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Select the outlier in the data set. $\newline$ $6, 73, 77, 78, 81, 84, 85, 89, 99$

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Calculate quartiles and interquartile range

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Q. Select the outlier in the data set.

\newline

6, 73, 77, 78, 81, 84, 85, 89, 99

Arrange Data Set: Arrange the data set in ascending order. $\newline$ The data set is already in ascending order: $6, 73, 77, 78, 81, 84, 85, 89, 99$ .
Calculate Quartiles: Calculate the first quartile ( $Q_1$ ), the median ( $Q_2$ ), and the third quartile ( $Q_3$ ) of the data set. $\newline$ To find $Q_1$ , we need to find the median of the lower half of the data set. The lower half is $6$ , $73$ , $77$ , $78$ . The median of this half is the average of $73$ and $77$ . $\newline$ $Q_2$ $0$ $\newline$ To find $Q_2$ , which is the median of the data set, we have $Q_2$ $2$ as the middle value since it is the fifth number in the ordered list. $\newline$ To find $Q_3$ , we need to find the median of the upper half of the data set. The upper half is $Q_2$ $4$ , $Q_2$ $5$ , $Q_2$ $6$ , $Q_2$ $7$ . The median of this half is the average of $Q_2$ $5$ and $Q_2$ $6$ . $\newline$ $Q_3$ $0$
Calculate IQR: Calculate the interquartile range (IQR). $\newline$ $IQR = Q3 - Q1 = 87 - 75 = 12$
Determine Bounds: Determine the lower and upper bounds for potential outliers. $\newline$ Lower bound = $Q1 - 1.5 \times IQR = 75 - 1.5 \times 12 = 75 - 18 = 57$ $\newline$ Upper bound = $Q3 + 1.5 \times IQR = 87 + 1.5 \times 12 = 87 + 18 = 105$
Identify Outliers: Identify any values that fall outside the bounds determined in Step $4$ . $\newline$ The value $6$ is below the lower bound of $57$ , so it is considered an outlier.

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