Question

What “error” should we report? The standard deviation of the mean or twice the standard deviation of a single measurement?

Answer 1

According to some sources, when one performs NNindependent measurements of a Gaussian-like distributed values, one should report the mean value ±the uncertainty±the uncertainty where the uncertainty is the standard deviation of the mean or twice the standard deviation of the sample. These are completely different values, so I'm not sure which one to report. The former tells us both how far the mean is from the "true" value (i.e. mean of the population) with a confidence interval of about 68% which is also equivalent to the confidence interval we can expect the mean to be calculated if we repeat the same measurements under the same conditions. The latter is a measure of the spread of the data and tells us that we can expect with a confidence of about 95% that the next measurement will have a value within [x¯−σx,x¯+σx][x¯−σx,x¯+σx] (I personally do not like the notation \sigma when it means twice the standard deviation, because it's also the notation of the standard deviation, but hey, apparently that's how it's used). Many times the author of an article doesn't mention which one it is, so which should we, in general, assume it is

Answer 2

A low standard deviation indicates that the data points tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the data points are spread out over a wider range of values.