Handbook

Repeated Measurements []

Measurements are repeated so that one gets an idea about the variability of the measurements. Given enough repeated measurements, these measurements will follow the normal distribution with a mean value and a standard deviation. The question arises of how many measurements are required.

Reasons for repeated measurements []

Some scientists have developed an intuition as for to when to stop, others use rules of thumb, again others continue measuring to increase precision. If one is interested in the variability of the measurements, one will want to know the mean and the standard deviation. Since the standard deviation is not affected by the number of measurements (see Type A), one can stop taking measurements as soon as the distribution adequately fits the normal distribution. The latter is usually done with a statistical test, which is beyond the scope of this unit.

Alternatively, one can look for changes in the value of the standard deviation and the mean value with additional measurements. When the first two significant digits of the standard deviation (and the corresponding digits of the mean) do not change with additional measurements, this is an indication that the data reasonably fits a normal distribution.

Students' ideas about repeated measurements []

The scientific reasons to repeat measurements can only be understood once a correct understanding of measurement uncertainties has been established. It is therefore no surprise, that the reasons students give for why they need to repeat their measurements differ greatly from those described above.

Taking students' belief in a ⓘ "true value"true value: Hypothetical construct that would portray an exact true result. into account, it becomes clear that students do not see a need to repeat measurements. If you measure correctly, you will get the correct value. Some students will measure again to confirm their result [1]. When the second measurement is not numerically identical, some students become perplexed or get the feeling of having done something wrong. Others will continue to measure until recurring values are found.

For some students, the order of the measurements has an important meaning. Some students place more trust in the first measurement since the setup might wear after measuring [2]. Although this might actually happen, this means that the setup needs to be fixed after every measurement.

Another practice found is that students see repeated measurements as a way of practicing their measurement technique [3]. Afterward, these students report the last measured value since they had the most practice in measuring when recording this measurement. In itself, taking some measurements for practice is a good strategy—one knows what to expect and how to take the measurements properly. However, if these measurements are truly practice rounds, they should not be part of the dataset, since mistakes are expected to occur.

Still, other students take repeated measurements by routine: they take six measurements because they were told to do so, because they did that last time, to be able to calculate a mean value, or because "more is just better" [2, 4, 5].

All these reasonings can be associated with the point paradigm. Students rely on single values and hence see no reason for a set of repeated measurements.

The reasons to take repeated measurements and when to stop are very complex. Probably too complex for secondary education students, especially knowing when to stop taking measurements. However, students can have a conceptual understanding of why repeated measurements are taken: to get an idea about the variability of the measurements. As for when to stop, simple rules of thumb can be used.

Some rules of thumb []

The most simple rule of thumb is to simply state how many measurements to record. Usually, six to ten measurements can give a pretty good idea about the variability of the data and start to approach a normal distribution. Simulations of several alternative quantifications of the uncertainty showed eight repeated measurements to be a favorable number [6, 7]. The exclude extremes quantification, see Type A, approximates the value of the standard deviation with this many repeated measurements.

A more sophisticated rule of thumb is to stop taking measurements until three consecutive measurements are not smaller/larger than the smallest and largest measurement in the dataset. Table 2 shows an example dataset of seven repeated measurements. After measurement number four, the next three consecutive measurements (5–7) are between the minimum (2.18 s) and maximum (3.61 s) of the first four measurements in the series. One could decide to stop measuring after the seventh measurement.

Table 2: Example dataset: with the rule of thumb, one can stop measuring after seven measurements.

MeasurementTime [s]
12.18
23.02
32.82
43.61
52.47
63.31
72.24

One major issue with this rule is that the order of measurements becomes an (explicitly) important aspect. Despite this issue, the dialog of why this rule is used will pave the way toward set paradigm understanding: one wants to see the variance and one wants to know when this is good enough. After learners have fully grasped the concepts of the mean value and the uncertainty, they can start to see the reasoning behind the underlying reasoning regarding repeated measurements.

Literature