# Number of samples required to get mean of unknown function up to some standard error

So I have some code simulating the behavior of a complex system. The system works perfectly (with a "fidelity" of $$f=1$$) for some parameter $$\beta=0$$. For other values of $$\beta$$ the fidelity will be some number between 0 and 1. I know that $$f(\beta)$$ peaks for $$\beta=0$$ (not sure how sharply exactly) and then drops off to some kind of noisy floor for larger absolute values of $$\beta$$. I don't have any analytic insight into $$f(\beta)$$ as a function.

Now, if I draw $$N$$ values $$\beta$$ from a Gaussian distribution with mean 0 and a fixed standard deviation $$\sigma_\beta$$, and then calculate the mean fidelity as $$\bar f(\sigma_\beta) = \frac{1}{N}\sum_{n=1}^N f(\beta_n)$$, is there a way for me to estimate the standard error on $$\bar f$$? Or, alternatively, is there a good way for me to figure out how many samples $$N$$ I need to take to get a value of $$\bar f$$ that I feel confident about within 1%?

This might be a fairly basic question, but it's been a long time since my undergraduate days ;-). In the absence of anything quantitative, I'd probably just try pulling 100 samples and check how stable the resulting $$\bar f$$ is compared to if I just used 50 samples (i.e., throw out every other sample).