What's the purpose of significant figures in measurements?

Question

I have read about significant figures in my chemistry and physics books, but none of them pointed out the significance of the significant figures.
Searching the Internet I found two possible answers: Some people are saying that it is to denote the accuracy and others are saying that it is to denote the precision of the measurement.
Both of these arguments are unsatisfactory, because in my opinion significant figures are not the only thing that governs the accuracy or precision of a measurement.

Layman's definition of accuracy and precision

Given a set of data points from repeated measurements of the same quantity, the set can be said to be accurate if their average is close to the true value of the quantity being measured, while the set can be said to be precise if the values are close to each other.

My arguments regarding significant figures, accuracy and precision:

Accuracy. Suppose I measure the weight of a candy using some device and get the measurement as 1.2 grams and then I measure it again using some other sophisticated device and got the answer 1.436575383 grams.

The first measurement is having two significant figures whereas the latter one is having 10 significant figures.

How can we say which measurement is accurate, if we don't know the actual mass of the candy? Here many might go for the latter measurement and will do all the calculations with that value. However, what if the second device was faulty, which was not known at the time of measurement.

Precision. Does a higher number of significant figures make a measurement more precise? How?

The larger the object I measure using the same device the higher is the number of significant figures displayed, but that does not make the measurements more precise.

Don't both, the precision and accuracy, depend on the operator of device?

So what is exactly the purpose of significant figures?

Answer 1

1. While estimating uncertainty can be quite involved, the purpose of significant figures is quite simple: They are a shorthand method of communicating the uncertainty of a measurement from those who are in a position to be able to estimate the uncertainty (those who made the measurement) to those who need to know the uncertainty, but can't determine it themselves (the reader/audience).

   I say shorthand because the uncertainty is more completely communicated by explicitly specifying it, e.g.:
   $1.0342(86)$, which is equivalent to $1.0342 \pm 0.0086$

2. You wrote:

   Both of these arguments are unsatisfactory because in my opinion significant figures are not the only thing that govern the accuracy or precision of a measurement.

   You seem to have it backwards. Significant figures do not govern the accuracy or precision of a measurement, in any way. Rather, it's the accuracy/precision that governs the number of significant figures.
   I.e., it's not significant figures ⟹ measurement uncertainty. Rather, as explained above, significant figures communicate what the measurement uncertainty is. Thus it's measurement uncertainty ⟹ significant figures.

   First you determine the uncertainty. Then you determine the appropriate number of significant figures to use to express that uncertainty. The fact that your scale reads out, say, $\pu{0.285746 g}$ doesn't mean that the uncertainty is in the last digit (which, here, would be micrograms), and thus doesn't mean your measurement should be expressed to six significant figures. It could be that you know the uncertainty is 100's of micrograms, regardless of the number of digits in its readout. In that case, you'd express the answer as $\pu{0.2857 g}$. Again: Uncertainty ⟹ significant figures.

   To give a car analogy: Reviews of a new sports car in various enthusiast magazines don't determine the car's performance. Rather, assuming the reviews are sufficiently competent and unbiased, it's the car's performance that determines the content of the reviews. The reviews don't determine performance, they merely communicate it.

Answer 2

You are absolutely correct in that there are several kind of uncertainties (often called errors). These are referred to as systematic and statistical errors and the difference between them can be conveniently be described in terms of a target with a distribution of shots. The distribution of the shots with respect to the bullseye represents the kind of errors as also illustrated in the figure below. The spread of shots represents the statistical error and is related to the precision of the measurement. The less spread the more precise. This says something about the reproducibility of the measurement. The deviation of the average of the shots is a measure of the systematic error (e.g. a calibration error). This is related to the trueness of the measurement. If both precision and trueness are high, the measurement is said to be accurate. (Confusingly, sometimes accuracy is used for trueness instead and no separate term is used for a measurement that is both precise and true).

Whereas you can get an idea of the magnitude of the statistical error by doing many measurements and using standard statistical analysis methods, the systematic errors are very difficult to spot. One way is, as you pointed out with the example of weighting candy, is to change the measurement instrument. Other means are by repeating measurements over long time scales or manipulating your measurement in a known way and check the response of the thing you want to measure. For the determination of fundamental constants for instance, several independent measurement methods are required.

