So the "measured" mass is not a single value but a set of samples with varying masses? And those samples have a standard deviation from the theoretical value?

As I understand it, the mean of those values is seven standard deviations away from the theoretical value, and you should only expect a result that extreme to occur by chance roughly once per 3.9e11 reruns of the entire experiment.

But I would caution that precision is so important to both physics and statistics that what I think I understand may be quite different from what’s actually going on.

No, it's the average value taken from (naturally) slightly varying measurements of (hopefully) the exact same mass. In the standard model, all particles of a given type are assumed to share the same properties, including the same mass, and so far we have no reason to believe otherwise.