If the median of an even number of distinct integers is ten, then the two middle numbers have to sum to 20.
Making them 9 and 11 would be ideal so we can use the rest of the small one-digit numbers for the below numbers and the small two-digit numbers directly above 11(what I'm saying is using these numbers will minimize the other numbers)
To minimize the others, we make them 1, 2, 3, 4, 9, 11, 12, 13, 14, and 15.
The average of those numbers is 84/10 or 8.4.