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Select the outlier in the data set. $\newline$ $29, 39, 49, 68, 83, 95, 99, 744$ $\newline$ 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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Identify an outlier and describe the effect of removing it

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Q. Select the outlier in the data set.

\newline

29, 39, 49, 68, 83, 95, 99, 744

\newline

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

\newline

Choices:

\newline

(A)increase

\newline

(B)decrease

Calculate Mean: Calculate the mean of the data set. $\newline$ Mean = $(29 + 39 + 49 + 68 + 83 + 95 + 99 + 744) / 8$ $\newline$ Mean = $(1106) / 8$ $\newline$ Mean = $138.25$
Calculate Standard Deviation: Calculate the standard deviation of the data set. $\newline$ First, find the squared differences from the mean for each data point. $\newline$ Squared differences: $(29-138.25)^2$ , $(39-138.25)^2$ , $(49-138.25)^2$ , $(68-138.25)^2$ , $(83-138.25)^2$ , $(95-138.25)^2$ , $(99-138.25)^2$ , $(744-138.25)^2$ $\newline$ Calculating each: $11934.5625$ , $9801.0625$ , $(39-138.25)^2$ $0$ , $(39-138.25)^2$ $1$ , $(39-138.25)^2$ $2$ , $(39-138.25)^2$ $3$ , $(39-138.25)^2$ $4$ , $(39-138.25)^2$ $5$ $\newline$ Sum of squared differences: $(39-138.25)^2$ $6$ $\newline$ Sum of squared differences = $(39-138.25)^2$ $7$ $\newline$ Variance = Sum of squared differences / (number of data points - $1$ ) $\newline$ Variance = $(39-138.25)^2$ $8$ $\newline$ Variance = $(39-138.25)^2$ $9$ $\newline$ Standard deviation = $(49-138.25)^2$ $0$ $\newline$ Standard deviation $(49-138.25)^2$ $1$ $\newline$ Standard deviation $(49-138.25)^2$ $2$
Identify Outliers: Use the standard deviation to determine if there are any outliers. $\newline$ Typically, an outlier is a data point that is more than $1.5$ times the interquartile range (IQR) above the third quartile or below the first quartile. However, since we do not have the quartiles, we can consider a data point that lies more than $3$ standard deviations from the mean as an outlier. $\newline$ Outlier threshold: Mean $\pm 3 \times$ Standard deviation $\newline$ Lower threshold: $138.25 - 3 \times 239.71 \approx -580.88$ (which is not possible since all data points are positive) $\newline$ Upper threshold: $138.25 + 3 \times 239.71 \approx 857.38$ $\newline$ The data point $744$ is above the upper threshold and is considered an outlier.
Effect of Removing Outlier: Determine the effect on the mean if the outlier is removed. Removing the outlier will decrease the sum of the data set without significantly reducing the number of data points, which will result in a lower mean.

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