Quantity and Measurement
The importance of quantityIodine is a deadly poison. Iodine is an essential nutrient. The key is amount of iodine. According to the Merck Corp website, 20 micrograms (one fifty thousandth of a gram) daily is needed for good health. At one hundred times that amount, one five hundredth of a gram, it starts to get toxic. So if somebody tells you "There's iodine in that drinking water," it isn't very useful information unless you know how much. One amount may be essential your health while another amount may be deadly. Very often, information isn't much good unless there is some measurement associated with it.
Unfortunately, it's common for people to act as if some amount of something is important without regard to how big that amount is.
After shopping at a store I was told that if I filled out a questionnaire at their web site I would be eligible for a chance to win a $100 gift certificate. That sounds nice, but how nice depends on how big the chance is. If it’s one chance in ten, then on the average I make $10, which would probably be worth spending 15 minutes on my computer. On the other hand, if the chances were one in 10,000, I’d only be getting an average of one cent – effectively a complete waste of my time. Since the store doesn’t want to give money away unnecessarily and people don’t ask how likely they are to get the $100, the store will probably not be giving much to the average responder.
The fact that there is a “chance” of something happening tells us very close to nothing at all. If we’re going let the chance of something happening influence a decision we make, we ought to have some idea of what that chance is.
During a recent drought in my area I saw the comment that the value of water was like gold. The economy had no doubt suffered many millions of dollars in losses because of lack of water. So, was water like gold? Gold costs close to $1000 an ounce. Water provided by my city water system is less than a hundredth of a cent per ounce. That would make gold ten million times more valuable. What’s the problem? The problem is that we take for granted that huge amounts of water are available. If an area 100 miles by 100 miles gets twelve inches of rainfall less than average in a year, that’s a shortage of 279 billion cubic feet, or more than two trillion gallons. At a penny a gallon (about what large scale users pay where I live) that would amount to twenty billion dollars. So even though water is actually very cheap by the ounce or gallon, it still becomes valuable when you’re talking about trillions of gallons.
The point is that saying “water is valuable” is close to meaningless unless we specify some quantity. Even in a drought, wasting a few ounces of water is unimportant - something we wouldn’t say about a few ounces of gold. When billions of gallons are involved, it becomes very important.
Accuracy of Measurement
All measurements, excluding things that are actually counted, have some degree of inaccuracy. If we measure the length of a board using a tape measure, the marks on the tape won’t usually be closer than a sixteenth of an inch or a millimeter, and the lines making the marks have some width, so there could be a slight variation even if the end of the board seemed to line up perfectly with the mark. The board itself may vary in length with humidity and temperature and where on the board the measurement is made. If we buy an eight foot board, we shouldn’t expect it to be the correct length to within less than a sixteenth of an inch, and for most purposes a tiny variation won’t matter. We take for granted that there will be some error in any measurement, and this isn’t a problem unless we have some job that requires an unusually precise length.
We expect some degree of error in all kinds of measurements. The distance between two cities might be off by several miles without causing concern. When the doctor weighs me he’s off by several pounds since he does it with my clothes and shoes on. When timing a runner with a stopwatch or measuring ingredients for a cake or the temperature of a fever, we assume the amount or number we get will not be exact.
When doing measurements for ourselves, we get along fine because we know what kind of accuracy we need, but if we are using measurements made by other people, we may have a problem since we don’t know how accurate their figures are. This can be very important in a large engineering project or when gathering data that’s to be published for other people to use.
Scientists and engineers and pollsters have ways of indicating how accurate their measurements are. Sometimes they include explicit error bounds. This is very common in public opinion polls. “Smith and Jones are effectively tied in the latest poll,” we might read. “47% favored Smith and 44% favored Jones, but there is a margin of error of 4%.”
Of course the people providing the figures don’t know the actual amount of error. If they did they could just use it to modify the figures and publish the correct numbers. Instead they calculate the typical error based on the number of people in the poll. If only a small number of people are questioned, it wouldn’t be unusual that by luck most of them would be on one side even if in reality more people were on the other side. If a huge number were asked, it would be surprising if the percentage on one side was very different from the percentage of the population as a whole.
In other cases scientists base error estimates on an awareness of the accuracy of the tests they make. If they are using carbon dating to find the age of a piece of cloth and their calculation tell them it is 982 years old, they might know from experience that various unknown factors might make it seem as much as 50 years younger or older, so they would report the age as between 932 and 1032 years old. In fact they might say it was between 930 and 1030, since the last digit isn’t very important.
In many situations, error bounds aren’t given specifically, but there is a general convention not to report digits that aren’t likely to convey information. Suppose we measure a tree and find it has a diameter of 18 inches. If we want to know the distance around the tree, the circumference, we would multiply 18 by pi. Using 3.1416 for pi, we get 56.5488 inches. This is pretty silly, since the original measurement of 18 might easily have been half an inch off, and a tree isn’t a perfect circle anyway. Even the first digit after the decimal point is in reality as likely to be any other digit as it would be a 5 if we actually measured the distance around the tree. The other decimal places are completely meaningless. When engineers make measurements, they have a standard practice of not reporting any digits that aren’t meaningful. In this case a reasonable figure to state would be 57 inches, rounding to the nearest inch. An actual measurement might give 56 or 58, but 57 is more meaningful than rounding to 50 or 60. On the other hand including the first decimal place, 56.5, is pointless since even the 6 in front of the decimal is in doubt.
The general rule when seeing a number is that the last digit reported may not be the correct one, but it’s close. Providing more digits would be useless. When whole numbers wouldn’t add useful information, then the number is rounded with zeros. If we’re told the dinosaurs died out 65,000,000 years ago, we’d be foolish to assume that the actual number is exactly 65 million, or that it’s even between 64,900,000 and 65,100,000. Once the zeros start, the real values are unlikely to be known. It is reasonable to assume scientists think the number is closer to 65 million that it is to 66 or 64 million.
Sometimes we may think we can’t think in terms of quantities because we don’t have access to good measurements. Often all we need is a ballpark estimate. Suppose a friend has a nice boat and we wonder if we’d like to get one like it, but we don’t know anything about what boats cost. “How much does a boat like that cost?” we might ask.
“I couldn’t really say. I got this five years ago and the prices have surely changed.”
“Give me a ballpark estimate,” we might ask.
“Oh, maybe between 30 and 60 thousand,” the friend might say. Now this isn’t a very accurate amount, with the maximum being twice the minimum, but if we only could afford $10,000, that’s all we’d need to know.
One might think that a measurement that could be off by as much as five billion years wouldn’t be very useful. But that was the situation a few years ago concerning the estimated age of the universe – it was commonly estimated that the universe was between 10 and 20 billion years old. So we could say it was 15 billion years old plus or minus 5 billion years. For an astrophysicist trying to figure out how galaxies were formed, this still provides very useful information. Any theory that required more than 20 million years could be thrown out.
If someone says “I’m not sure how big it is – I might be off by a factor of 10,” our first reaction would be to think their information was entirely useless. However there are many cases where this is the best we can do, and it’s a lot better than nothing.