You can calculate this by reversing the CR tables. For example CR1 is a medium encounter for a party of 4 level 1 characters. CR1 has 71–85hp, if you assume it takes 3-5 rounds to kill an encounter that means you can assume 3.5-7 DPR per character.
If you continue through the table in the DMG you will arrive at an answer that is a rough average across all classes, builds, and party composition. It's certainly better than nothing, and is in line with the designers' intent.
This methodology will give you an answer that seems to agree with longstanding optimization artefacts like The Optimists' Guide to D&D 5E Damage by ClassThe Optimists' Guide to D&D 5E Damage by Class.