Happy families are all alike; every unhappy family is unhappy in its own way.
Before I begin, let me emphasize I'm talking in very large generalizations. This is for clarity purposes; please don't mistake my simplifications for being rude.
I was born and raised in a "functional family." I always got enough to eat, I got help with school work, and my parents showed love and affection. They never hit my brother or myself and their fights were rare and civil. I had an excellent upbringing (thanks, by the way, Mom and Dad). For my brother's and I's part, we were good kids. We weren't angels by any means, but we stayed out of major trouble, did well in school, etc.
So when I wonder if most families are functional--perhaps little a better, perhaps a little worse than mine--I naturally think back to my childhood and conclude my childhood was typical. This is, of course, a poor way to draw a general conclusion. You can tell because when I would ask my friends who come from dysfunctional families, they would make the opposite conclusions: "we're not unusual, David, you are." They are, in all likelihood, making the same mistake I am.
So I might say "But we had many family friends who are also functional families." Of course, functional families like to hang out with other functional families so that's not very good. It leads to dysfunctional families drawing one of two conclusions:
(a) They were somewhat dysfunctional, which means they would be friends with other somewhat dysfunctional families (functional families won't have them and the somewhat dysfunctional won't hang out with the really dysfunctional one). Thus they cite the other families they know and use that as more evidence of how common people like them are.
(b) They were really dysfunctional, which means, based on this rough model, that they will have few to none family friends. So they would probably say "look, my family was really bad; it seems reasonable that something somewhat better than what I experienced would be the norm."
(This says nothing about the families that are functional to a level that's annoying. As this episode of South Park suggests, such families would have trouble finding outside friends as well.)
So how common are functional families? I have no idea and unless someone did a random sample and came up with a good measure of functionality (good luck), then I'd say no one really knows. But what learning about others' families have taught me is that while the average is hard to figure, the standard deviation is huge.