I can intuitively understand that as I take more samples from a random variable $X$ (gaussian distribution), the mean would approach $E(X)$.
But what I don't get is if I look at it mathematically. Say I took two samples $x_1$, $x_2$. The mean would be $(x_1+x_2)/2$. But variances add, which is contrary to the previous idea that the variance for the mean should get smaller as I increase the number of samples.
I've asked this question elsewhere and from what I understand variances in this case do not add. Somebody talked about there being a difference between "i.i.d. and independence". But I don't get it, either something is independent or it is not. The way I see it acting like $x_1$ and $x_2$ are dependent would be committing the gamblers fallacy.
Or is there a difference between a) the process of first sampling $x_1$, then $x_2$ and b) viewing the sampling of $x_1$ and $x_2$ as a single process?
I tried following this proof this for the sum of variances and using twice the same random variable $X$ but that resulted in nonsense.
In short: What's the deal with independence in this context and how does it influence $var(x_1 +x_2)$?