Measurement is a most important part of the sciences. Not only do scientists use measuring instruments of many kinds, they also have been in the forefront of developing new measuring tools and standards.
SI Units
While some countries continue to use their own customary measuring units (in the United States we use pounds, feet, gallons, and so on), the International System of Units (SI Units, from the French Systéme International) has been adopted worldwide for both commercial and scientific purposes. The basic SI units include the following:
- Meter: length or distance
- Kilogram: mass, more commonly called "weight"
- Second: time
- Ampere: electric current
- Candela: light and other radiation
- Mole: comparing other substances to the molecular weight of a certain form of carbon
- Kelvin: temperature
From these basic units come many "derived units." An example is the measurement of area, for which the unit, called the square meter, is derived from the basic length unit or meter. Another example is the measurement of velocity, commonly called "speed" for which the unit, called meters per second, is measured from the basic meter traveled.
How to Measure
To measure something usually means taking a widely accepted tool, such as a meter stick, and comparing it with the thing to be measured. Practical measuring methods abound, ranging from astronomers' methods for estimating the Earth's distance in light-years from celestial objects to micrometers and other tools used for measuring very small objects. Scientists are even finding ways to measure the sizes of atomic particles, their "spin," and other characteristics.
For this discussion of measurement, the metric system will be used in all examples. If you have a meter stick at hand for reference it will be useful. If you do not have one, the commonly available foot ruler with a metric scale along one side (about 30 centimeters, each divided into tenths or millimeters) will do.
Let's say that you want to measure the width of a room with a meterstick. You agree to measure it to the nearest millimeter (or 0.001 meter). If you are not familiar with measuring in the metric system, note that a millimeter is equal to roughly 1/32-inch.
If the room you are measuring has a hard floor, you can make a pencil mark at the end of the meter stick each time you lay it out across the room. Or, if there is a carpet, you can stick a pin in it to mark the end of the stick. Let's say that you find there are five whole meters and a part of a meter that looks like figure 10.1 at the arrow point. This would read 5.823 meters.
Multiple Measurements
But one measurement is not enough if you intend to understand the problems of measurement. You should try to get an independent second measurement. To do this, have a friend make a measurement without telling him or her ahead of time what result you got.
Suppose your friend measures the room using the same tools and methods and gets a result of 5.834.You now have two measurements: 5.823 and 5.834 meters. Which is right? No one knows. What if you measure it again? Good, but would that prove to be the correct measurement any more than the first two? Not likely. No one can know the "true" measurement. We must face the problem that there is no perfect way to measure it. Two people will probably not measure anything, even the width of a room, exactly alike. Even the same person is not likely to get the same measurement twice, especially if one allows enough time between measurements to forget the first measurement.
Just as important, no two meter sticks or other measuring instruments are exactly alike, either. There is no such thing as a perfect measuring instrument, just as there is no perfect measurement.
To get as good a measurement as is reasonably possible of the width of the room, it is necessary to take several independent measurements. Let's say you end up with five different values: (1) 5.823, (2) 5.834, (3) 5.829, (4) 5.830, and (5) 5.825. Now we have a statistical question: Which one shall we choose to be the width of the room? If we take the arithmetical average, or mean, we get 5.828 meters. This average, we see, is not anyone of the values we got by actual measurement. Is it the "correct" measurement? All we can say is that it is probably very close to it.
If you were measuring the diameter of a marble with a micrometer, you would find the same problems. Several independent measurements would probably give you several different values. Again, no one would know which was the correct diameter. Once again, there would be a statistical problem in choosing a number to represent the diameter, the "true" diameter, which no one can know.
Scientists have understood this uncertainty about measuring for a long time. It seems to surprise others, however. Some people react to this discovery by saying, "Well, if I can't really know what the measurement is, why bother trying to make it exact?" The reason is that scientists and others who keep working out better ways to measure things are trying to communicate better. Reporting scientific findings to others is an important part of scientific method, as you know, and communication among scientists has been helped immeasurably by better and better systems of measurement.
