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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?

MSulkanen
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1 Answers1

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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.

rob
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