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The exact distance between the upper lip and the tip of the dorsal fin will forever be hidden in a fog of uncertainty. The angle at which we hold the calipers and the force with which we close them on the object will never be exactly reproducible. A more fundamental limitation occurs whenever we try to compare a continuously-varying quantity such as distance with the fixed invervals on a measuring scale; between 59 and 60 mils there is the same infinity of distances that exists between 59 and 60 miles!

The "true value" of a measured quantity, if it exists at all, will always elude us; the best we can do is learn how to make meaningful use (and to avoid mis-use!) of the numbers we read off of our measuring devices.

Image by Stephen Winsor;
used with permission of the artist.

Uncertainty is certain!

In science, there are numbers and there are "numbers". What we ordinarily think of as a "number" and will refer to here as a pure number is just that: an expression of a precise value. The first of these you ever learned were the counting numbers, or integers; later on, you were introduced to the decimal numbers, and the rational numbers, which include numbers such as 1/3 and π (pi) that cannot be expressed as exact decimal values.

The other kind of numeric quantity that we encounter in the natural sciences is a measured value of something– the length or weight of an object, the volume of a fluid, or perhaps the reading on an instrument. Although we express these values numerically, it would be a mistake to regard them as the kind of pure numbers described above.

Confusing? Suppose our instrument has an indicator such as you see here. The pointer moves up and down so as to display the measured value on this scale. What number would you write in your notebook when recording this measurement? Clearly, the value is somewhere between 130 and 140 on the scale, but the graduations enable us to be more exact and place the value beteween 134 and 135. The indicator points more closely to the latter value, and we can go one more step by estimating the value as perhaps 134.8, so this is the value you would report for this measurement.

Now here’s the important thing to understand: although “134.8” is itself a number, the quantity we are measuring is almost certainly not 134.8— at least, not exactly. The reason is obvious if you note that the instrument scale is such that we are barely able to distinguish between 134.7, 134.8, and 134.9. In reporting the value 134.8 we are effectively saying that the value is probably somewhere with the range 134.75 to 134.85. In other words, there is an uncertainty of ±0.05 unit in our measurement.

All measurements of quantities that can assume a continuous range of values (lengths, masses, volumes, etc.) consist of two parts: the reported value itself (never an exactly known number), and the uncertainty associated with the measurement.

Scatter and error in measured values

All measurements are subject to error which contributes to the uncertainty of the result. By “error”, we do not mean just outright mistakes, such as incorrect use of an instrument or failure to read a scale properly; although such gross errors do sometimes happen, they usually yield results that are sufficiently unexpected to call attention to themselves.

#### Random error

When you measure a volume or weight, you observe a reading on a scale of some kind, such as the one illustrated just above. Scales, by their very nature, are limited to fixed increments of value, indicated by the division marks. The actual quantities we are measuring, in contrast, can vary continuously, so there is an inherent limitation in how finely we can discriminate between two values that fall between the marked divisions of the measuring scale. The same problem remains if we substitute an instrument with a digital display; there will always be some point at which some value that lies between the two smallest divisions must arbitrarily toggle between two numbers on the readout display. This introduces an element of randomness into the value we observe, even if the "true" value remains unchanged.

The more sensitive the measuring instrument, the less likely it is that two successive measurements of the same sample will yield identical results. In the example we discussed above, distinguishing between the values 134.8 and 134.9 may be too difficult to do in a consistent way, so two independent observers may record different values even when viewing the same reading. Each measurement is also influenced by a myriad of minor events, such as building vibrations, electrical fluctuations, motions of the air, and friction in any moving parts of the instrument. These tiny influences consititute a kind of "noise" that also has a random character. Whether we are conscious of it or not, all measured values contain an element of random error.

