I don't really think so.
I guess normality implies a majority or an average which I don't think exists in terms of people or personalities.
A teacher at our school once talked about "the average Australian household" which she thought to be a mother and a father with two children and a mortgaged house. But in reality, this "normal household" is not that common at all.
As for personalities, the statistics for MBTI frequencies (not that MBTI typing is really accurate anyway) show that ISFJ is the "majority"... but represent only 13.4%. I don't think 13.4% represents any kind of average or commonality in a group.
Classification aside, our experiences as people are so varied that surely none of it is normal. Someone who seems perfectly average may have a silent illness, or a financially disadvantaged background, or have indigenous ancestry - none of which are considered "normal".
I definitely don't count myself, or any other person, as normal!