It is the mark of an instructed mind to rest assured with that degree of precision that the nature of the subject admits, and not to seek exactness when only an approximation of the truth is possible.- Aristotle
Estimation is one of the most useful skills a science student can use in everyday life. With some basic facts, a diligent person with good estimation skills can compute some amazing things. Scientists are forever doing "back-of-the-envelope" calculations to see if an idea is worth pursuing. You can find answers to questions that are really market research questions like "How many bicycles are sold in the US each year?" or one I looked at before writing this book, "How many science teachers are there in the U.S.?"
There are about 300 million people in the U.S. (this is a fairly common statistic). To get at the number of science teachers, consider that science teachers exist principally to teach 6 to 18 year olds. One route to the answer then would be to first estimate the number of 6 to 18 year olds. Try dividing the population into eighteen year groups and then making educated guesses of the fraction within each group:
|
Age Range |
Est. Fraction |
|
0 - 18 |
20% |
|
19 - 36 |
20% |
|
37 - 54 |
20% |
|
55 - 72 |
20% |
|
72 -90+ |
20% |
The chart at left is obviously a wild guess, but not a stupid one (it could certainly be massaged or even measured by simply counting people in a community). Calculating gives us:
Note that none of the "input" numbers are very accurate, so we always round to about one or two significant digits. These estimates are called "order-of-magnitude" estimates and are just that, good to about a factor of three or so (A factor of three above and a factor of three below makes a total range of 9, which is about 10, which is one order of magnitude). The author checked this particular estimate through a National Science Teachers Association report that there are 307,000 science teachers in the U.S. The estimate above would have hit this "right on" had we guessed 5 teachers per science teacher, for example. The point is, with a reasonable "educated guesses" or estimates, one can get a useful result.
Consider the more generally interesting example, "How much does a new car cost to drive each mile averaged over its entire lifetime?" Suppose your car costs $20,000 initially and lasts 10 years. Each year the car is driven 10,000 miles. This means that the car's total useful life is 100,000 miles, therefore:
20,000 ($/car) * (1car/100,000 miles) = $0.20 per mile
If the car is salvaged for $3000, then the actual "capital" cost was only $17,000 and so the capital cost is only $0.17 per mile.
Now suppose that gasoline costs $1.30 per gallon and the car gets 25 miles per gallon, then (since 1 gallon "equals" 25 miles in an extended sense of the term "equal"):
100,000 miles * (1 gallon/25 miles ) = 4,000 gallons of gasoline.
4000 gallons gas * ($1.30/gallon ) = $5200 for gasoline over the entire life of the car. Finally,
5200 ($/car) * (1car/100,000 miles) = $0.052 per mile or about 5 cents per mile.
Next, the maintenance on the car is not free. Suppose that it costs $20 per oil change every 3000 miles.
$20/3000 miles = $0.007 per mile.
Finally, licensing and tags cost about $150 to $300 per year and insurance costs another $500 to $1500 per year. Roughly, then
10 years/car *( ($200+ $1000)/year ) = 12000 $/car, so
12000 ($/car) * (1 car/ 100,000 miles) = $0.120 per mile.
Finally, adding all these costs together (capital, gasoline, maintenance, legal) gives a total average cost per mile of
$0.170 + $0.052 + $0.007 +$0.120 = $0.349 per mile or about 35 cents per mile. It is very interesting that only about one seventh of this is gasoline cost (even at $1.30 per gallon)! This calculation can be compared to a "correct" answer as well. The 1995 U.S. Tax Code allows $0.30 per mile for business use of an automobile.
Estimating requires practice. My father and I used to have contests in the car trying to estimate things like "How tall is that telephone pole?" or "How long or how wide do you suppose those lines on the highway are?" As we could, we would check our answers later by measuring. My father became very good at this, he can guess a person’s weight within a pound or two every time (and he gets to check his estimates because he is a physician). Checking your estimates is a good follow-up procedure for several reasons; it gives builds self-confidence and helps steer future estimation.
Zero / Infinity Analysis:
In addition to estimating, a very useful skill and technique is what a friend of mine calls a "zero-infinity analysis." This is a technique for determining whether you got an estimate right or not. For example, if my tires are too big, will that cause my speedometer to read too high or too low? The technique is to imagine extreme cases. If I imagine VERY large tires, so big that they go around only once each mile, then what would my speedometer read when traveling sixty miles per hour. At 60 mph, a mile takes one minute and my VERY large tires rotate just once. Consequently my speedometer would indicate a very low value. If, on the other hand, I imagine very small tires. They would have to spin much faster to keep up with the normal sized tires; hence, the speedometer would read too high. The conclusion then is that oversized tires cause the speedometer reading to be too low or, when the indicated speed is one thing, the actual speed is higher. Albert Einstein did a lot of this kind of "experimentation" – imagining experiments in his head –gedanken experiments.