Dealing with Uncertainty in Measurement
How do we deal with this uncertainty, besides pretending it does not exist? In the most ordinary measuring of things, one reasonably careful measurement is all that is needed. Most people will settle for that, believing that they know the length of a thing, or its time, or its weight, and so forth. However, many scientific and technical workers must work as nearly as they can to the limits of the accuracy of their instruments and methods. They must also report to others what those limits of accuracy are. That is why there is need for ways to express these limits.
Here is one method: Let's say that you are measuring across a piece of paper with a scale such as a ruler divided into centimeters (cm) and tenths of a centimeter (or millimeter). You want to express your measurement as centimeters to the nearest tenth, such as 23.7 cm. (See figure 10.2.)
As you measure, you will see that the edge of the paper is not quite on one of the marks showing tenths, so pick the tenth mark that the edge is nearest to. That is, pick 23.7, not 23.6. (See figure 10.3.)
However, you do not want anyone to think that you are claiming that the paper measured exactly 23.7 cm. Instead, you would like to tell others that you are reading the scale to within one-half of a division either way from a scale mark. (See figure 10.4.)
One-half of one-tenth is 0.05 in decimal notation. This is why you put "± 0.05 cm" after your measurement figure. For example: 23.7 ± 0.05 cm (which reads, "23.7 plus or minus five-hundredths of a centimeter").
For another method of expressing uncertainty, let's go back to the measurement of the width of a room. There we made five measurements and reported the mean as 5.828 meters. If we had wanted to express the same measurement in millimeters (mm) instead of in meters, we could have reported 5,828 mm. Remember, 5,828 mm is the average of five measurements of which the shortest was 5,823 mm and the longest was 5,834 mm. This spread from lowest to highest we call the "range" of the 5 measures. We express the range as 11 mm by subtracting the shortest measurement from the longest.
Our mean of 5,828 mm is 6 mm below the top of the range and 5 mm above the bottom. We may also want people to know the range when we report the mean of several measurements. This is recorded as: 5,828 ± 6 mm. Or, we could report it in its meter unit: 5.828 ± 0.006 m.
This system, using the mean to express the range, is an overly simplified explanation of the method, but it does show how the system works.
There is yet another way to express the uncertainty of measurements. As a simple example, let's say that you are measuring a rod with a meter stick and find that it is just under 1 meter. That is reported as 0.999 m (or 999 mm) because you wish to report that the measurement is nearer to 0.999 m than to 1.000 m (or nearer to 999 mm than to 1,000 mm). You are saying that the range is within 1 mm out of about 1,000 mm. Therefore, you report your measurement as 0.999 m with an "error" of 1 part per 1,000. The term "error" is not used in this method to mean that you made a mistake; it is another way of expressing the usual inaccuracy of any measurement.
Suppose, however, that you take the same meter stick and measure a room that is close to 10 m (or 10,000 mm) across. You find that several measurements range from 9,995 to 10,005, or a net range of 10 mm. This, too, shows an error of about 10 parts in 10,000 or 1 part in 1,000, as in the previous example.
To say that these measurements are accurate to 1 part in 1,000, we must assume that your meter stick is "exactly" 1 meter long. Of course, we cannot claim that because we are working on the assumption that no two things are exactly alike (including your meter stick and some master meter stick).
All measuring instruments, including wooden meter sticks, do change from their original dimensions. This unstable nature of things is part of our measurement problem. The changes that come with variations in temperature, humidity, and other factors must always be expected as part of our measuring and should be allowed for in our reporting if we are trying for high accuracy. Scientists working on one of the most important measuring problems, the measurement of the speed of light, using the best equipment and methods they can devise, must still allow for the uncertainty of their measurements. Thus, they report the velocity of light in a vacuum as 299,792,456 ± 1.1 meters per second.
We have discussed three different ways of reporting the uncertainty of measurements: (1) Reference to one-half of the smallest scale division; (2) Reference to a range of several values above and below the mean of those values; (3) The error in parts per 1,000, parts per million, or the like. Now that you better understand how the "pros" work with uncertainty in measurement, you may be able to handle it fairly easily while you do your own science investigations.
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