I have a data set expressed as in the figure:
Here $y$ is some measured quantity with known error and 'fit' is some attempt to fit a function with zero error.
In order to evaluate the quality of the fit (and thereby rank different fits) it is possible to calculate a Pearson Correlation Coefficient, in this case approx $0.8$.
My question is that although I can calculate a correlation coefficient is it possible to also calculate an error for the Pearson's correlation coefficient? i.e. $0.8\pm 0.05$?
If a Pearson Correlation Coefficient is not the best way to score the fit I would also be interested in alternatives.