2017 AMC 12B Problems/Problem 5

Problem

The data set $[6, 19, 33, 33, 39, 41, 41, 43, 51, 57]$ has median $Q_2 = 40$ , first quartile $Q_1 = 33$ , and third quartile $Q_3 = 43$ . An outlier in a data set is a value that is more than $1.5$ times the interquartile range below the first quartle ( $Q_1$ ) or more than $1.5$ times the interquartile range above the third quartile ( $Q_3$ ), where the interquartile range is defined as $Q_3 - Q_1$ . How many outliers does this data set have?

$\textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 4$

Solution

The interquartile range is defined as $Q3 - Q1$ , which is $43 - 33 = 10$ . $1.5$ times this value is $15$ , so all values more than $15$ below $Q1$ = $33 - 15 = 18$ is an outlier. The only one that fits this is $6$ . All values more than $15$ above $Q3$ = $43; + 15 = 58$ are also outliers, of which there are none so there is only $\boxed{\textbf{(B) 1}}$ outlier in total.

2017 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 4	Followed by Problem 6
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2017 AMC 12B Problems/Problem 5

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