## Related questions with answers

The frequency table summarizes the distribution of time that 140 patients at the emergency room of a small-city hospital waited to receive medical attention during the last month.

$\begin{array}{lr} \hline \text { Waiting time } & \text { Frequency } \\ \hline \text { Less than } 10 \text { minutes } & 5 \\ \text { At least } 10 \text { but less than } 20 \text { minutes } & 24 \\ \text { At least } 20 \text { but less than } 30 \text { minutes } & 45 \\ \text { At least } 30 \text { but less than } 40 \text { minutes } & 38 \\ \text { At least } 40 \text { but less than } 50 \text { minutes } & 19 \\ \text { At least } 50 \text { but less than } 60 \text { minutes } & 7 \\ \text { At least } 60 \text { but less than } 70 \text { minutes } & 2 \\ \hline \end{array}$

Which of the following represents possible values for the median and IQR of waiting times for the emergency room last month? (a) median = 27 minutes and IQR = 15 minutes (b) median = 28 minutes and IQR = 25 minutes (c) median = 31 minutes and IQR = 35 minutes (d) median = 35 minutes and IQR = 45 minutes (e) median = 45 minutes and IQR = 55 minutes

Solution

VerifiedLet us first determine the total frequency in the given table:

$5+24+45+38+19+7+2=140$

The $\textbf{median}$ is the middle value of the sorted data set. Since the total frequency is 140, there are 140 data values and then we expect the median to be the average of the 70th and 71st data value.

Since the frequency of the first two categories is $5+24=29$ and since the frequency of the first three categories is $5+24+45=74$, the 70th and the 71st data value lie in the third category and thus the $\textbf{median}$ also lies in the third category, which is $\textbf{between 20 and 30 minutes}$.

Thus answer options (a) and (b) are then possible.

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