I have two measurements $x_1\pm s_{x_1}$ and $x_2 \pm s_{x_2}$. The error intervals ($\pm$ one standard deviation) do not overlap. Does this necessarily mean that these are two samples from normal distributions with different means?
Or should I run a test to exclude (to some degree) the fact that they might still be from the same distribution?
Both means have an equal amount of measurements (about 2000). It is given that both are pulled from normal distributions.
There is no reason to suspect that the two samples are from normal distributions based on the information you have provided.
Even if the data are from normal distributions, the error intervals do not tell us useful information without knowing the number of samples you took. For example, if you took exactly one measurement for x1 and x2, then the standard deviations would be 0. In spite of this, there is no reason to suspect that x1 and x2 come from distributions with different means.
It sounds like want you want to build are confidence intervals. If you know how x1 and x2 are distributed, there are many formulas for building these. The wikipedia entry has examples for a normal distribution, and a good stats textbook will have many others. If you do not know the distribution, you can use bootstrapping techniques to determine it and create useful confidence intervals. If the confidence intervals do not overlap, then you have reason to believe that the data come from distributions with different means.
