Given a set of $N$ iid samples, $x_k$, of random variable $X$ and the expectation estimator,
$$E[X] \approx \frac{1}{N} \sum_{k=1}^N x_k,$$
how do I find the variance of this estimator (not the variance of $X$) as a function of $N$?
I would only like to know the error in the estimate of $E[X]$ and what I would expect is that more samples (larger $N$) drives the variance down, so that
$$\lim_{N \to \infty} \bigg[\frac{1}{N} \sum_{k=1}^N x_k - E[X]\bigg] = 0.$$
Am I right about this? Perhaps variance is not the correct way to measure the error in the estimate? What I would like to understand is, how many $N$ is sufficient because there is a cost for every extra $x_k$ that is processed.