Monday, December 15, 2008

Back of the Envelope

A physics professor my wife knows complains that few of his students have any idea of how to do plausibility calculations, how to figure out whether quantitative claims could be true. My wife suggests that it would be a good topic for a class in elementary school, since such calculations usually require nothing more than arithmetic and demonstrate one reason why arithmetic is useful. It occurs to me that it might also be a good topic for a book.

In physics, economics, and I expect some other fields, these are sometimes referred to as “back of the envelope” calculations, because they can be, sometimes are, done on the back of an envelope destined for the waste basket. They use approximate data, approximate models, and are expected to give the right answer to within about a factor of ten in either direction. Here are a few examples--readers are invited to provide more.

How long cars last: It’s not uncommon to hear complaints about how quickly cars, especially American cars, wear out, sometimes linked to the claim that they are designed to wear out so you will have to buy a new one. Along similar lines, American consumers are sometimes pictured as routinely buying a new car every two years.

If the average car lasts two years, the cars currently on the road represent about two years' production. It is easy enough to look up how many cars are on the road and how many are produced each year; for 2005 the figures (from the Statistical Abstract of the United States) are 137 million automobiles registered and 17 million new cars sold. Dividing the first number by the second tells us that the average car lasts about eight years. Some of those, of course, are made by Japanese or Korean or European firms. But it is easy enough to repeat the calculation for an earlier year, back when most cars driven in America were built in America by American firms.

Population density: One occasionally sees concerns that, as population grows, housing swallows up land needed for farming. To see how plausible they are, I start by estimating the amount of land actually occupied by housing. A large house nowadays has 3000 square feet of floor area, typically on two floors, and is occupied by four people, which means 1500/4 square feet of land per person. That’s surely a high figure, since most people don’t live in houses that large and a considerable part of the population is in urban areas in housing with more than two stories. Round numbers make calculation easier, so call it 200 square feet per person. An acre is 200'x200', so that gives you 200 people per acre. There are 640 acres in a square mile, so (rounding down for simplicity) about 100,000 people per square mile. The U.S. population is about 300 million, so the total area occupied by housing should be about 3000 square miles.

The U.S. is, very roughly, 3000 miles east to west and 1000 miles north to south. So housing occupies about 1/1000 th of its area.

This is a very approximate figure, produced without looking up anything. And it only includes housing, not lawns, streets, grocery stores, and the like. But it is still enough to show that mental images of people packed in like sardines due to too many people in not enough space, along with the associated arguments about how rats behave when there are too many of them in a cage, have to be wrong. There are areas where human construction occupies a large fraction of the land area. But the reason—at least in the U.S.—is not that there are too many people for too little land but that many people, for a variety of reasons, prefer to live in densely populated areas.

One reason for this particular error is that people form their opinion based on what they see around them—and spend most of their time in places where there are people. They are averaging population density over people rather than over acres, asking how densely populated an area the average person lives in rather than how densely populated the average acre is. One way of correcting that error is to observe a sample that is not biased in that way—by, for instance, looking down from an airplane window when flying across the country. What you see is much more likely to be corn fields, mountains, desert or forest than a sea of rooftops.

Asteroid Strikes: The risk of asteroid strikes has two relevant dimensions: How much damage would an asteroid of a given size do and how likely is an asteroid of that size to hit the earth.

Start with the one case we have good evidence on—the Tunguska event. In 1908 something caused a very large explosion, roughly equivalent to a hydrogen bomb; the current preferred theory seems to be that it was an airburst of a large meteor or comet fragment. It knocked down trees over an area of about 2000 square kilometers. Dropping a hydrogen bomb from time to time at some random location sounds pretty scary; perhaps we were just extraordinarily lucky that it hit Siberia instead of Manhattan. A simple back of the envelope calculation can tell us about how lucky we were.

