Interestingly, there doesn't appear to be any specified height for target placement in the NRA Conventional Pistol regulations. International Pistol specifies 55 inches, plus or minus 8 inches (which seems like a pretty large tolerance, but that's what it is).
In my experience shooting at perhaps a dozen or so different ranges in two different leagues, most of the target centers seem to be about shoulder height for an average (say 5' 10" or so) person, which I guess makes some sense in that when you're on target your arm will be more or less level.