Now to come to your question about significant figures. Personally I think the best way to address this issue is to not only state a certain number of significant digits, but also the error in this digit. Often the statistical and systematic errors are combined into one by adding them in quadrature (in which case it is implicitly assumed they are independent)

$$ \Delta_\text{tot.}=\sqrt{\Delta_\text{stat.}^2+\Delta_\text{sys.}^2}, $$ But even better, in my opinion, is to give both the systematic and statistical error separately.

Image source

Answer 3

Once upon a time, calculations were done on paper and on slide rules. It would take considerable effort to carry 10 digits in a calculation instead of 3 (which is what most slide rules are capable of). Then, students started using calculators, and would just copy the result they got, with 10 or more digits. Science teachers thought that was ridiculous, and started teaching error propagation using significant figures.

Usually, when students start working in the lab, they have cheap or moderately expensive glassware and instruments, and there is always a measurement going into the calculation not known to more than 3 or 4 significant figures (unless you just determine a mass, where you might get 6 or 7). Some teachers, for that reason, implement a 3 significant figures rule, and move on to more sophisticated treatment when the students gain experience and access to better instrumentation.

Suppose I measure the weight of a candy using some device and get the measurement as 1.2 gram and then I measure it again using some other sophisticated device and got the answer 1.436575383 gram.

An analytical balance that reads off to the nanogram will be calibrated back to the SI unit definition, otherwise you would not pay for all those digits (so you expect it to be accurate). Also, there will be more sophisticated measures to prevent drafts and vibrations than in a cheap balance (so you expect it to be precise). So for an instrument that is not broken and is used properly, you expect the precision and accuracy to correllate to a certain degree, and you would not average results from a cheap and a sophisticated instrument.

If you are wondering about the skill of the operator, you would measure a standard multiple times and compare results to an experienced operator or to the specs that came with the instrument.

Answer 4

A number itself doesn't inherently precision or accuracy. It is the representation of a measurement: the output of a measuring instrument. That instrument is what has precision and accuracy. We accept the number representing an instrument's output on the basis that we trust the calibration and quality of the instrument.

Establishing a certain number of significant figures is a way of capturing the precision of the measurement based on what we know about the properties of the instrument. If we know that some scale weighs to the milligram, and we weigh some object that weighs 230 milligrams, then we have three significant figures. To clarify that the 0 counts as significant, and isn't merely a placeholder, we use scientific notation: 2.30 × 10² mg.

This figure could be wildly inaccurate. Suppose someone weighed a 1000 mg object on the scale previously and zeroed it out, so that it reads -1000 unloaded. Our 230 mg object is actually 1230 mg. We have misinterpreted the output of the instrument based on a false belief about its calibration state.

Suppose the scale is of poor quality. If we measure the same object multiple times, we get different figures: 220, 227, 239, 231, ... in that case, our acceptance of three significant figures of the 230 measurement had been based on the false belief that all digits on the scale's display actually mean something.

The absolute value of a measurement, and the number of significant figures, are just representations, which are based on beliefs about the instrument. If those beliefs are true, the measurement's value is useful, and the number of significant figures is an additional important, meaningful attribute which gives a coarse idea about the limits of the measurement's precision.

The primary value of the concept of significant figures is that it provides an at-a-glance sanity check on the honesty of scientific and engineering measurements and calculations. We know that if some calculation has several inputs, the least accurate of which has only two significant figures, but the result is being reported with four significant figures, that something is wrong. Suppose we see some final or intermediate result $x$ being calculated via some function $f$ like this:

$$x \approx f(1.1\times {10}^2, 5.763\times 10^{-3}, 8.59\times 10^3)$$ $$x \approx 2.253\times {10}^{-2}$$

We know right away, "wait a minute, how is this researcher getting four significant figures when the least precise input has a mantissa of 1.1? Oops; the honest result is actually 2.2 something, not 2.253". Note that we can perform this sanity check at a glance just by looking at the list of input arguments and the output; we do not have to know anything about $f$.

That's the value of significant figures. Numbers do not attest to the accuracy of a measurement, and significant figures do not vouch for it accuracy. What significant figures do is provide a representation of the measurement's precision and help eliminate or reduce false inflation of precision in subsequent calculations. They cannot ensure that the precision of the measurement itself isn't falsely inflated relative to the actual abilities of the instrument, but that's a separate problem which doesn't negate the value of significant figures.

Like all computing mechanisms, significant figures obey the GIGO principle: garbage in, garbage out. If a measurement appears to give us 10 significant figure, but 7 of them are garbage, then the same is true of our calculation; it may look like we have 10 significant figures in the result, but at least 7 of them are garbage.