The earth is a globe with a radius of about 4000 miles, roughly 7000 kilometers. The area of a sphere is 4π times the radius squared, making the surface area of the earth about six hundred million square kilometers. So the area over which trees were knocked down by the Tunguska explosion represents about 1/300,000 of the area of the earth. The current population of the earth is between six and seven billion. If we assume that the area over which trees were knocked down is about the same as the area over which humans would be killed, the average death toll from a Tunguska event would be about 20,000. That is a lot of people but hardly a global catastrophe—about half the number killed in the US each year in auto accidents, about one three hundredth of the number killed in the Holocaust.

How likely is a Tunguska event? It is unlikely that one would have occurred in the past century without being observed, given the seismographic effect, which registered as far off as Washington D.C. How much farther back one can push that argument I don’t know, so I will assume that such events happen at a rate of one a century. If so, the average mortality from such events is about 200 deaths/year. Every death matters, but there are a lot of problems in the world that do a great deal more damage than that. There is a good deal left out of these calculations—for one thing I don’t know how the area of damage from a sea strike would compare with that from a land strike or how easily it would be observed if it happened during the past century, and a sea strike is considerably more likely than a land strike. But they are enough to give us a rough scale for the problem.

So far I have considered only things on the scale of the Tunguska event, but we know that there have been, at very long intervals, much larger meteor strikes. One famous one about sixty million years back is sometimes referred to as the Dinosaur Killer, on the theory that its effects killed off the dinosaurs. My geologist wife objects to that label on the grounds that lots of other things went extinct at the same time; the technical term is apparently the K-T event. The evidence for several earlier large strikes with less drastic consequences is preserved as astroblemes, geological structures believed to be the result of asteroids hitting the earth. So let’s guess that they occur at a rate of one every sixty million years. We don’t know how many people would be killed by a strike on that scale, but the upper limit is everyone, so use that for a very rough calculation. Dividing about six billion people by about sixty million years gives us a mortality rate of about a hundred people a year.

Here again, my calculations leave a lot out. Mass extinctions on the scale of the K-T event occur at a rate considerably below one every sixty million years; there are fewer of them than there are astroblemes. That suggests that perhaps I should have divided by 300 million or so instead of sixty million. On the other hand, I have not considered events intermediate between the two categories, infrequent enough to be left out of the historical record and small enough to be left out of the fossil record but still bigger than Tunguska and more frequent than K-T. But I think my calculations are sufficient to show that anual mortality due to asteroid strikes is tiny compared to other sources.

One final question is whether annual mortality is all that matters. Perhaps we ought to consider not only individual lives but the survival of our species and our civilization. Seen from that standpoint, if an asteroid strikes really does kill everyone the cost, as evaluated by those presently living, might be considerably larger than the number of lives lost. My own guess is that even something on the scale of the K-T event wouldn't wipe out either our species or our civilization, but I might be wrong.

A correspondent points to the signs currently appearing on the tables of local restaurants, explaining that, because of the water shortage, they will only bring drinking water if you ask for it. The obvious question is whether the amounts involved are large enough to matter. It's straightforward to estimate how much water is being saved per person per year. You can go from that either to an estimate of the size of the local reservoir and the number of people it serves, or the amount of water used for some other purpose, such as watering lawns or flushing toilets, or—with a little searching—to per capita water consumption in the U.S., and compare the numbers.

Readers are invited to suggest further examples of such calculations. They should involve claims that people might make and care about and that can be evaluated without any expert knowledge, using information lots of us already have or can easily find. Bonus points for an example that would work for a twelve-year old.


Anonymous said...

These are also known as "Fermi Problems."


Garg Unzola said...

I don't have a maths background and it seriously stunts my growth in computer science.

Regardless, while trying to wrap my head around Bayesian reasoning I found this site which put it so absurdly simple that I nearly couldn't believe it:

They also have a book that analyses the physics in Hollywood films which might appeal to 12 year olds to start doing back of envelope calculations:

chriscal12 said...

