I have ~20 datasets which contain data that I assume is, conditional on a single variable, binomially distributed. I have built a physical model of the process generating the data and can simulate data from this model. I am interested in quantifying how well this simulated data match the conditionally binomially distributed data that I have observed. Additionally, for each dataset the physical model generating the data is slightly different.
So what I essentially end up with is ~20 pairs curves like this
and I am looking for an appropriate and easily interpretable metric for assessing essentially how close these curves are to one another. My first thought is to calculate the Pearson-correlation between them but this is a terrible metric because I want to quantify exactly how well the data matches the observations, rather than some shifted and scaled version of this. My next thought is to just calculate the MSE between the two curves but this can be a bit hard to interpret without knowing the full data distribution.
I already know that my physical model does not perfectly reproduce the observed data, so doing a $\chi^2$ test or something like this will not be super productive.
The measure should ideally also accommodate sampling error in the values of the data curve, estimation error in the simulated curve (as pointed out in comment below).
What would be an appropriate measure in this case? Is there something very obvious I am missing?