Zombies aggro up to 32 blocks away, as soon as they can move. Other mobs will not aggro until the player moves to within about 16 blocks.

I reproduced this effect in the images I submitted. I built a cage for the zombie and markers to show distance. While the player stands on the red block, the red line at the cage mouth is 32 blocks away. While standing well behind the red block, the zombie in the cage is outside of the mob mobility range and it is frozen in place. I hadn't even reached the red block yet when the zombie immediately came to life and started heading toward me.

http://imgur.com/a/iL0Fw