Problem 5 · 2013 AMC 8
Easy
Arithmetic & Operations
mean-vs-median
Hammie is in the 6th grade and weighs 106 pounds. His quadruplet sisters are tiny babies and weigh 5, 5, 6, and 8 pounds. Which is greater, the average (mean) weight of these five children or the median weight, and by how many pounds?
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Answer: E — Average, by 20.
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Hint 1 of 2
One giant value (106) sits among four tiny ones. The median ignores how big that outlier is — it only cares about position — but the mean gets dragged toward it. So expect the average to win.
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Hint 2 of 2
Median = middle of the sorted list (resistant to outliers); mean = total ÷ count (sensitive to outliers). One lonely large number is what separates them.
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Approach: find the resistant median, then the outlier-pulled mean
- Sort the weights: 5, 5, 6, 8, 106. The median is the middle one = 6 — notice the 106's size never mattered, only that it sits at the end.
- Mean = (5 + 5 + 6 + 8 + 106) ÷ 5 = 130 ÷ 5 = 26 — the lone 106 hauls the average up to 26.
- The average is larger, by 26 − 6 = 20 pounds.
- You'll see this again: whenever one value is wildly bigger than the rest, the mean exceeds the median — that's exactly why incomes are reported by median, not average.
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