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The five-number summary for the weights (in pounds) of the players on a high school soccer team is {113, 143, 152, 165, 210} Are there any outliers? Explain.


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Given five-number summary: 113, 143, 152, 165, 210

The five-number summary\textbf{five-number summary} consists of the minimum, the first quartile, the median, the third quartile and the maximum.

Minimum=113Q1=143Median=152Q3=165Maximum=210\begin{align*} \text{Minimum}&=113 \\ Q_1&=143 \\ \text{Median}&=152 \\ Q_3&=165 \\ \text{Maximum}&=210 \end{align*}

The interquartile range\textbf{interquartile range} IQR is the difference of the upper and lower quartile:


Outliers\textbf{Outliers} are observations that are more than 1.5 times the IQR above Q3Q_3 or below Q1Q_1.

Q3+1.5IQR=165+1.5(21)=196.5Q11.5IQR=1431.5(21)=111.5\begin{align*} Q_3+1.5IQR&=165+1.5(21)=196.5 \\ Q_1-1.5IQR&=143-1.5(21)= 111.5 \end{align*}

We then note that the maximum of 210 exceeds 196.5 and thus the data set contains an outlier (as the maximum is an outlier).

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