MathDB
Means, Medians, and Modes

Source: 2019 AMC 10A #12 / AMC 12A #7

February 8, 2019
AMC 10AMC2019 AMCAMC 122019 AMC 12A2019 AMC 10Astatistics

Problem Statement

Melanie computes the mean μ\mu, the median MM, and the modes of the 365365 values that are the dates in the months of 20192019. Thus her data consist of 1212 1s1\text{s}, 1212 2s2\text{s}, . . . , 1212 28s28\text{s}, 1111 29s29\text{s}, 1111 30s30\text{s}, and 77 31s31\text{s}. Let dd be the median of the modes. Which of the following statements is true?
<spanclass=′latex−bold′>(A)</span>μ<d<M<spanclass=′latex−bold′>(B)</span>M<d<μ<spanclass=′latex−bold′>(C)</span>d=M=μ<spanclass=′latex−bold′>(D)</span>d<M<μ<spanclass=′latex−bold′>(E)</span>d<μ<M<span class='latex-bold'>(A) </span> \mu < d < M \qquad<span class='latex-bold'>(B) </span> M < d < \mu \qquad<span class='latex-bold'>(C) </span> d = M =\mu \qquad<span class='latex-bold'>(D) </span> d < M < \mu \qquad<span class='latex-bold'>(E) </span> d < \mu < M
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