Expressing a Number and its Uncertainty

Reporting the Best Representation of a Series of Measurements

Scientific experimentation frequently involves making measurements with some kind of instrument, and this most often means measuring something to as high a precision and accuracy as is necessary to draw valid conclusions from an experiment. Precision refers roughly to how tightly clustered a sequence of measurements is. Accuracy refers to how closely the measurements agree with the actual value of the quantity being measured. It is possible to have high precision and poor accuracy. An analogy would be a tightly clustered bunch of bullet holes way off to one side of a target. High accuracy and poor precision would be a analogous to target peppered all over with holes, but with the center of the distribution of holes in the bull’s eye. Obviously one would like to have as much of both precision and accuracy as possible.

It is an experimental fact that, as the measurement scale of an instrument gets finer, there is eventually a point at which several measurements of a given quantity (a length, for example) will give different values. The question then arises, which value should be reported? The most frequently occurring value (the mode)? The one that has half of the values above it and half below (the median)? The arithmetic average or mean? There are actually good reasons for giving any of these, but the context in which the measurement is made and the kind of information desired determines which is used. These numbers are often referred to as measures of central tendency because they each represent a value about which the set of measurements tend to cluster. When using instruments for measuring physical quantities such as length, mass, volts, time, etc., it is usually the mean that is reported. However, along with this mean, there is reported another number that expresses the uncertainty in the measurement. It is this additional number that gives a great deal of information about the quality of the measurements made during an experiment, and it allows an experimenter to judge whether or not the measurements are adequate for drawing conclusions from an experiment. We will discuss how this is done shortly, but first let’s consider the arguments for reporting the mean value.

Suppose we take N measurements of a quantity, and have before us N numbers that are different from each other. Let’s call these N measurements . We are looking for a single number that best represents these N numbers. Whatever value we choose to report and however we choose to calculate it, let’s call this number Q. Now we can subtract this number, Q, from each of the N measurements to form what are called deviations (sometimes called residuals). Thus we obtain a set of numbers . Some of these might be positive, some might be negative, and some zero. If we square each of these deviations, then we will have a set of numbers that are greater than or equal to zero. The sum of the squares of the deviations is a measure of how much the individual measurements are spread about the number, Q. If this sum is large, the measurements are spread out and quite different from each other. If it is small, the measurements must be tending to cluster more closely to each other. Let’s call the sum of the squares of the deviations S, that is

or, more compactly,

We now ask, “What is the number, Q, that minimizes the sum of the squares of the deviations?” What we are suggesting here is that the number that best represents this set of measurements is, in effect, the one about which the measurements are most tightly clustered. Finding the number, Q, that minimizes the sum, S, is a calculus exercise. If you have not had calculus yet, don’t worry about the following calculation; it is the result of the calculation that is important.

To minimize S with respect to the variable Q, we simply take the derivative of S with respect to Q, set this derivative equal to zero, and solve for Q. Therefore

From the last line we find that

What we have shown is that the number, Q, that minimizes the sum of the squares of the deviations is the number we get when we add up the N measurements and divide by N, namely the mean or arithmetic average of the measurements. In other words, the number that best represents the set of measurements is the mean of the measurements (if what we imply by “best represents” is the number about which all of the measurements are most tightly clustered).

There are various notations for the mean of a series of numbers, . The most common are . The last notation is often called the expectation value of x. In any case, each of these usually refers to the sum of the measurements divided by the number of measurements .

Reporting the Standard Deviation of a Sequence of Measurements

In scientific or engineering measurements, one usually does more than just report the mean of a sequence of measurements. An additional number is reported (a measure of dispersion) that gives some indication of how much the measurements are spread out around this mean, thus giving an indication of the precision of the measurement.

We already mentioned that the sum of the squares of the deviations is a measure of how tightly clustered individual measurements are to a number Q, and we found that the Q that minimizes this sum is the mean of all of the measurements. Therefore, to indicate how precisely the several measurements are clustered, it might seem reasonable to report the sum of the squares of the deviations about the mean. In other words, along with the mean, we might report

Here we have used as the mean of the x_i’s. There are problems with using this sum, however. Firstly, the number, S, gets larger and larger as the number of measurements increases (this is as useless as quoting the sum of a set of measurements instead of the mean of the measurements). Secondly, the sum is made up of the squares of the deviations, and therefore does not have the same units as the measurements themselves. Instead, it has units which are the squares of the units in which the measurements are made. Thus, if the measurement is in meters, S has units of square-meters.

A better choice would be to find the mean of the squares of the deviations. In fact, this has a name called the variance. Thus

The variance is a useful number in that it does tell us what the mean square deviation is, and this is a measure of the spread about the mean. However, it still has units which are the squares of the units in which the measurements are made, but if we take the square-root of the variance, the units are consistent. Therefore it has become customary to report the square-root of the variance as a measure of the precision of a measurement. The square-root of the variance is called the standard deviation, and has been given the Greek letter sigma (s) as its most common designation. Sometimes you will see S.D. or s.d. as the designation of the standard deviation. (Another name for the standard deviation is the root-mean-square deviation or rms deviation.) In summary, then, the standard deviation is given by

We will learn later that there are two different standard deviations that can be calculated from a series of measurements, one called the population standard deviation, or just s, and the other called the sample standard deviation, or s. The only difference is that N is replaced by N-1 in the denominator of the calculation of the sample standard deviation. The important distinction between a population and a sample of the population will be discussed later. The different notations, or s, and or s, are so common that it is important to get used to them. Generally and are used together in some texts and applications, and s and s are used together in others.

