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The frequency table below summarizes the optical magnification power of lenses on 19 models of cameras classified as "superzoom."

Maximum Zoom161820212224304042Frequency415212211\begin{matrix} \text{Maximum Zoom} & \text{16} & \text{18} & \text{20} & \text{21} & \text{22} & \text{24} & \text{30} & \text{40} & \text{42}\\ \text{Frequency} & \text{4} & \text{1} & \text{5} & \text{2} & \text{1} & \text{2} & \text{2} & \text{1} & \text{1}\\ \end{matrix}

Find the five-number summary for these data and identify any outliers.


Answered 2 years ago
Answered 2 years ago
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Using the definition (The five-number summary):

(minimum, Q1 ,median, Q3 ,maximum)\begin{align*} (\text{minimum, $Q_1$ ,median, $Q_3$ ,maximum}) \end{align*}

The total of the frequencies is 19, the median of 19 numbers will bi 9.5. We sum the frequencies from left to right until we come to 9.5 (the quartiles will be the fifth numbers from each end of the distribution) . Median is 20; Minimum is 16, maximum is 42;

Q1=18;Q3=24;Q1 is fifth from leftQ3 is fifth from rightThe five-number summary: 16,18,20,24,42IQR=1.5(Q3Q1)=1.5(2418)=1.56=9;Q19=189=9Q3+9=24+9=33;Outliers:40,42\begin{align*} Q_1&=18;\\ Q_3&=24;\\ Q_1~\text{is fifth from left}\\ Q_3~\text{is fifth from right}\\ \text{The five-number summary:}~{16,18,20,24,42}\\ IQR=1.5\cdot (Q_3-Q_1)=1.5\cdot (24-18)=1.5\cdot 6=9;\\ Q_1-9=18-9=9\\ Q_3+9=24+9=33; \text{Outliers:} {40,42} \tag{Numbers greater than 33} \end{align*}

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