Let's say we have n normally distributed random variables all with the same median and variance.
Do we have a possibility to estimate the distribution law of the sum of those variables?
I assume that the law will be also normally distributed with the same median but with variance multiplued by the square root from n. But I didn't find any confirmation for that.
I suspect that the answer could be figured out if I spend some time reading common articles on the subject but I suspect that the simple answer on this particular case as a separate article can be worthwhile for people like me: not strong enough in math looking for this solution.
The sum of independent normal random variables is normal. That assumption of independence is very important: don't leave it out! The mean of the sum is the sum of the means (expected value is always additive), the variance is the sum of the variances (variance is additive for independent random variables).