Is there a test to show that your RMSE is sufficently small for a dataset? I am currently experimenting with some machine learning, which is going well. However, I am curious if there is a way to show that the $RMSE$ I am achieving is sufficiently small enough for the data set.
For context, the data I am working with is on a scale from $10^0$ up to a scale of $10^2$. With an SD on the order of $10^1$, this is also the scale of my $RMSE$ I am achieving. When I compare the $\frac{RMSE}{SD}$, I get that: $\frac{RMSE}{SD} \approx 1$, and when I compare the $RMSE$ of both the training data to the testing data, they also are similar. i.e.: $$RMSE_{Train} - RMSE_{Test} \approx 0$$
So would both of these conditions satisfy that the $RMSE$ I am achieving, is sufficiently small enough for my data set?
Or is there a test I can do to specifically assess this? Thank you for your help.

There is no rule to say what is a good value. This is not like school where there is some cutoff for an A, a lower standard for a B, and even lower standard for a C, etc. It will depend on the context. For instance, even $R^2$, which has some scaling built in (related to what you’re doing by comparing to the overall standard deviation), is not interpretable as a grade in school. In some fields, $R^2=0.4$ might be outstanding performance. In other fields, $R^2=0.9$ might be rather pedestrian performance.
With RMSE, the situation is even worse. Your RMSE also has some units, and if you just want to make the RMSE number smaller, change the units. The distance from New York to Sydney is a lot of meters but a lot smaller of a number of light years!