Another important feature of the asteroid example is the level of population over time. We have some evidence on the human population on earth in the past 60 million year interval (0 for most of it), and some evidence suggesting what it will be in the next few decades maybe. But we have no ability to predict population levels in the next few thousand years, let alone 60 million. Wouldn't those numbers affect the calculation dramatically?

Arthur B. said...

Regarding population density, offers a not-too-biased sample of the earth. Humans are pretty much nowhere to be seen.

Anonymous said...

Here's a good book on this topic:

Guesstimation: Solving the World's Problems on the Back of a Cocktail Napkin, by Lawrence Weinstein and John A. Adam. Princeton University Press, 2008. (paper, 301pp) Guesstimation at Amazon

Michael Hartl said...

My Ph.D. advisor, Sterl Phinney, has for many years taught a course on Order-of-Magnitude Physics at Caltech (often with fellow astrophysicist Peter Goldreich). Around ten years ago, one of his doctoral students even got a degree out of it, essentially writing a book inspired by the course for half of his dissertation. Happily, it's available for free online: Order-of-Magnitude Physics by Sanjoy Mahajan. It's rather more advanced than would be appropriate for the elementary or secondary level, but it's definitely in line with the kind of book you envision.

Anonymous said...

"A large house nowadays has 3000 square feet of floor area, typically on two floors, and is occupied by four people, which means 1500/4 square feet of land per person"

Having been involved a bit with zoning issues, the above numbers are significantly out of whack. In the Eastern suburban US, a truly *large* house can easily have over 5000 sq ft of living space, while a quite small one has perhaps 1200. But more importantly, the post assumes that only the house itself occupies land. There is also the surrounding grounds. Large houses are often built on sites of 1.5 to 3 acres or larger, by law (that is: you may not use a smaller lot size in the areas where such houses are built). 25 housing units per acre (generally apartments) is considered a high density, so if you assume 4 people per housing unit, that gives you 100 people per acre for high-density housing, and the average will be significantly loer, and this is in a fairly densly populated area of the US. An average is probably soemthing more like 25 people per acre used exclusively by housing, increasing your estimate by a factor of 8. But none of this affects the main point much.

-David Siegel

Anonymous said...

It occurs to me that it might also be a good topic for a book.

I thought Innumeracy by John Allen Paulos was about this... not that I've gotten around to reading it.

Anonymous said...

The simple order of magnitude rule of thumb for urban population densities is 100 m2/person. It's a pretty good average for most countries and cities over the whole of history. (There are occasional exceptions down to ~10 m2/person, and some extreme suburban sprawl up to ~1000 m2/person.)

Chris Bogart said...

You might like Douglas Hofstadter's essay, "On Number Numbness" (it's in his book, Metamagical Themas). He talks about this problem being particularly acute for really large and really small numbers.

Gary McGath said...

A Tunguska equivalent at sea could be quite nasty, owing to the tsunamis it would generate.

Anonymous said...

It is interesting to apply the reasoning you apply to meteor strikes to major terrorist attacks. These are, of course, very minor killers (annualised) but have a very substantial effect on public policy.

Anonymous said...

Jon Bentley discussed this in "Programming Pearls," in a section titled "The Back of the Envelope" (originally published as a column in the Communications of the ACM). For further reading, he suggested "How to Lie with Statistics" by Darrell Huff and the Paulos book. He related the origin of the "Fermi problem" to a story that Enrico Fermi roughly calculated the yield of the first atomic bomb test by tossing a handful of paper shards into the air when he saw the flash, and measuring how far they were transported by the shock wave.

Bentley noted that responses to the column included some appropriate for children, citing two of the form "How many leaves did we rake this year?" and "How long would it take you to walk to Washington, DC?" When I was at the University of Maryland some years ago, the campus newspaper had an article about two students who accepted a bet that they could walk to Baltimore and back in a day. They lost. Given the distance (about 30 miles, depending on the specific end points), the definition of a "day" (the daylight hours, if I recall correctly) and the likely condition of the students when they accepted the bet, you can probably figure out why.

