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.