#### Systematic error

Suppose that you weigh yourself on a bathroom scale, not noticing that the dial reads “1.5 kg” even before you have placed your weight on it. Similarly, you might use an old ruler with a worn-down end to measure the length of a piece of wood. In both of these examples, all subsequent measurements, either of the same object or of different ones, will be off by a constant amount. Unlike random error, which is impossible to eliminate, these systematic errors are usually quite easy to avoid or compensate for, but only by a conscious effort in the conduct of the observation, usually by proper zeroing and calibration of the measuring instrument. However, once systematic error has found its way into the data, it is can be very hard to detect.

### More than one answer in replicate measurements

If you wish to measure your height to the nearest centimetre or inch, or the volume of a liquid cooking ingredient to the nearst “cup”, you can probably do so without having to worry about random error. The error will still be present, but its magnitude will be such a small fraction of the value that it will not be detected. Thus random error is not something we worry about too much in our daily lives.

If we are making scientific observations, however, we need to be more careful, particularly if we are trying to exploit the full sensitivity of our measuring instruments in order to achieve a result that is as reliable as possible. If we are measuring a directly observable quantity such as the weight or volume of an object, then a single measurement, carefully done and reported to a precision that is consistent with that of the measuring instrument, will usually be sufficient.

More commonly, however, we are called upon to find the value of some quantity whose determination depends on several other measured values, each of which is subject to its own sources of error. Consider a common laboratory experiment in which you must determine the percentage of acid in a sample of vinegar by observing the volume of sodium hydroxide solution required to neutralize a given volume of the vinegar. You carry out the experiment and obtain a value. Just to be on the safe side, you repeat the procedure on another identical sample from the same bottle of vinegar. If you have actually done this in the laboratory, you will know it is highly unlikely that the second trial will yield the same result as the first. In fact, if you run a number of replicate (that is, identical in every way) determinations, you will probably obtain a scatter of results.

To understand why, consider all the individual measurements that go into each determination; the volume of the vinegar sample, your judgement of the point at which the vinegar is neutralized, and the volume of solution used to reach this point. And how accurately do you know the concentration of the sodium hydroxide solution, which was made up by dissolving a measured weight of the solid in water and then adding more water until the solution reaches some measured volume. Each of these many observations is subject to random error; because such errors are random, they can occasionally cancel out, but for most trials we will not be so lucky– hence the scatter in the results.

A similar difficulty arises when we need to determine some quantity that describes a collection of objects. For example, a pharmaceutical researcher will need to determine the time required for half of a standard dose of a certain drug to be eliminated by the body, or a manufacturer of light bulbs might want to know how many hours a certain type of light bulb will operate before it burns out. In these cases a value for any individual sample can be determined easily enough, but since no two samples (patients or light bulbs) are identical, we are compelled to repeat the same measurement on multiple samples, and once again, are faced with a scattering of results.

As a final example, suppose that you wish to determine the diameter of a certain type of coin. You make one measurement and record the results. If you then make a similar measurement along a different cross-section of the coin, you will likely get a different result. The same thing will happen if you make successive measurements on other coins of the same kind.

Here we are faced with two kinds of problems. First, there is the inherent limitation of the measuring device: we can never reliably measure more finely than the marked divisions on the ruler. Secondly, we cannot assume that the coin is perfectly circular; careful inspection will likely reveal some distortion resulting from a slight imperfection in the manufacturing process. In these cases, it turns out that there is no single, true value of either quantity we are trying to measure.

### The mean and its meaning

When we obtain more than one result for a given measurement (either made repeatedly on a single sample, or more commonly, on different samples), the simplest procedure is to report the mean, or average value. The mean is defined mathematically as the sum of the values, divided by the number of measurements:

If you are not familiar with this notation, don’t let it scare you! Take a moment to see how it expresses the previous sentence; if there are n measurements, each yielding a value xI, then we sum over all i and divide by n to get the mean value xm. For example, if there are only two measurements, x1 and x1, then the mean is (x1 + x2)/2.

Problem Example

Calculate the mean value of the set of eight measurements illustrated here.

