Mr. Chen gave a test to his class of $20$ students. The dot plot below shows the distribution of test scores.

Later Mr. Chen discovered that there was a scoring error on one of the questions. He regraded the tests, awarding some of the students $5$ extra points, which increased the median test score to $85$. What is the minimum number of students who received extra points?
(Note that the median test score equals the average of the $2$ scores in the middle if the $20$ test scores are arranged in increasing order.)