Summary of Reporting a Measurement and its Uncertainty

We have just seen that the best representation of a series of measurements of a quantity is the mean of the measurements. The justification for this is that the mean is the number that minimizes the sum of the squares of the deviations. We have also seen that the variance is the mean of the squares of the deviations, and the standard deviation is the square-root of the variance. Thus the mean value of the measurements also minimizes the standard deviation. This suggests that we can capture the essence of a measurement of a quantity by quoting the mean of a series of measurements of that quantity followed by another number that represents the amount of spread about this mean. The standard way of reporting this is as follows:

where the mean of the series of measurements is given by

and the standard deviation, s, is either

When N is large, the population and sample standard deviations do not differ by much. However when N is small, they could be quite different. In fact, if only one measurement were made, the population standard deviation would be , suggesting that the measurement was infinitely precise. This doesn’t make sense, especially if we know that subsequent measurements would vary. For a single measurement, the sample standard deviation would be (which is, at least, undefined) suggesting that no information about precision was obtained.

This latter observation suggests that the sample standard deviation is the one to use for a series of measurements of a given quantity that is presumed to be fixed (such as the length or weight of a single item). The justification for this is that the measuring instrument and measuring process are giving us varying results about an exact value which we do not really know but can only discover approximately through our imperfect measuring instruments and procedures. A single measurement gives us no information about how our measurements would vary, and the sample standard deviation reflects this by being undetermined. We shall say for the moment that the sample standard deviation (the one with the N-1 in the denominator) is the one to use for expressing the uncertainty in a measurement, however, it depends on what we are measuring, and we will need to explore this further after we discuss a few more topics.

An Example of Reporting a Measurement Properly

Suppose we have the following set of measurements of a length in centimeters: 17.15, 17.42, 17.34, 17.27, 17.19, and 17.30. Table 2 shows the details of the calculations of the mean and standard deviations. We have deliberately chosen some numbers that illustrate the calculations but raise some additional issues as well.

Table 2: An Example of the Calculations of the Mean and Standard Deviations

i	x_i
1	17.15	-0.128	0.016
2	17.42	+0.142	0.020
3	17.34	+0.062	0.004
4	17.27	-0.008
5	17.19	-0.088	0.008
6	17.30	+0.022

N = 6	103.67		0.049
	17.278		0.008
			0.010

The first column of Table 2 contains the measurement number. The second column shows the individual measurements, their total, and their mean. The third column shows the individual deviations from the mean. The fourth column shows the squares of the deviations from the mean, the total of these, and both a population and a sample standard deviation. According to standard practice, the measurement of the length should be reported as 17.28 ± 0.10 centimeters (that is, ). However, a careful look at the numbers raises some questions about how many digits to retain in the answer.

Notice that each measurement is reported to four digits. This suggests that the instrument used to make these measurements was capable of measuring to this precision, but note also that the individual measurements are varying in the first decimal place (the third digit). The object itself apparently has a roughness that is larger than the resolution of the instrument. If this is the case, what is the meaning of the second decimal place? How can we assume that the second decimal place has anything to do with the length of the object if the first decimal place is varying between 1 and 4? Notice also that the calculations were carried out to three decimal places but rounded down to two. If the first decimal place is varying, certainly the third decimal place cannot have any meaning, so we drop it. But how do we handle that second decimal place? We need to discuss the concept of significant figures and how we decide what digits to keep, but we can give a preliminary rationale for the answer we just stated.

One possibility would be to acknowledge that the object itself has a roughness in its length that affects the first decimal place in its measurement. Since it is this first decimal place that is varying, we could argue that the measurement should be reported as 17.3 ± 0.1 centimeters because this does not give the impression that anything is known about the second decimal place, and should thus be a fair report of the length of the object and its variation. In fact, this is the way the measurement might be reported if no other information were to be conveyed. However, by reporting the measurement as 17.28 ± 0.10 centimeters, we are implying something additional about the measuring process, namely that the resolution of the measuring instrument was more than sufficient for the measurement at hand. We have gained some additional confidence that the variation in the first decimal place is indeed real, and was easily measured using this instrument.

We now begin to see how much information can be conveyed in reporting a measurement, but we must also be careful that we do not convey any false impressions about our measurements. For example, if (by using the un-rounded calculations from the table) we were to report the measurement as 17.278 ± 0.099 centimeters, we would be claiming that we could detect changes in the third decimal place in our measurements, and this is not evident in the individual measurements shown in the table. The additional digits are artifacts of the calculations, especially the process of division. They do not represent any knowledge beyond the second decimal place.

It may seem that the calculations shown in Table 2 are tedious and time-consuming. However, modern graphing calculators and computer software contain built-in procedures for doing these routine kinds of calculations and much more. One merely enters the data into appropriate lists and calls up the programs to do these calculations. Many of these built-in computer and calculator routines do several standard calculations simultaneously. So the process of generating this information about measurements is not as difficult as it once was. The fact that so many data handling routines are now standard features in the firmware and software of calculators and computers attests to the frequency and importance of these kinds of calculations in scientific and engineering applications.

Responsible reporting of measurements is required of scientists and engineers, and they can get very irritable when reviewing reported measurements that do not make sense. Novices are easily spotted by the inconsistent ways they report their measurements, and their data and experimental results immediately become suspect when this happens. One of the biggest faux pas that marks a novice is the reporting of more digits than is warranted by a measurement. This not only demonstrates a lack of understanding of the measuring process on the part of the novice, it also suggests that the novice is not sufficiently aware of limitations and pitfalls in the experiment itself. Sometimes these novices believe they are covering up sloppiness and laziness by reporting many digits, or that these extra digits somehow make the experiment more accurate. However, there is almost always an internal consistency in a good set of measurements that is a strong indication of good experimentation. This is because the nature and behaviors of measurements follow physical laws, and good experimenters are keenly aware of this. Novices who report inconsistent measurements blatantly reveal their ignorance of scientific experimentation.