AMC 10-2019 (A12)
Melanie computes the mean µ, the median M, and the modes of the 365 values that are the dates in the months of 2019. Thus her data consists of 12 1s, 12 2s, ... , 12 28s, 11 29s, 11 30s, and 7 31s. Let d be the median of the modes. Which of the following statements is true?
(A) µ < d < M (B) M < d< µ (C) d = M = µ (D) d < M < µ (E) d < µ < M
d obviously must be less than M, so we can rule out (B) and (C). Since there are 365 entries, M is the 183rd number, which lies between 15*12 + 1 (180) and 16*12 (192). Therefore, M=16. We can also see that d = 14.5. If we ignored the 11 29s, 11 30s, and 7 31s, µ would also be =14.5. Because there are larger elements given, though, we can see that µ is greater than d. On the other hand, since there are fewer 29's, 30's, and 31's than the rest of the numbers, the mean has to be lower than the median. So, the answer is E (d < µ < M).