They may be equivalent, but neither makes any sense as written.
An average is a single number that summarizes a set of numbers.
It would make sense to say "The graph shows the number of patients using public hospitals by month from 1990 to 2000." This singular formulation stresses that the graph shows each individual number. Those numbers have an average that may differ from each individual number.
It would also make sense to say "The graph shows monthly patient numbers at public hospitals from 1990 to 2000." This plural formulation stresses that the graph represents a large set of numbers.
Which formulation to use depends in part on what you want to stress. Sense and intent determine word choice.
My personal preference would also depend heavily on the detail in the graph. If you can in fact distinguish individual numbers, I would probably use the singular. If, as is usually the case, the graph does not have sufficient detail to discern accurately the individual numbers from which it is derived, I would probably use the plural.
EDIT: A comment by the OP is causing me to amend this answer somewhat. It now seems that the numbers are not monthly numbers at all, but rather monthly averages by year. That is the only way to make sense of the comment. It is also consistent with the lack of detail in the graph.