1. He is average height.

2. He is medium height.

Are both grammatical?

He is of average height.

He is medium in height.

Do you mean we cannot omit 'of'?

The word "of" is usually used with "average" but not with "medium."

Weird, huh?

Yes, both sentences, "He is average height" and "He is medium height," are grammatically correct. They are both expressing the same idea, that the person being referred to has a height that is neither exceptionally tall nor exceptionally short. Both sentences use the adjective "average" or "medium" to describe the height of the person.