Random draws from Gaussian, at what point (sample size) can I decide that the array of samples resemble the original distribution well? I am trying to simulate saccadic reaction times (for eye movement), with a Gaussian distribution with a true mean "mu" and a true standard deviation "sigma". Let's say I begin the experiment (by simulating draws from that normal distribution), is there a method for quantifying how many samples should I draw until the experiment data starts to approximate the original distribution sufficiently well?
I have tried to use the Maximum Likelihood estimator for the array of samples, but I haven't got any significant results.
Thank you!

The difficulty of your question lies in the definition of "resemble well". This depends on how, specifically, you will use the results. What counts as good resemblance for my product development methodology decision might be woefully insufficient for a defect inspection in critical-parts manufacturing.
There is no hard and fast rule.
What I would do if I were you is start with simulating 10 samples and then do whatever you need to do. Repeat that 500 times, and see how much the outcome varies. If the variation is too much, try 30 samples, repeat 500 times, see how much the outcome varies. If that variation is also too much, try 100 samples, then 300, then 1000, and so on.
Once you get to 10,000 samples (if you do) it might be painful to run the simulations and wait for the results. Then you can look at the trend of the variation of the outcome, and see if you can extrapolate some sort of curve. Figure out roughly where on that curve you need to go, try that, and use the outcome of that to improve the next guess.

How do you know if the variation is too much? Again, depends on the specific decision you're trying to make. Maybe you need to know a value within ±20, and you want to be 90 % confident of this. Or you want to predict something in a way that when you say it is true, it is indeed true 99.5 % of the time.
Either way, that depends on the decision you're trying to make, and the costs involved in getting it wrong.
If you're not making a decision and just reporting your results to someone else, you still need to know what precision they require.

If this sounds very numerical, that's because the numerical solution is the only general solution to your problem.
If you're extremely lucky, there might be a theoretical solution to your specific use case, but I wouldn't hold my breath.
Either way, you need to start from exactly what you will use the numbers for.
