To clarify, and this is way off topic at this point, but I personally wouldn't use the word normal, or tell them they're normal or abnormal, instead I would use healthy. It might be easier to explain using a medical example.
Earlier this month, I went to get my heart checked out. Turns out my heart is abnormal. By that, I have a low heart rate, lower than the norm, and my heart pumps a lot of blood with each pump. This is absolutely healthy, from what I was told, but if someone asked me if my heart was normal I would say no.
So for mental health: Let's take Tez's example of a pet dying. If I had a dog, and he died, I would probably be devastated. I would cope by talking with my fiancée, and spend a lot of time training at my dojo. Is this normal? My bet would be no, most people probably talk to their friends or significant other, but not a lot of people would handle it by getting thrown around/throwing other people around. If you think about it, it's downright weird that that's what I enjoy and is what gets me out of a funk. Is it healthy? I think so, and I certainly wouldn't need counseling because of it, even though I'm not 'normal'.
Again though, entirely off topic by this point, those words are just one of the things that I care far more than I should about their definitions/differences, like some people on here (not saying names) with what 'self defence' is.