Select the outlier in the data set. $\newline$ $3, 69, 71, 74, 78, 84, 87, 89, 97$ $\newline$ If the outlier were removed from the data set, would the mean increase or decrease? $\newline$ Choices: $\newline$ (A)increase $\newline$ (B)decrease

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Grade 8

Identify an outlier and describe the effect of removing it

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Q. Select the outlier in the data set.

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3, 69, 71, 74, 78, 84, 87, 89, 97

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If the outlier were removed from the data set, would the mean increase or decrease?

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Choices:

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(A)increase

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(B)decrease

Identify Outlier: Identify the outlier in the given data set. $\newline$ The data set is: $3, 69, 71, 74, 78, 84, 87, 89, 97$ $\newline$ An outlier is a data point that is significantly different from the rest of the data. We can use the interquartile range (IQR) method to identify outliers.
Calculate Quartiles: Calculate the quartiles ( $Q1$ , $Q2$ , $Q3$ ) of the data set. $\newline$ To find the quartiles, we need to arrange the data in ascending order (which it already is) and then find the median ( $Q2$ ), the lower quartile ( $Q1$ ), and the upper quartile ( $Q3$ ). $\newline$ The data set in ascending order is: $3$ , $69$ , $71$ , $74$ , $Q2$ $0$ , $Q2$ $1$ , $Q2$ $2$ , $Q2$ $3$ , $Q2$ $4$ $\newline$ Since there are $Q2$ $5$ data points, the median ( $Q2$ ) is the middle value, which is $Q2$ $0$ .
Find Quartile Boundaries: Find the lower quartile ( $Q_1$ ) and the upper quartile ( $Q_3$ ). $\newline$ The lower half of the data set (below the median) is: $3, 69, 71, 74$ $\newline$ The upper half of the data set (above the median) is: $84, 87, 89, 97$ $\newline$ Since there are $4$ numbers in each half, the lower quartile ( $Q_1$ ) is the average of the middle two numbers of the lower half: $(69 + 71) / 2 = 70$ $\newline$ The upper quartile ( $Q_3$ ) is the average of the middle two numbers of the upper half: $(87 + 89) / 2 = 88$
Calculate IQR: Calculate the interquartile range (IQR). $IQR = Q3 - Q1 = 88 - 70 = 18$
Determine Outlier Boundaries: Determine the outlier boundaries. $\newline$ The lower boundary for outliers is $Q1 - 1.5 \times IQR = 70 - 1.5 \times 18 = 70 - 27 = 43$ $\newline$ The upper boundary for outliers is $Q3 + 1.5 \times IQR = 88 + 1.5 \times 18 = 88 + 27 = 115$ $\newline$ Any data point below $43$ or above $115$ is considered an outlier.
Identify Outliers: Identify the outlier(s) in the data set. Looking at the data set, the number $3$ is below the lower boundary of $43$ , so it is an outlier.
Calculate Mean with Outlier: Calculate the mean of the data set with and without the outlier. $\newline$ First, calculate the mean with the outlier: $\newline$ Mean with outlier = $(3 + 69 + 71 + 74 + 78 + 84 + 87 + 89 + 97) / 9$ $\newline$ Mean with outlier = $642 / 9$ $\newline$ Mean with outlier = $71.33$ (rounded to two decimal places)
Calculate Mean without Outlier: Now, calculate the mean without the outlier: $\newline$ Mean without outlier = $(69 + 71 + 74 + 78 + 84 + 87 + 89 + 97) / 8$ $\newline$ Mean without outlier = $649 / 8$ $\newline$ Mean without outlier = $81.13$ (rounded to two decimal places)
Determine Mean Change: Determine if the mean would increase or decrease upon removal of the outlier. Since the mean without the outlier ( $81.13$ ) is greater than the mean with the outlier ( $71.33$ ), the mean would increase if the outlier were removed.