Solution:

There are eight data points (10.4 was found in three trials, 10.5 in two), so n=8. The mean is

(10.2+10.3+(3 x 10.4) + 10.5+10.5+10.8)/8 = 10.4.

Accuracy and precision

We tend to use these two terms interchangeably in our ordinary conversation, but in the context of scientific measurement, they have very different meanings:

Accuracy refers to how closely the measured value of a quantity corresponds to its “true” value.

Precision expresses the degree of reproducibility, or agreement between repeated measurements.

Accuracy, of course, is the goal we strive for in scientific measurements. Unfortunately, however, there is no obvious way of knowing how closely we have achieved it; the “true” value, whether it be of a well-defined quantity such as the mass of a particular object, or an average that pertains to a collection of objects, can never be known– and thus we can never recognize it if we are fortunate enough to find it.

Thus we cannot distinguish between the four scenarios illustrated above by simply examining the results of the two measurements. We can, however, judge the precision of the results, and then apply simple statistics to estimate how closely the mean value is likely to reflect the true value in the absence of systematic error.

You would not want to predict the outcome of the next election on the basis of interviews with only two or three voters; you would want a sample of ten to twenty at a minimum, and if the election is an important national one, a fair sample would require hundreds to thousands of people distributed over the entire geographic area and representing a variety of socio-economic groups. Similarly, you would want to test a large number of light bulbs in order to estimate the mean lifetime of bulbs of that type. Statistical theory tells us that the more samples we have, the greater will be the chance that the mean of the results will correspond to the “true” value, which in this case would be the mean obtained if samples could be taken from the entire population (of people or of light bulbs.)

This point can be better appreciated by examining the two sets of data shown here. The set on the left consists of only three points (shown in orange), and gives a mean that is quite far removed from the "true" value, which is arbitrarily chosen for this example.

In the data set on the right, composed of nine measurements, the deviation of the mean from the true value is much smaller.

Deviation of the mean from the "true value" becomes smaller when more measurements are made.

Absolute and relative uncertainty

If you weigh out 74.1 mg of a solid sample on a laboratory balance that is accurate to within 0.1 milligram, then the actual weight of the sample is likely to fall somewhere in the range of 74.0 to 74.2 mg; the absolute uncertainty in the weight you observe is 0.2 mg, or ±0.1 mg. If you use the same balance to weigh out 3.2914 g of another sample, the actual weight is between 3.2913 g and 3.2915 g, and the absolute uncertainty is still ±0.1 mg.

Although the absolute uncertainties in these two examples are identical, we would probably consider the second measurement to be more precise because the uncertainty is a smaller fraction of the measured value. The relative uncertainties of the two results would be

0.2 ÷ 74.1 = 0.0027 (about 3 parts in 1000 (PPT), or 0.3%)

0.0002 ÷ 3.2913 = 0.000084 (about 0.8 PPT , or 0.008 %)

Relative uncertainties are widely used to express the reliability of measurements, even those for a single observation, in which case the uncertainty is that of the measuring device. Relative uncertainties can be expressed as parts per hundred (percent), per thousand (PPT), per million, (PPM), and so on.

What you should be able to do

Make sure you thoroughly understand the following essential ideas which have been presented above. It is especially imortant that you know the precise meanings of all the highlighted terms in the context of this topic.

• Give an example of a measured numerical value, and explain what distinguishes it from a "pure" number.
• Give examples of random and systematic errors in measurements.
• Find the mean value of a series of similar measurements.
• State the principal factors that affect the difference between the mean value of a series of measurements, and the "true value" of the quantity being measured.
• Calculate the absolute and relative precisions of a given measurement, and explain why the latter is generally more useful.
• Distinguish between the accuracy and the precision of a measured value, and on the roles of random and systematic error.

Concept Map

You can download a pdf document suitable for viewing or printing that contains
all five sections of the "Matter-and-measure" unit.