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i have a small set of data. I'm sure that their distribution is normal, with standard deviation $\sigma=10$. I'm interested in the mean value, $\mu$ and its uncertainty. Which is the minimum sample size i must have ? And which will be the uncertainty on the mean value ? I think the uncertainty on the mean value is $\sigma_\mu=\sigma/\sqrt{N}=10/\sqrt{N}$, where $N$ is the sample size. Which is the min value for $N$ i should have ? Thank you Andrea

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One approach is to use a 95% confidence interval of the form $\bar X \pm 1.96\sigma/\sqrt{n},$ where $\pm 1.96$ cut probability 2.5% from the upper and lower tails, respectively, of the standard normal distribution. (This leaves 95% in the central part of the distribution.)

Then. as you say the 'standard error of the mean' is $SD(\bar X) = \sigma/\sqrt{n}.$ Also, $M = 1.96\sigma/\sqrt{n}$ is called the 'margin of error' for the sample mean $\bar X$ as an estimate of the population mean $\mu.$

So, if you know $\sigma$ and you know what size $M$ you want, then you can solve for $n$ to get $n = (1.96\sigma/M)^2.$ It is customary to round up to the next larger integer.

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