What are the Exact Rules for Significant Figures, Precision, and Uncertainty?

Question

In the physical sciences (which are physics, chemistry, astronomy, materials science, etc.), we learned that the uncertainty is +/- the smallest unit (which is 1) of the last significant figure if the uncertainty is not given in a recording of data. So, if we have a digital measuring device that measures to the nearest millimeter, has a manufacturer's stated uncertainty of +/- 1 mm, and gives a reading of 914 mm, then it will obviously be recorded as just "914 mm".

However, does the true value actually lie somewhere between exactly 913 mm and exactly 915 mm, or may it stray outside even those numbers if higher precision is used? For example, if go down to the micrometer, is the uncertainty actually +/- 999 μm or +/- 1,499 μm according to the rules of significant figures? If we measure the same sample using a micrometer, is the reading guaranteed to be somewhere between 913,001 microns and 914,999 microns, or is it instead only guaranteed to be somewhere between 912,501 microns and 915,499 microns, respectively?

Answer 1

The rules of significant figures are rule-of-thumb way to communicate errors and should only be seen as a primite first step to talk about uncertainties and measurement errors.

You gave the excellent example of trying to find the "true value" of distance by doing a distance measurement via a digital device, with a stated uncertainty of +/- 1. The measurement could be noted as : $d \approx 914~\text{mm} \pm 1mm$. In most written communication this is the way to go: explicitly state your error bounds in (SI) physical units, and only state a reasonable amount of digits.

I will extend your example from here on.

Sometimes your device will give you more digits than you need, or would be reasonable by your error bounds. To give an example:

$d \approx 914.001223223223 \text{mm} \pm 1000mm$

Here, the inclusion of ever more digits after the comma holds decreasing information, when you take into account that the stated errors are quite large. I write decreasing because it still holds information if you assume that your errors follow a statistical distribution that you assume to know.

Lets say you instruct $10000 \pm 1$ scientists to measure the object by their means of choice and give you their results. Propably the results they will report will differ slightly and follow a distribution like this:

plot of normal distribution

Gaussian Error distribution: background

What I am trying to get at is, that the stated error bars are stated under the assumption that we may assume that the real error follows a simple (e.g. gaussian) distribution. More often than not the errors stated are understood as:

"We assume that the real error distribution is somewhat gaussian, and that the center of that distribution lies at $914$mm, and that the first standard deviations away from the stated medians are $\pm 1mm$".

If you want to go even deeper into the rabbit hole you may start to ask yourself if there even is such a thing as a "true value of X" and read some Karl Popper.

To go back to your initial question, I would suggest you distance yourself from finding and applying the "exact rules" for stating measurements by some fixed rule while disregarding the context. In more cases than not these "exact rules" lead to more ambiguity and you mask the real errors underlying your measurements.

Here are my liberal rules of communicating measurement data:

1. always state your assumptions on the sources and relative strenght of errors within your measurement setup to the best of your knowledge. (Floating Point error / physical parameter inputs /Model/Thermal/Human(!)/ etc.)
2. state your assumptions on how large these respective errors are in physical units, or the units relevant to your field.
3. State your explicit error bars so that you are confident that other people with different setups may reproduce your results to the stated precision.

I will have a go at your example:

"We measured the length of object x with a digital distance meter of company X, model X with a stated uncertainty of $\pm 1 mm$ at room temperature of 22 $\pm$ 2 °C. The measurement may have been impacted by the temperature of the object, the device, the contact pressure of the clamps as well as perpendicularity of the clamps at the surfaces. We assume the majority of the measurement uncertainty to be the thermal stresses and report a measurement of: $914 \pm 1 \text{mm}$."