I have a statistics question that I need some help with. It’s really a question of “what is the statistical importance of a deviation from a fit?”
Let me illustrate with plots:
1. Here we have some data and a line fit to the data. Everyone I’m sure agrees that this is a good fit.

2. Now, the same data and fit, but with one point that’s off by 1-sigma. 1-sigma events happen all the time (well, roughly 1/3 of the time) so we’d still assume that the fit matches the data well.

3. Now we have a point that’s 2-sigma from the fit. Assuming a normal distribution, that should only happen by chance ~5% of the time, so we start wondering if the deviation of that point is actually a significant event.

4. Now, the real question. Instead of a single deviant point, we have two points. Both of them are 1.3-sigma from the fit. If taken individually, there’s a ~20% probability that each one does match the fit. However, we “know” that they’re correlated in that the depression of both points is related to something physically going on. How would I determine how statistically significant this depression is?

Needless to say, my real data isn’t faked and is more complex than the example, but I need to figure out the same sort of answer. I’d appreciate any help that anyone can offer. And I apologize for using my astronomer’s imprecise statistical descriptions.