The list of distinct, positive integers that has a median of 10 and the smallest average is:

1, 2, 3, 4, 9, 11, 12, 13, 14, 15

We know the first four numbers have to be 1, 2, 3, 4 because they are the four smallest distinct, positive integers.

Then the two middle numbers (9 and 11) must sum to 20 to make the median 10. And we can see why we should use 9 and 11 to minimize the average because, for example, if we used 5 and 15, that would force all of the last four numbers to be greater.

Then the last four numbers are the four smallest, distinct positive integers greater than 11.

average = (1 + 2 + 3 + 4 + 9 + 11 + 12 + 13 + 14 + 15) / 10 = 84 / 10 = 8.4