How to measure if a mean is stable Background
I am a physics major, however I am currently interning at a psychiatry/neuro-imaging laboratory. The primary area of research in my lab is diffusion tensor imaging (DTI). A lot of the studies that are conducted here are group comparisons between "normal" controls, and patients whose brains are to some degree "abnormal" (Schizophrenia, Traumatic Brain Injury, etc.). A common procedure is to segment the brains into several regions of interest (ROI) and then calculate some characteristic numbers for each ROI, most importantly the Fractional Anisotropy (FA). To put it very simple, every region of the brain is is assigned a number between 0 and 1.
Since the FA of an ROI is calculated as the mean of all voxels in that ROI, it can be assumed that the FA value for a specific ROI is normally distributed among all healthy controls (Central Limit Theorem).
What happens next is that for each ROI, the mean FA is calculated from all healthy controls, i.e. for every ROI in the brain we find a number that is the "standard value" for this ROI. It is then investigated how much the patients with "abnormal" brains differ from this "standard value".
Question/Problem
An important question here that has been given to me as a summer project is how many healthy controls it takes to make this mean (i.e. the "standard value") stable. "Stable" means that the mean of $N$ and $N+1$ controls does "not differ much", e.g. adding another control contributes only negligible new information.
I know that this is a very vague formulation, and this is also the reason why I have come here:
What do you think would be a suitable way of characterizing if the mean of $N$ and $N+1$ controls does "not differ much"?
My supervisor has described to me what she would ideally like to have at the end of my internship here: A program into which she can put the data of all the controls she has so far, then the program does its magic, and in the end it tells her "You have enough controls, your mean is stable", or "Your mean is not stable, and you need at least $X$ more controls to make it stable."
I realize this probably sounds pretty wishy-washy and not very well-defined to you, but it's the same for me :-/
I have already started to do some research myself, and tried a couple of things, but nothing so far seemed particularly promising.
Therefore, I would be very grateful for any kind of advice how to tackle this problem. Algorithms, references to literature, wild ideas, ... anything that gives me a starting point for further research would be greatly appreciated!
Thanks in advance for your efforts!
 A: Means don't switch from unstable to stable. 
Given some amount of variation in the population (which itself can be estimated, of course), you can compute the standard error for a mean. The expected variation from $N$ to $N+1$ will be a function of that, but note that a wildly different next person will move the mean more than a next person whose values are very typical.
That standard error of the mean decreases as a smooth function of sample size. 
In some situations it might be better to work with a margin of error rather than the standard error (an interval half-width), but the two will be closely related. It might also be better to work with relative error rather than absolute (a percentage margin of error, perhaps).
Either way, someone in the domain will still need to say what "stable" is in those terms; it's really in-domain knowledge, not statistics, that determines that. But they may be more willing to say "we regard a 2% margin of error as stable" than to plump for a raw standard error or a raw $N$.
