're doing, rather than just making up numbers.
posted by Dr.Enormous at 8:45 AM on April 1, 2009 [1 favorite]

That's why you have to add a sanity-check step at the end. In your example, said students should only need to ponder for a moment what the implications of a person standing in the sunshine would be, if photons were so energetic. In the piano tuner example, if you arrive at a figure of thousands or more you would have to figure that you would regularly see piano-tuner vans around any urban area, and possibly even know one or two tuners. Conversely, if you arrive at a number in the single digits, you probably have never even known someone who's used the services of one.
posted by Jakey at 9:18 AM on April 1, 2009

12 piano tuners

Your guess for the number of piano tuners per unit population is less than an eighth of the other guy's, using the same methodology as he did. If anything, your example backs my point.

True, but as a former non-member of the Piano Technicians Guild I can tell you there are far more piano tuners who are non-members than members. Maybe somewhere between 1 in 5 and 1 in 20 active piano technicians is a member of the guild.

However (and somewhat contradicting what some have said above) you have to accept that when you make estimates of this sort you can easily be off by an order of magnitude in either direction.

If you can refine the numbers that go into your estimate, or come at it from a completely different direction, you might be able to narrow it down.

So your estimate of 12 is not so much "wrong" as well to the low end, and the check of Guild members (which are only a portion of all piano technicians) confirms both that you're in the right ballpark and that you're on the low end of that ballpark.

But (as others have pointed out) even being able to estimate within an order of magnitude can be very, very useful in many situations. If you can narrow it down a little, so much the better.
posted by flug at 9:31 AM on April 1, 2009

True, but (...) you have to accept that when you make estimates of this sort you can easily be off by an order of magnitude in either direction.

I'm sorry, but is it not clear that that's exactly my point?

That the only thing I've been doing is contradicting someone who said exactly the opposite?

That is, that all I've been doing is contradicting someone who said that you certainly won't be off by an order of magnitude?
posted by Flunkie at 9:37 AM on April 1, 2009

In this particular example, the person guesses that 20% of Chicago households have a piano. I seriously doubt that.

Fermi used this example when he was at the University of Chicago in the 1940s and 50s. Pianos were a lot more popular in households back then.
posted by xil at 10:54 AM on April 1, 2009

Quick question re: #2 : who among you has lotto tickets that are thicker than a piece of paper? All the lotto tickets I have ever purchased were printed on paper that might reasonably be estimated as thinner than copier stock.
posted by lucasks at 11:08 AM on April 1, 2009

this is a more interesting guessing game. it's not about guesstimating right, but about how good you think your guess is.
posted by fistynuts at 1:48 PM on April 1, 2009 [1 favorite]

There are some more of these order-of-magnitude estimations in the "How Big is That Number" series on Ask Doctor Math.

My favorite is the "How much volume would a googol grains of sand take up?" question.
posted by albrecht at 2:03 PM on April 1, 2009

If you're interested in slightly more technical versions of this same kind of thing, there are a few places where "order of magnitude physics" classes are commonly taught. Material that was used in an OOM class at Caltech some years ago is here. Problem sets from a similarly styled course taught by Eugene Chiang at Berkeley are on Eugene's website, and are awesome.
posted by chalkbored at 2:19 PM on April 1, 2009 [2 favorites]

I am reminded of the (apocryphal?) question that was supposed to be asked at Microsoft employee interviews: How many gallons of water flow through the mouth of the Mississippi each second?
posted by Drasher at 2:33 PM on April 1, 2009

Q: How much does New York weigh?

A: NYC weighs between 302,584,000,000,000 kg and 111,342,281,879,194,624 kg

weight_of_new_york.py
posted by jcruelty at 11:12 PM on April 1, 2009

(p.s. weight in the common sense of 'mass')
posted by jcruelty at 11:13 PM on April 1, 2009

Isn't NYC an abstract object, weighing less than a gram (estimating the weight of the memories used to create an instance of the object)

and no, you peeps in NYC may not look up from your screens to check if it's still there
posted by fistynuts at 5:22 PM on April 2, 2009