In which league would one have to be, just to be considered average (statistically)?

According to the statistics of sc2ranks.com, currently over 50% of all players are either in silver or bronze league. This means the median of skill is somewhere at the top of silver league.

Those statistics are only for the current ladder season, so inactive players aren't counted.

From Liquipedia's page on the Battle.net leagues:

League        Targeted Ratio of *Active* Players
----------------------------------------------
Grandmaster   200 players
Master        2%
Diamond       18%
Platinum      20%
Gold          20%
Silver        20%
Bronze        20%

This means that the average (and median) player should be in the Gold league.