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# The five-number summary for the number of males (in thousands an a company's fleet of cars is {7, 49, 66, 72, 98}. Are there any outliers? Explain.

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Given five-number summary: 7, 49, 66, 72, 98

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

\begin{align*} \text{Minimum}&=7 \\ Q_1&=49 \\ \text{Median}&=66 \\ Q_3&=72 \\ \text{Maximum}&=98 \end{align*}

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

$IQR=Q_3-Q_1=72-49=23$

$\textbf{Outliers}$ are observations that are more than 1.5 times the IQR above $Q_3$ or below $Q_1$.

\begin{align*} Q_3+1.5IQR&=72+1.5(23)=106.5 \\ Q_1-1.5IQR&=49-1.5(23)= 14.5 \end{align*}

We then note that the minimum of 7 is less than 14.5 and thus the data set contains an outlier (as the minimum is an outlier).

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