It is difficult to get any hard facts here, as there some preferences of certain avatar types might correlate with some other properties which might influence how good the posts are, or otherwise influence the votes.
You might want to do an experiment here:
- Create three new accounts.
- One account gets a "funny" image, one stays with the default Gravatar pattern, one gets a serious looking photo.
- Each time when you are about to post a question or answer, throw a dice to decide which of the three accounts to use.
- After about 100 (or 1000?) posts each, compare which got most upvotes/downvotes.
To exclude other factors, all three should have similar names. Or you might want to add different types of names, too: the default user920248 type, a real sounding name (not one already active on SO), a pseudonym like "OffBySome" (but again, not one already in use here).
This would be nine accounts, quite some work.