In molecular dynamics simulations of proteins, the radius of gyration is often used to assess the compactness of a protein. When comparing two protein radius of gyration, what difference can be considered significant 1 Angstrom, 5 Angstroms, 10 angstroms? Significant meaning: yes it is definitely more compact or no it is definitely not more compact. I have been searching through but can't find any reference related to that.
It depends on your actual scenario. If we assume that you have two Molecular Dynamics (MD) replicas for the same protein (and their respective trajectories), then when you perform a Radius of Gyration ($R_g$) computation from a MD trajectory you commonly end up with a time series for the values for it, $R_g(t)$.
From $R_g(t)$ you can already plot the time series of the values (always good to visualize beforehand) and, once the simulation has converged for each replica, compute a histogram/distribution, so you should end up with something like the following for each MD trajectory (so, two of them in total).
Then you can use a statistical two-sample test like the Kolmogorov-Smirnov test to compare both distributions and actually assess if there is a significant statistical difference between both of them.
Now, on the other hand, if what you have is just two individual values for $R_g$ then I suspect you can just directly use the relative error between them (surely one of the samples can be considered a reference) to have an idea how one deviates from another. Maybe something like $>5$ percent is significant, this is subjective.