On Uncertainty in Science

I’ll let you in on a bit of a secret. For most of my life, I hated doing experiments in science.

It didn’t really matter if the experiments were in physics, chemistry, or biology class (though I enjoyed the fact that physics experiments tended not to be as messy). In fact, when I was in secondary school, my grade was asked at the end of the year to vote on what kind of science class they wanted the next year. There were two choices. One was to keep the material more theoretical and from the textbook. The second was to introduce the content in a much more “hands-on” sort of way, which meant more laboratory experiments. If I recall correctly, I was one of the only students who chose the first option.

I didn’t really understand why everyone wanted to do the hands-on program. In my eyes, it just made things seem less exact and more messy. Other students seemed to like the idea that they could do experiments, but it wasn’t my idea of a fun time.

Moving into CÉGEP, I kept this attitude of not enjoying lab experiments. They were annoying to do, and completing the lab reports after were the worst. One had to deal with uncertainties and significant figures and sources of error that made everything seem much more messy than the theoretical predictions that were made using mathematics. I longed for simple relations without error bars.

From reading the above, it may seem like I think science should be all theoretical. Of coure, this is not the case, and I think, if anything, we need to talk more about the uncertainty and messiness in science. If we want to have a society that understands the way we get results in science, we need to communicate this uncertainty more clearly.

Science is not mathematics. Sure, we want to describe the world using mathematics as our language, but we need to keep in mind that nature will not bend to our will. There will always be fluctuations, imprecise measurements, and sheer randomness in some data. We use mathematics to make these uncertainties as small as possible, but we can never fully eliminate them. As such, it’s crucial to realize that a measurement means nothing without its corresponding uncertainty. The reason is simple: we take measurements in order to compare them. If we just dealt with measurements as precise quantities that have no uncertainty, than we would find a lot less agreement with our predictions. This would make it near impossible to do science.

Let’s take a very simple example. Imagine we wanted to measure an object that is said to be 4.500 metres long. To verify this claim, we take a metre stick that has granulations every centimetre and measure the object. Say it comes out to 4.52 metres. Do we say that these two measurments are different?

The answer is, it depends. To find out for sure, we need to know the uncertainties that are associated with each measurement. When the object was measured to be 4.500 metres long originally, what were the uncertainties on that measurement? Was it $\pm \ 1 mm$? These are critical questions to ask when making comparisons.

If we imagine that the metre stick has an uncertainty of $\pm \ 1 cm$ (because this metre stick is only marked off in centimetres), then the two values we are comparing are: \begin{align} (4.500 \pm 0.001) \ m, \\ (4.52 \pm 0.01) \ m. \end{align} The question now becomes: do these two measurements overlap? This is the key question, and in our case, the measurements don’t overlap, since the first measurement could be at most 4.501 m and the second measurement could be at least 4.51 m. Since these two measurements don’t overlap, we consider them to not be in agreement.

As you may notice, this isn’t a trivial matter. It may have seemed like the two measurements did agree at first glance, but without knowing their associated uncertainties, we have no idea. This means that if someone tells you some figure that came from experiment and wasn’t just a theoretical calculation, you need to know their uncertainty if you want to compare the figure to anything else. Without it, the measurement is meaningless.

What I want to stress here is that uncertainty is inherent in science. There’s no getting around this fact, no matter how precise and careful your experiment is. This is why I find it so amusing when people attack scientific results on the basis that they are simply uncertain. Of course they are! This isn’t mathematics, where results have infinite precision. In science, we have this inherent uncertainty, but we use the tools of mathematics to make sure that the uncertainty is as small as possible, and we make our claims using this uncertainty. We make do with what nature presents us.

If there’s one thing I want to ask of you, it is this: make sure you’re aware of the inherent uncertainty in science, so that you aren’t worried when you see scientists saying that the measurements agree with theory, despite the seeming non-equivalence. Chances are, the uncertainties in the measurement is what allows scientists to make this claim. Conversely, look for those who try to exploit this characteristic of science to push information that simply isn’t supported by the scientific method.