Are the encounter rates found by data mining the games, by trial and error with data, or from the physical guides they made for the games? Just a general curiosity after looking at different sites for encounter rates and noticing one was off (due to a typo most likely).

Most likely datamining, because it would take an unbearably long time to calculate those from trial and error like you mentioned. Datamining would give the exact answer and would be way simpler than calculating.