Geek Vader said...

as far as the asteroid strike - yes, on average, they have a quite modest impact. There's no doubt that we should not be pouring all of our national resources into asteroid collision detection and prevention.

However, we should not treat asteroid strikes as 'no big deal'. They are a heluva big deal if we happen to be around when it hits.

For example, do you have medical insurance? Life insurance? Why? Back of the envelope calculations clearly indicate you'll spend more in premiums than the cost of the health care you're likely receive.

Oh, I get it - you're spending more now, to lessen the impact of catastrophic failure in the future. Interesting.

Anonymous said...

I'm not sure if it would 100% qualify as a "Back of the Envelope" concept since it requires prior knowledge of a simple energy formula, but it's a common thing to do among those with an interest in ballistics.

I began doing this in 8th grade, it involves taking a particular cartridge of bullet, and determining the energy level the projectile has at a given velocity (usually the muzzle velocity, just after the bullet leaves the barrel).

Let's say you're speculating about how much kinetic energy a round will have, you take the velocity squared, multiply it by the bullet's weight in grains, and divide the product by 450240. The result is the foot pounds of energy the bullet has.

So a typical (sometimes police-issued) 147 grain 9x19mm round ( coming out of a Glock pistol would have it's energy level calculated as follows:

Velocity at the muzzle- 1,000 feet per second

Weight in grains- 147

Calculation- 1,000 x 1,000 x 147= 147,000,000/450240= 326.5 Ft/Lbs

It's something I do out of sheer boredom, which isn't uncommon for me.

happyjuggler0 said...

I'm pretty sure that I got my example from a Julian Simon book, probably The Ultimate Resource, but I am not sure. My apologies if it is from elsewhere.

Imagine every single person standing in their own respective 18 inch by 18 inch square. How much space is required to fit them all in?

Answer: They can fit in Atlanta, all on the ground. Imagine how much smaller a space is needed if they were in skyscrapers instead of on the ground.

Simon's point of course is that we aren't "running out of space" due to population growth. Now obviously squares this small mean that you can't move around and actually live, but hopefully it puts in perspective that one aspect of the so called "population problem" is not in fact a problem, nor will it likely ever be a problem. The book explains in more detail all kinds of other aspects of the alleged "population problem", but that is off topic and I won't go there.

Anonymous said...

There is a book about how to do such calculations: Consider a Spherical Cow: A Course in Environmental Problem Solving by John Harte.

Amazon shows a series of sequels to it, also.

Eric Rasmusen said...

My child is actually doing plausibility problems in school (4th grade)! They aren't word problems-- just trying to see whether an arithmetic answer is plausible.

I suspect that statewide multiple choice tests should get the credit for this.

Damien Sullivan said...

I've tried to popularize the acronym/coinage BOTEC or botec, since "back of the envelope calculation" is pretty long and yes, at times I've had to type it a lot.

The "maths background" needed is generally just arithmetic and a feel for magnitudes, not even algebra. Though doing them well takes a fluency with arithmetic and approximation, especially when doing them in one's head -- knowing when to round off, and which way to do so.

But the housing botec is a good example of going horribly wrong. I'm in an unusual position, since I know off-hand that Manhattan's density is 65,000/mi2, NYC's is 25,000, and "urban" for the US can mean 5000 or less, maybe even 1000 though I'm not sure of that bit. So that 100,000 figure trips flags for me as being off by potentially 100 -- meaning 1/10th of land usage, not 1/1000th. And on average it's probably wrong by 10, so 1% of land usage.

Which might not seem like much, except that concern is largely about *new* land usage, suburbs and exurbs and such, which are lower density and often indeed being built right on good farmland -- which itself is only a fraction of the total US land. (E.g. most of the Rockies and of Alaska are suitable for neither agriculture nor efficient human habitation.)

So the botec itself is wrong, and the issue itself may be more complicated than trivial botecs can handle.

Damien Sullivan said...

