“You’re as stubborn as my brother”
“But your younger brother isn’t stubborn at all”
“I was talking about my older brother”
The average is a generic concept covering multiple more specific concepts like mean and median. If you say something about the generic concept it should not depend on any properties of just one of the specific concepts, in order to hold generally.
Your brother is a term for a single person that is simply under-determined and could turn out to apply to either one, but not both. What you say about your brother should apply to the brother you mean, in order to hold.
I don’t even understand what you mean. If it covers more than one concept that can contradict each other, how can you expect that it is true for all of them?
Take the set 1, 7, 10. When I say the average is 7, you can say “no, the mean is 6” and when I say the average is 6 you will answer “no, the median is 7”