Arrange the 9 distinct observations in ascending order

Let the observations be $x_1 < x_2 < x_3 < x_4 < x_5 < x_6 < x_7 < x_8 < x_9.$

Use the definition of median for 9 observations

Since there are 9 observations (an odd number), the median is the middle term, i.e. the 5th observation.

Hence, $x_5 = 20.5.$

Modify the largest 4 observations

The largest 4 observations are $x_6, x_7, x_8, x_9.$

Each of these is increased by 2, so the new set becomes $x_1, x_2, x_3, x_4, x_5, x_6+2, x_7+2, x_8+2, x_9+2.$

Check the median of the new set

The first five observations remain unchanged: $x_1 < x_2 < x_3 < x_4 < x_5.$

Also, since only the last four terms were increased, the 5th term is still $x_5$.

Therefore, the median of the new set is still $x_5 = 20.5.$

Conclusion

The median does not change.

Therefore, the correct option is: $\boxed{\text{D: remains the same as that of the original set}}$

Comparison with stored correct answer

Stored correct answer: D

Derived answer: D

These match.