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I have always understood significant figures to be those figures which we know with certainty. Wikipedia (https://en.wikipedia.org/wiki/Significant_figures) provides a related but less rigid definition of "[the] digits that carry meaning contributing to its measurement resolution."

To use an example from another post on this topic, suppose a pencil were measured to be somewhere between 86 and 87 millimetres, but closer to 86. I understood that we would round to 86, and the value would be written with its two significant figures. We would then understand that to mean 86 millimetres +/- 0.5 millimetres. This is consistent with our definition because the two figures, 8 and 6, are known with certainty (assuming that our measurement was correct). However, various other posts, including Definition of Significant Figures to which I refer above, provide a slightly different methodology. It proposes that we estimate the next digit (3 in the example), and write the value with three significant figures: 86.3. Recognising that the last figure was an estimation, we take our uncertainty to be +/- 0.1.

The examples above are obviously incompatible. The method that I learnt is consistent with both of the definitions. We do know both figures with certainty, and they carry meaning contributing to their measurement resolution. However, in the example above, it is clear that we have introduced more uncertainty that is probably warranted. Indeed, by recognising that the pencil was between 86 and 87 millimetres, and closer to 86, we can presumably be certain that the real value is between 86 and 86.5, removing half of the uncertainty introduced by the first method. The second method improves upon our understanding of the real value, but it introduces an estimate and proposes it as significant. Consequently, the first definition of significant figures does not apply. Furthermore, it is impossible to make such an estimate with most measurements (with electronic devices, or where the value is not so easily perceptible), and in such cases one would presumably have to revert to the whole number +/- 0.5 of the unit value of the last significant figure.

Both methods, incidentally, collapse when one introduces operations - something I intend to address in another post.

Is there a gold standard as to the method that is used in professional circles? Indeed, are significant figures used in any professional circle at all, and if so, which?

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  • $\begingroup$ Significant figures are a really poor concept that should never be used. The decimal system has absolutely nothing to do with the error distribution of a physical quantity. Why would it? I can write the same number in binary, base 5, base 7, octal, duodecimal, hex or whatever base I want. Which ones represents the error distribution of what you are measuring? $\endgroup$
    – CuriousOne
    Jun 15, 2016 at 19:23
  • $\begingroup$ The gold standard is to do a proper error analysis. Significant figures are a underpowered and highly approximate substitute for the real thing. But real error analysis is costly in terms of both time and mental effort so we don't do it in introductory teach contexts. $\endgroup$ Jun 15, 2016 at 20:09
  • $\begingroup$ Granted, I understand that significant figures are less powerful than an error analysis. I should note, however, that one could theoretically apply the approach to significant figures to an analysis in any base. So, am I to understand that they are never used professionally? And am I also to understand that there are various interpretations of the approach?Odd then that the method is taught at all. $\endgroup$
    – POD
    Jun 15, 2016 at 20:18
  • $\begingroup$ In my experience the notion "don't write digits that don't mean anything" is ubiquitous but no one ever argues over the rules for 'significant figures' as such. And I see different treatments in different texts. $\endgroup$ Jun 15, 2016 at 20:53
  • $\begingroup$ Thank you, dmckee. I think that you have answered my question. $\endgroup$
    – POD
    Jun 16, 2016 at 4:55

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