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In a thread I posted recently about significant figures, someone said that they are defined as follows. I will use an example to save time trying to generalize the definition.

The person said, for example, that 2.34 has 3 significant figures not because we can count three digits in the usual way, but because significant figures are defined by the ratio of the uncertainty to the number. Here, we have $2.34\pm 0.005$, and the required ratio is $\frac{0.005}{2.34}$ which is roughly $\frac{10^{-3}}{10^0}=10^{-3}$. Thus, the number of significant figures is $3$ (as per the exponent of the result, which is -3).

Another example is 0.00002. Here the ratio is $\frac{0.000005}{0.00002}$ which is roughly $\frac{10^{-6}}{10^{-5}}=10^{-1}$, hence there is 1 significant figure in 0.00002.

My problem is that this seems to be a much more complicated definition than the usual one which is defined as "the number of digits ignoring all leading zeros". Since definitions should be simple, it seems to make more sense to define significant figures in terms of the simpler definition. Of course, both would suffice as a definition, since they are equivalent: one can show that each definition implies the other.

So what is the actual definition of significant figures? Is it the simple one related to simply counting the digits, or is it the complicated one involving ratios?

Also, another question that is still bugging me is regarding the motivation behind the definition of significant figures. How was the definition discovered? Ie what was the motivation for defining significant figures in this way?

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    $\begingroup$ "So what is the actual definition of significant figures? Is it the simple one related to simply counting the digits, or is it the complicated one involving ratios?" For the purposes of high school curriculum, the definition is the simple one, but the point of the definition is that it approximates the complicated one. $\endgroup$
    – knzhou
    Feb 6, 2019 at 16:44
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    $\begingroup$ I haven't heard this complicated and, it seems to me, not that accurate definition before. I have always heard the simpler "count digits" definition you mention. Which also is intuitive: Our number system is build up so that zeros before a number add zero new factors of 10, such as in $003$, while zeros after do have influence, as in $300$, because they move the number to another factor of ten. This is the basis of the decimal system, where the placement of each number represents a number of a particular magnitude of 10. This simple structure is inheritted to thesignificant figures. $\endgroup$
    – Steeven
    Feb 6, 2019 at 16:45
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    $\begingroup$ In higher physics, nobody uses sigfigs. They are an approximate tool used to give high school students a taste of error analysis. The real error analysis rules are quite different. $\endgroup$
    – knzhou
    Feb 6, 2019 at 17:12
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    $\begingroup$ In actual error analysis, you explicitly keep track of uncertainties for every quantity, rather than just leaving it up to the digits. Sometimes you even keep track of the entire probability distribution. $\endgroup$
    – knzhou
    Feb 6, 2019 at 17:12
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    $\begingroup$ Related: physics.stackexchange.com/questions/458045/… $\endgroup$
    – exp ikx
    Feb 6, 2019 at 18:19

3 Answers 3

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The significant figures (also known as the significant digits) of a number are digits that carry meaning contributing to its measurement resolution.

This is what you can read on Wikipedia about significant figures before details start. I think this statement is an excellent starting point to define significant figures, because it tells you what the two ingredients for a reasonable definition of significant figures are:

  1. You have a number representing the result of a measurement
  2. You have a number representing the measurement resolution (or better: the measurement uncertainty).

Example: The result $x$ of the measurement is 2.768547 and the uncertainty is 0.048584 (we omit the units here, for ease of notation). We may write this as $$ x = 2.768547\pm 0.048584, \tag1$$ which is scientifically perfectly correct.

In fact, this notation is shorthand for $$ \mbox{pdf}(x) = \frac{1}{\sqrt{2\pi\times 0.048584^2}}\exp\left(\frac{(x-2.768547)^2}{2\times 0.048584^2 }\right),$$ meaning that the measurement value $x$ has a gaussian probability distribution (pdf stands for probability density function) with mean 2.768547 and variance $0.048584^2$. This distribution quantifies our uncertainty about the true value of $x$.

Still, writing the result like in eq. (1) would not satisfy some of us. The reason is that although this is perfectly correct, it is not economic. The reason is this: if the uncertainty in the number 2.768547 is 0.048584, how certain are we that the 6th digit after the decimal point is the 7? Intuitively we would all agree that given the uncertainty, we are completely uncertain about this digit. It could equally well be 6 or 8 or any other integer between 0 and 9. The same reasoning applies to the 5th, and the 4th digit. So why, is the reasoning, should we then write these digits down at all? The economic notation would be $$ x = 2.768\pm 0.049. \tag2$$

However, some could still think this is a clumsy notation. Why don't we just agree that we approximate the uncertainty by 0.05 (i.e. by half a digit) and quote even more economically $$ x = 2.77,\tag3$$ which then implies the $\pm 0.05$ error by convention. Our result is a notation with three significant digits.

