Uncertainty in Physical Measurements: Introduction
Table of contents
Module 0 – Introduction
To the Student
A crucial part of the way science describes the physical universe is quantitative. In general there are two different types of numbers that we use in the description of some physical system:

An exact value. For example, there are exactly three bananas shown to the right.

An approximate value. If we want to determine the total mass of the bananas we will need to use some sort of scale. However, in the real world there are no perfect scales, so the measured mass is always an approximate value.
Say we measure the mass of the bunch of bananas with a digital scale, and the reading on the scale is 324.1 grams. The question is what is the uncertainty associated with the value of 324.1?
The answer to this type of question often depends on the context. For example, if a carpenter says that some length is “just 8 inches,” she probably means that she thinks the length is closer to \(8 \frac{0}{16}\) inches than it is to \(7 \frac{15}{16}\) inches or \(8 \frac{1}{16}\) inches. If a machinist says a length is “just 200 millimeters” he probably means that he thinks the length is closer to 200.00 mm than it is to 199.95 mm or 200.05 mm.
In general, the best way to describe the uncertainty in some measurement is to assign it a quantitative value. The following Modules concentrate on how to determine and communicate such a number. Two examples may help make it clear to you why this material is important to anybody doing in work in the sciences, whether or not the science is Physics.
Example 1
Sadly, many otherwise good scientists are unaware of the material you will be learning here, which sometimes causes some wrong conclusions.
For example, in the early 1970’s some researchers reported that a diet that was high in fiber reduced the incidence of polyps in the colon. Polyps are a precursor to cancer, so people began cramming as much fiber down their gullet as they could.
In January 2000 a massive study in the New England Journal of Medicine showed that fiber in the diet has no effect on the incidence of polyps.
There were two problems with the earlier study: First they had a fairly small number of people in the two samples they studied. Second, they neglected to do an analysis of the uncertainties in the results based on the sample sizes.
If they had calculated the uncertainties in their numbers, they would have known that although the measured rates of polyp formation were different for the two groups, those differences were zero within the uncertainties of the measured rates.
Note that there are benefits to fiber in your diet, but not as a way to reduce the formation of polyps in your colon.
Example 2
As you may know, the neutrino is one of the elementary particles that make up the universe. Between 2009 and 2011, in a very complex and difficult experiment, a collaboration of 178 physicists based in CERN in Switzerland and the Laboratori Nazionali del Gran Sasso in Italy measured the speed of 16,111 neutrinos, using an instrument called the Oscillation Project with EmulsiontRacking Apparatus (OPERA). The neutrinos travelled from CERN to the lab in Italy, a distance of 721278.0 m. Although the measured speeds of the individual neutrinos varied, the mean value of all speeds was:
\(\bar{v} = (1.000\ 024\ 8) c > c \) (1)
where c is the speed of light in a vacuum.
According to Einstein’s 1905 Special Theory of Relativity, no object can move faster than the speed of light. So if this experimental result is correct, then that theory must be wrong! Therefore it is crucial to know what the uncertainty is in the number of Eqn. 1.
Because the discrepancy from the speed of light occurs in the onehundredthousandths position, it is easier for us humans to “read” the experimental result by defining \(t_c\) as the time for an object to travel the distance at exactly the speed of light and \(\bar{t}_m\) to be the mean measured time of the neutrinos. Then we define \(\delta t = t_c \bar{t}_m\) and:
\( \delta t\left\{ {\begin{array}{*{20}{c}} { < 0,\,\,\,\,v < c} \\ { = 0,\,\,\,v = c} \\ { > 0,\,\,\,v > c} \end{array}} \right.\) (2)
Using this notation, the experimental result is:
\(\delta t = 60\ ns\) (3)
If the uncertainty in the value of \(\delta t\) is, say, 500 ns, then the experimental result is consistent with the theory of relativity: the data only tells us is that the value of \(\delta t\) is probably between (60 – 500) = −440 ns and (60 + 500) = +560 ns. So the actual value of \(\delta t\) could well be a negative number and it could well be that neutrinos do not travel faster than c.
However, when the experimenters evaluated all the individual uncertainties in all the various parts of the experiment they calculated that the total uncertainty was 10 ns, so the actual value of is probably between (60 – 10) = 50 ns and (60 + 10) = 70 ns. We write this result as:
\(\delta t_{OPERA} = (60 \pm 10)\ ns\) (4)
This shocking result caused the experimenters to spend months rechecking everything they could think of in the experiment. They found no mistakes so in September 2011 they published their result, which made headlines around the world. Being excellent scientists, they also released all of their raw data and invited others to check their work.
Simultaneously to the OPERA experiment, another team at CERN and Italy was doing measurements of the speed of neutrinos. The experiment was called Imaging Cosmic And Rare Underground Signals (ICARUS). In March 2012 they published their final result, which was:
\(\delta t_{ICARUS} = (0.3 \pm 9.8)\ ns\) (5)
This result is completely consistent with neutrinos travelling at less than or equal to the speed of light.
Later, the OPERA team discovered a loose fiber optic cable and a wrongly functioning oscillator. When they revised their calculations, they published a new result:
\(\delta t_{OPERA,\ revised} = (6 \pm 11)\ ns\) (6)
This new result is consistent with the ICARUS one within experimental uncertainties. So, at least for now, the Special Theory of Relativity has been saved.
Note that the experiments have not proved that the Special Theory is correct. They have also not proved that neutrinos do not travel faster than the speed of light. All they have shown is that the data are consistent with neutrinos travelling at speed < c.
You may have noticed above that we were careful to use the word probably in, for example, “… the actual value of \(\delta t\) is probably between (60 – 10) = 50 ns and (60 + 10) = 70 ns.” This indicates, correctly, that our study of uncertainty in physical measurements will require understanding some elementary statistics. We will begin that study in Module 1.
Each Module contains the material that you need to know, and some Questions and Activities. You should definitely read the materials before you come to the Practical. In the Practical you will be working with your classmates in a small Team to answer the Questions and do the Activities. Reading the Questions before is a good idea, so that you and your classmates will already have an idea of what they are about.
Enjoy!