Do you mean average of all shots taken or just successful shots?
That will vary widely.
My average bow shot on elk is probably 20. From 3 to 35.
I'm sure many others average much farther. But I'm not
good enough to ethically take 45 or 60 or 80 yard shots.
It would be easy to average shots at 1000 on antelope.
Your average shot distance will depend a lot/mostly on you.
Few hunting rifles are incapable of being accurate enough to
make ethical kills on game at 500
Proficient at 500 under hunting conditions?  You need to find  that not just
specify it. " By next season I'm going to be proficient at 850". Yeah,
great goal, only time will tell if I can achieve it. Odds are I can't.
Watch a few of these
Note these are frequent and pretty experienced shooters under ideal conditions
with rifles you probably aren't going to be packing into the woods.