You see that going from notation (1) to (3), we lost precision in quoting the mean and the variance of our measurement result. You also see that we made these steps in a desire to make our notation more economic, but doing that we sacrificed precision.

My plea to students is: Do not try to memorize the definitions and rules for determining significant digits like the ten commandments. Rather remember the reasoning that I gave you above, and decide on the need to be more or less economic given the specific requirements of your problem at hand.

My plea to physics teachers is: Do not overdo it with forcing your students into obeying rules for finding significant digits. Rather make them critical thinkers who can figure out themselves by reasoning how many digits are significant in their measurement results. And remember: it is more precise to quote a larger number of digits for mean and variance, but it may be less economic. What are we striving for in science?

A last remark: Given the mean and the variance of your measurement result, you can actually work out quantitatively, what the probabilities are to have specific digits at certain positions (i.e. powers of 10) in your decimal number. This is a nice exercise in elementary probability theory (which certainly every teacher of physics should do).

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    $\begingroup$ Good answer, and welcome to Physics.SE! For your plea: it's definitely not heeded in practice. One time my little sister came home with a 30-question quiz on significant figures, where all of the questions were about nasty edge cases. (For example, if you need to round a decimal that ends in 5, you should round up only if the previous digit is odd, and the number is not negative.) Predictably, she did poorly, and devoted that evening to rigorously memorizing a list of fifteen sig fig edge case rules. High school teaching is way out of control. $\endgroup$
    – knzhou
    Feb 7, 2019 at 0:27
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    $\begingroup$ Note that you have revealed one of the big flaws with the "significant figures" approach, which is that it appears to quantize uncertainties in powers of ten. In the original question the OP assumes that "2.77" is shorthand for "$2.770\pm0.005$," but you come to the same abbreviation with a much larger uncertainty. You (correctly) include the third digit, for which we have a best guess, but round away the fourth digit, for which we really have no information at all. People who treat the smallest significant figure as more than a suggestion are usually new to the laboratory. $\endgroup$
    – rob
    Feb 7, 2019 at 0:53
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If you're thinking about it this hard, you may be ready to think about "real" uncertainty analysis.

The significant figures approach to uncertainty analysis is a low-quality heuristic that gets across the idea of uncertainty analysis without having to do complicated arithmetic or partial derivatives. It works for simple arithmetic, but needs minor modifications when dealing with numbers raised to powers (including fractional powers, like square roots), and totally fails when you're dealing with nonlinear functions. For a pathological example, consider

\begin{align} \sin(0.50^\circ) &= 0.008727 \\ \sin(0.51^\circ) &= 0.008901 \\ \cos(0.50^\circ) &= 0.999962 \\ \cos(0.51^\circ) &= 0.999960 \\ \end{align}

These (small) angles are different in the second significant figure. The sines are different in the second significant figure, as you might expect, but the cosines are different only in starting in the sixth significant figure. For an even more troublesome example, try it yourself with $90.50^\circ, 90.51^\circ$.

A calculus-aware analysis of the uncertainties explains this discrepancy immediately. But in an introductory physics course (or chemistry, which is where I learned about significant figures first), we don't want to teach every student to do partial derivatives so that we can reassure them that their $0.50^\circ$ and their neighbor's $0.52^\circ$ really are consistent with each other and they can stop worrying. So we tell them how to count the significant figures and not to get too fussed about discrepancies that are limited to the least significant / first insignficant figure.

I've written elsewhere about using anger management to decide whether a trailing digit is significant. But my anger-management approach doesn't address your previous question about leading zeros, which can be added or removed by changing units.

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The rules for significant figures are based on actual measurements taken during physics experiments. The number of significant figures that are reported tells readers of that experiment something about the instrument that was used to take that measurement. This is important because the scientific method requires that any published experimental results be something that can be replicated by other scientists, using the same experimental techniques. With this being the case, even though the rules for significant figures may seem a bit arbitrary, they have been discussed among scientists for quite some time, and the majority of those scientists agreed to the particular set of rules that exist today.

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    $\begingroup$ I actually made a survey of the treatment of sigfigs in about a dozen college-level, introductory texts (mostly physics but a few chemistry and engineering) books a few years ago in an effort to nail down just what the consensus rules were. As soon as you look beyond a superficial treatment what you find is a deep lack of consensus. John Denker has much (that is not complementary) to say on the subject in his treatment of uncertainty. I mostly hammer away at the notion of representing what is reasonably well known and let it go at that. $\endgroup$ Feb 7, 2019 at 7:14

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