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Here is a question that combines uncertainties in measurements and significant figures.

Consider the following results to measure the value of g:

$$g={9.7,9.8,9.7,10.0,10.1,10.3}$$

Then to four decimal places $\bar{g} = 9.9333$, while the uncertainty of the mean is $$0.2422/\sqrt{6} = 0.0989$$

But clearly we would not quote these values as they stand.

So, what would be the statement of the mean value of g and its uncertainty to the correct number of significant figures? Would it be $\bar{g}=9.9 \pm 0.1$ or $\bar{g} =9.93 \pm 0.10$? The original measurements had two significant figures, but it would seem using the mean and its uncertainty would permit an additional significant figure. And what is the general rule for the significant figures for mean values and their uncertainties?

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marked as duplicate by Kyle Kanos, sammy gerbil, Rob Jeffries, John Rennie, Emilio Pisanty Aug 10 '18 at 23:54

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    $\begingroup$ Possible duplicate of Multiple measurements of the same quantity - combining uncertainties $\endgroup$ – Kyle Kanos Aug 6 '18 at 16:49
  • $\begingroup$ shouldn't it be dividing by sqrt(5), i.e. sqrt(n-1)? $\endgroup$ – Walter Aug 6 '18 at 17:18
  • $\begingroup$ Isn't uncertainty built into significant figures in the last written digit? To explicitly show the uncertainty in a measurement, I would completely avoid sig figs and instead write the most precise estimate $\pm$ the error. $\endgroup$ – Quantumness Aug 6 '18 at 17:21
  • $\begingroup$ Related : Significant figures vs. absolute error $\endgroup$ – sammy gerbil Aug 6 '18 at 18:00
  • $\begingroup$ Walter: the Standard Error of the Mean (SEM) is given by $$\sigma_\bar{x} = \cfrac{\sigma}{\sqrt{n}}$$, where $\sigma$ is the standard deviation of the population (that's where $n-1$ comes in) and $n$ is the number of samples to determine the mean. $\endgroup$ – MSulkanen Aug 6 '18 at 19:20
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The original measurements had two significant figures, but it would seem using the mean and its uncertainty would permit an additional significant figure.

This is the entire point of making multiple measurements. Well, that and the necessary side process of checking whether your data are consistent with the distribution you expect.

Here's a significant-figures-only way to think of it. When you add your six values, you get $\sum g = 59.6$, where the third significant figure is trustworthy because it came from the trustworthy digits in each of the six individual values. So your average, $\overline g = \sum g / N$, should also have three significant figures.

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  • $\begingroup$ Well put. OK, so this also translates to the uncertainty as well, consistent with the significant figures of the mean? $$\sigma_\bar{g} = \cfrac{0.242}{\sqrt{6}} \rightarrow \sigma_\bar{g} = 0.10$$ That seems to hang together with the significant figure of the mean. $\endgroup$ – MSulkanen Aug 7 '18 at 13:56
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    $\begingroup$ I think reporting an uncertainty of $\pm0.10$ is more appropriate than an uncertainty of $\pm0.1$, because the former implies that small differences in the data might have shifted the result to $\pm0.09$ or $\pm0.11$, while the latter suggests a small difference in the data might have shifted the result to $\pm0.02$. Keeping an extra significant figure if the leading digit is a one (or sometimes a two) is a good housekeeping idea. $\endgroup$ – rob Aug 7 '18 at 15:30
  • $\begingroup$ Per your earlier comment, by taking multiple measurements you should be able to improve on the uncertainty - at least one more significant figure. It's interesting to note that in teaching introductory physics labs there is discussion of uncertainty and significant figures but no discussion about how one is used in conjunction with the other. $\endgroup$ – MSulkanen Aug 7 '18 at 17:07
  • $\begingroup$ One rule I've read is that uncertainties should never be quoted with more than 2 SFs. $\endgroup$ – MSulkanen Aug 9 '18 at 18:46
  • $\begingroup$ The Particle Data Group publications have a nice rule for rounding uncertainties; see section 5.3 of the introduction to the review. $\endgroup$ – rob Aug 9 '18 at 19:16

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