Uncertainty

Dividing the circumference of a circle by its diameter always gives the same value of $\pi = 3.14159\ldots$ In theory, we should be able to measure $\pi$ ourselves. Wrap a measuring tape around a mug, place a ruler across it, and divide our two results.

Trouble shows up the moment you read the tape or ruler: the edges never lands exactly on a mark. A measurement is always a best estimate together with an uncertainty, written as

When you read a single mark off a ruler, the edge falls somewhere inside the smallest division. Half of that smallest division is a standard choice of uncertainty.

Something like $\pm 0.5\ \text{mm}$ expresses the absolute uncertainty, which carries units. Dividing it by the measurement gives the relative (percent) uncertainty: $0.5\ \text{mm}$ on a $200\ \text{mm}$ length is only $0.25\%$, but on a $5\ \text{mm}$ length it is a hefty $10\%$. It can be difficult to measure small things.

Furthermore, the number of digits used implies something about the uncertainty as well. Reporting $24.8\ \text{cm}$ implies the uncertainty is on the order of $0.1 \ \text{cm}$. We often round the uncertainty to one significant figure, then round the measured value to the same decimal place: $3.097 \pm 0.083$ is reported as $3.10 \pm 0.08$.

Propagation of uncertainty

To get $\pi$ we have to divide one uncertain number by another. So how do we quantify the uncertainty of the resulting quotient?

The simplest method is to just try the extremes. Take the smallest and largest plausible value of each input, recompute the result every which way, and see how far apart the answers land. The range of values quantifies the uncertainty; the half-width of this range is the new $\pm$.

Once you’ve done this procedure a few times, you may start to notice a pattern:

Addition/Subtraction: absolute uncertainties add. If $R = A \pm B$, then $\delta R = \delta A + \delta B$.

Multiplication/Division: relative uncertainties (approximately) add. If $R = A \times B$ or $R = A / B$, then

Note that when you subtract two measurements, the absolute uncertainties still add. If the two measurements are nearly equal, the result itself is tiny — so the relative uncertainty can blow up. Measuring a thin gap as the difference of two lengths is a good way to get a useless answer.

Comparing your result to theory

If you’ve followed these procedures, each measurement isn’t a single value. Instead, it’s a range around your best estimate of the value: from $\text{estimate} - \text{uncertainty}$ to $\text{estimate} + \text{uncertainty}$. Drawn on a graph, that range is an error bar.

This is a good time to clarify the scientific meaning of two similar words: error and uncertainty. Error is the difference between a measured value and the true value. Since the true value of something is rarely known, we don’t know the error in our measurements (if we did, we’d just take better measurements). The best we can do is estimate the error, i.e. express uncertainty.

Rather than compare results to some true value, we compare them to a theoretical prediction. In this case, theory predicts that $\pi \approx 3.14$ (nevermind that this is a rare instance in which we know the true value). Theory agrees with your measurement when it falls within your uncertainty range.

By random chance, it’s common to have some measurements in disagreement with the theoretical value. If the magnitude of this disagreement is improbable (statistics can help quantify this probability), either you made a mistake or there’s a systematic error — a consistent bias, like a tape that was read off-center every time — that your uncertainty didn’t account for.

It’s helpful to quantify any discrepancy as a multiple of the uncertainty:

If this quantity is about $1$, the uncertainty is likely a good estimate of the error; very close to $0$ or much greater than $2$ typically means something is wrong. This is also where precision and accuracy differ: small values of the numerator correspond to accurate measurements, while small values of the denominator are precise. As an experimentalist, the goal is usually to make both as small as possible while keeping a ratio close to 1.