In the grand scheme of things, a 1200 player is probably pretty good indeed. A 1200 player could walk around a populated area (like a mall) all day long and perhaps not encounter anyone better at chess. Maybe a few on an unlucky day. My friends who are 1200 have no problem winning nearly 100% of the time when they're challenged to play by players outside of the serious chess scene.
It all depends on context. On a site like chess.com, where everyone is at least a chess player, a 1200 is better than average*, but not by a large margin, and such a player certainly wouldn't be seen as "accomplished" by most.
And at a typical open OTB tournament or chess club, a 1200 might be one of the weaker players, since the people who go to chess clubs and OTB tournaments are a self-selected group - usually it's the most serious players who would be willing to invest time and money to travel to a place just to play chess.
If I go to a local tournament I'll probably be the top player, but if I go to a major open tournament like a World Open or Millionaire Chess, nobody would notice me amongst the sea of GMs.
It's all relative :)
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*I'm going by chess.com blitz ratings, since blitz is the most popular game on this site. The average rating is currently 1095. http://www.chess.com/livechess/players?type=Blitz
<NM Happytoad> that's quite unbelievable. It makes people rated 1200 and 1450 already seem like pretty accomplished players.Â
lol