Handbook

True Value []

Suppose one is interested in the falling time of an object that is dropped from a height of 1 m. This time can easily be calculated using the equation: $$\displaylines{h &= \frac{1}{2}g t^2 \\ \rightarrow t &= \sqrt{\frac{2h}{g}}\\ &= \sqrt{\frac{2\cdot 1\text{m}}{9.81\text{m/s}^2}}\\ &= 0.451\,523\,6\dots\text{s}}$$ In principle, this falling time can be calculated precisely. However, when experimentally verifying this exact number, problems emerge. Human reaction time will influence the measurements and no perfect measurement instruments exist. One can start to wonder if the theoretical result can be confirmed experimentally.

Measuring to absolute precision []

To confirm this falling time, an experiment needs to be done. The setup of the experiment has to be such that the object can fall 1 m. This is easier said than done. Using a tape measure, the height can only be measured up to a precision of 1 cm. This means that one can "only" be sure that the height is between 1.005 m and 0.995 m. One can decrease this interval by taking a more precise measurement instrument, but the precision will always have a limit due to the resolution of the instrument. Furthermore, this assumes that the measurement instrument has been calibrated exactly. This, of course, can never be the case since it would require another perfect instrument to be compared to.

Not only the instrument, but also the surroundings will affect the measurement. Suppose that in the setup a ball is dropped from a table. A temperature change will cause the steel legs of the table to expand or contract. This effect is negligible for everyday measurements, but to acquire absolute precision, this effect has to be taken into account.

Ultimately, one can never be one hundred percent sure that the experimental setup has a height of exactly 1.000 m.

And remember, these are only the problems for the setup. The falling time also needs to be measured to absolute precision, one needs to have an absolute value for g, account for air resistance, etc.

The measurand []

Of course, this example is a bit exaggerated. However, it illustrates the problems of measuring something to absolute precision. At some point, measurement uncertainties will always affect measurements in the real world. Therefore, these conditions have to be included in the definition of the quantity that is being measured: the falling time of an object from a height of 1 m, at a temperature of 20°C, at longitude x and latitude y, .... This description of the quantity that is being measured and its surrounding conditions that affect its value is called the ⓘ measurandmeasurand: The complete description of that which is bein measured. Since one is unable to define the measurand in absolute detail, i.e., all the experimental conditions (that can have uncertainties themselves), this means that a ⓘ "true value"measurand: The complete description of that which is bein measured of the measurand—an idealized result with zero uncertainty—cannot exist! Note that this inexistence goes beyond the impossibility of not being able to determine the "true value". Hence, the term "true value" is not used any more in metrology (the science that is concerned with measuring).

For good scientific practice, results have to be reproducible. Therefore, when reporting an experiment, one has to clearly define the measurand. This description is often done in measurement equations but should also include what was measured and report the conditions that could have affected the measurement result. Furthermore, the measurement uncertainty, a quantification of the variability in measurement results, should be reported. This shows the precision of the measurement result (a small uncertainty indicates a precise result) and is an indication quality of the experiment.

There are cases in which certain information about the measurand can be omitted. In the example before, the air temperature could have affected the falling time of the object. However, using everyday measurement instruments, the uncertainty in height and time measurements are much larger than the uncertainty due to temperature fluctuations. Therefore the influence of temperature changes will be impossible to measure and can be omitted from the measurand.

Conditions that can affect the measurement result should be reported. For instance, when stating the height of the table, one should include the uncertainty. When the time measurements are done by hand, an estimation of the reaction time should be reported and integrated into the evaluation of the uncertainty.

Students preconceptions about the "true value" []

The idea of the existence of a "true value" is very persistent among learners [1–4]. Students often pursue to obtain the "exact answer" and want to know whether their result is correct. This belief in a single and exact number coincides with point paradigm thinking.

Although some students might understand that their measurement result has an uncertainty, they still might think of reference values as a "true value" to compare their own result with. Unfortunately, the uncertainty of these reference values is very often not reported in school books. Although the uncertainty is not reported, this does not mean that the reference values do not have an uncertainty. Sometimes, the reported values are reported up to a decimal place that remains unaffected by its uncertainty. For instance the value G = 6.674 · 10-11 N m2 kg-2 has fewer decimal places than the known value of G = (6.674 30 ± 0.000 15) · 10-11 N m2 kg-2. Despite practical reasons to omit the uncertainty, students should be aware that all reference values (apart from seven fundamental constants that are defined as absolute quantities [5]) have been measured or rely on measurements and thus have an uncertainty. Some helpful prompts for students could be:

Using these prompts, students can be guided toward a set paradigm understanding. They start to look critically at their experiment. Ultimately, they start to describe in finer detail what they are measuring in terms of the measurand. Also, they start to express the limitations of their measurement result in the form of an uncertainty interval.

How long is a banana? []

For a practical example of how to introduce the measurand to students, see: Musold and Kok [6].

Literature