Harzer is right, but he left a lot out.
It is
only in certain very specific registers that you might find,
"I am average height", or
"I am of average height." Perhaps if you were talking on the telephone to a blind date (obviously someone you've never met) you might say something like that. Or if you were describing yourself to a potential employer (again, on the telephone). The key, here, is that it is such formal language, you would use it only if you wished to remain aloof, in some respect, from the listener. (
Please note, dear reader, the elemental use of 'remoteness' here...it is a very useful concept). There might be several different scenarios which could cause you to feel this way, and so I've only described two of them here.
In regular, natural, conversation, where you had no particular reason to distance yourself from your listener(s), you probably would say:
"I'm about average height", or
"I'm five feet eleven", or sometimes, in the United States, at least,
"I'm five foot eleven." (
Note the contractions).
But as Harzer said, never,
*"I have average height."
Larry Latham