Quite a few people use the Innvulnerable Rager archetype for barbarians, which gives you DR instead of trying to keep AC competitively high. The idea being, you hit harder and can take more punishment than your enemies, so you'll come out on top.
It's quite hard for barbarians to keep AC high enough, because they tend to 2H weapons so no shield, and only medium armor proficiency. So it seems better to pick another horse to bet on (HP).
Then again, I've seen Uncanny Dodge be very useful. Quite often barbarians are at the front of the party when a bunch of NPC rogues burst out from ambush and try to sneak attack the PCs. But with Uncanny Dodge the front PC is not flat-footed so no sneak attack damage.
So, suppose you're going Invulnerable Rager, how much dex to you need? Well, it still has a few uses: Acrobatics (jumping over obstacles), Reflex saving throws, ranged to-hit (thrown weapons are a good addition to a 2H weapon build). Also, if you're using a polearm and Combat Reflexes, you'll want some dexterity as well.
All in all I'd say 10-14, depending on how much use you have for ranged attacks and combat reflexes.