2022 AMC 8 Problems/Problem 19
Mr. Ramos gave a test to his class of students. The dot plot below shows the distribution of test scores.
Later Mr. Ramos discovered that there was a scoring error on one of the questions. He regraded the tests, awarding some of the students extra points, which increased the median test score to . What is the minimum number of students who received extra points?
(Note that the median test score equals the average of the scores in the middle if the test scores are arranged in increasing order.)
Before Mr. Ramos added scores, the median was . There are two cases now:
Case #: The middle two scores are and . To do this, we firstly suppose that the two students who got are awarded the extra points. We then realize that this case will have a lot of students who receive the extra points, therefore we reject this case.
Case #: The middle two scores are both . To do this, we simply need to suppose that some of the students who got are awarded the extra points. Note that there are students who got or less. Therefore there must be only student who got so that the middle two scores are both . Therefore our answer is .
A diagram to visualize better what I explained here:
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