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When performing $N$ independent measurements following a Gaussian distribution, it is sometimes said to report the mean and the standard deviation of the mean as value and its uncertainty. Now, the instrument has a non zero precision, and the standard deviation of the mean can be made arbitrarily close to 0 granted that the number of measurements can be made arbitrarily large. I had been taught to report the precision of the instrument instead of the standard deviation of the mean when this latter becomes smaller than the former, on the basis that "one cannot report the period of a pendulum with an error smaller than the precision of the chronometer used". I feel this rule is arbitrary and I don't find information about it. I also don't see reports where the uncertainty is smaller than the precision of the apparatus. My question is, what do scientists report as uncertainty when the standard deviation of the mean is smaller than the precision of the instrument? I see no problem reporting a number arbitrarily close to 0 as long as it's clear than I'm reporting the standard deviation of the mean. I'd also have no problem reporting another statistics, e.g. the standard deviation to indicate how spread the data is, but then I'm not reporting the uncertainty of the mean value anymore.

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Ideally, you should report both errors. Example: your measurement device makes systematic errors of 0.2 and your mean value is 1.5 with a standard uncertainty of 0.15, then you quote along the lines of:

$$x = 1.5 \pm 0.15\,(\text{statistical}) \pm 0.2\,(\text{systematic})$$

This way this allows people reading your work the freedom to use advanced statistical method to treat the two errors differently. This is especially important if people are trying to combine your measurements with somebody else who uses a different apparatus with, perhaps, a different systematic error. @Tajimura's is acceptable for crude reporting but it makes it hard to impossible to sensibly combine your results with others. When I say crude, this is a bit mean actually: if you have to report only one series of results and the measurement error of your apparatus is not correlated with the random errors of the measurement entering the statistical error above, then his formula is perfect. But one day your work might become relevant to other physicists and they will hate you if you did not separate systematic and statistical errors!

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Back in my university we used to report so-called "full uncertainty" which is a combination of standard deviation and device precision:

$\Delta x = \sqrt{\sigma ^2 + \delta ^2}$

Where $\sigma$ is a standard deviation and $\delta$ is a measurement device precision.

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