Botecs I like to do involve space colony construction, energy and water sustainability, and the average power of earthquakes and volcanoes. (The fact that human power usage is about half of total geothermal heat flux is a matter of simply looking it up.)

volcano: 1 km3 of rock launched say 10km into the air. That's 3e12 kg launched 1e4 meters, plugged into mgh for 3e17 Joules of potential energy. If the volcano can manage to go off like that once a century then the average power is 1e8 Joules/second -- 100 megawatts. A big urban powerplant is a gigawatt.
Related figure: the power provided by all that molten rock. Assume 1000 K temperature difference and specific heat of 1 J/K-gram, and get 3e18 J. Higher but still tame relative to modern civilization.

Yellowstone caldera: covered the Western US in 1.5 feet of ash 640,000 years ago (facts from geology field trip). Say half the US (5 million km2) one million years ago. Call it a meter of solid rock, which is overkill. 5e11 m3 of rock (500 km3; Wikipedia says 1000, which is a tip for me to Google and check my memories; turns out I was using the Bishop ash bed, from Long Valley caldera and only the third biggest eruption), or 15e14 kg. Upgrade that to 3e15 kg, in light of the parenthesis.

By the same calculations, 3e15 kg launched 10 km up had 3e20 J, and molten had 3e21 J. Over a million years (3e13 seconds) that gives power rates of 1e7 to 1e8 Watts. Squish it to 100,000 years and you still just hit a gigawatt.

At the time of the Indonesian tsunami I did similar though much more data-poor calculations for big earthquakes (along the lines of 1000x1000x1 km block going up a meter) and got similar numbers; the lesson I take away is that in sheer energy magnitudes, we're quite capable of wrestling with the geophysical cycle, even if we're not sure how. (I envision tapping magma chambers, to release lava and toxic gas in controlled manner, or at worse to set off volcanoes at controlled times with everyone in safe conditions. I can't envision much I believe in for dealing with earthquakes.)

Garg Unzola said...

My one varsity mate was incredibly obsessed with football. He started doing BOTEC to work out the odds of teams winning. Eventually he started his own gambling ring. He's currently involved with a community football team, because he's effectively retired from all the money he'd made.

He never studied maths. Shows all you need to get ahead in life are indecent obsessions and a large supply of envelopes with lots of space on the back.

Anonymous said...

Here's the problem: Are the UFOs real?

In my teen years I remember I read a book (by a Russian author, methinks) that speculated on whether or not we, the Terrans, are the only intelligent life-form in the Universe. It concluded that we may not be the only ones, but --at the same time-- that is very likely that in the "Surrounding" Universe (that is, the one that we can ever hope to reach by space traveling) there's no other life-form sufficiently human-compatible so that to interfere with them in any meaningful way.

(Human-compatibility was defined as being able to sustain life in certain temperatures, light spectrum and such other conditions that are necessary for human life.)

So, David: can you do a back of the envelope speculation on the question posted at the top of this comment?

TheVidra said...

Your estimation is useful for developing a quick big picture idea about a subject; however, hidden costs should also be taken into account. For example, one estimates that a terrorist attack kills (on average) very few people per year, directly - and thus our society pays more attention to such an event, at the expense of a much more serious peril (drunk driving, heart disease, etc).

However, a terrorist attack can kill or debilitate more people than one can estimate on the back of an envelope (based on the statistics of the attack themselves): for example, imagine the people going to emergency rooms for non-life threatening anxiety-related causes. People with severe injuries (not related to the attack) would suffer longer wait times and die as a result. Foreign policy can change due to a terrorist attack, resulting in war (perhaps) - killing many more people than the attack itself. People's behavior could change - for example, more people could switch to driving as opposed to taking public transportation (out of fear) - resulting in more clogged roads, and more accidents (or more ambulance and fire truck delays). These are just some examples of unforeseen yet lethal effects of a terrorist attack, which I am not sure one can estimate with a degree of accuracy. Even post-fact analysis could be tricky (and based on guesstimation).