Resistant Measures Listed below are 10 wait times (minutes) for “Rock ‘n’ Roller Coaster” at 10 AM (from Data Set 33 “Disney World Wait Times”). The data are listed in order from lowest to highest. Find the mean and median of these ten values. Then find the mean and median after excluding the value of 180, which appears to be an outlier. Compare the two sets of results. How much was the mean affected by the inclusion of the outlier? How much is the median affected by the inclusion of the outlier?


15 20 25 30 30 35 45 50 50 180 

Verified step by step guidance

1. Step 1: Calculate the mean of the original dataset. To find the mean, sum all the values in the dataset and divide by the total number of values. Use the formula: $\mathrm{Mean}=\frac{\sum x}{n}$, where $x$ represents each data point and $n$ is the number of data points.
2. Step 2: Calculate the median of the original dataset. Since the dataset is already ordered, locate the middle value(s). If the number of values is odd, the median is the middle value. If the number of values is even, the median is the average of the two middle values. Use the formula: $\mathrm{Median}=\frac{x+y}{2}$, where $x$ and $y$ are the two middle values.
3. Step 3: Exclude the outlier value of 180 from the dataset. Recalculate the mean using the same formula as in Step 1, but now sum the remaining values and divide by the new total number of values ($n$ becomes 9).
4. Step 4: Recalculate the median after excluding the outlier. With the new dataset, locate the middle value(s) as described in Step 2. Since the number of values is now odd, the median will be the middle value.
5. Step 5: Compare the mean and median from the original dataset to the mean and median of the modified dataset (after excluding the outlier). Determine how much the mean and median were affected by the inclusion of the outlier by calculating the difference between the original and modified values for both measures.

Key Concepts

Mean

The mean, or average, is calculated by summing all values in a data set and dividing by the number of values. It provides a central value that represents the data set but can be heavily influenced by outliers, which are extreme values that differ significantly from other observations. In this question, calculating the mean both with and without the outlier (180) will illustrate how the outlier skews the average.

Median

The median is the middle value of a data set when arranged in ascending order. If there is an even number of observations, the median is the average of the two middle numbers. Unlike the mean, the median is resistant to outliers, making it a better measure of central tendency when extreme values are present. This question requires comparing the median with and without the outlier to assess its impact.

Outliers

Outliers are data points that lie significantly outside the range of the rest of the data. They can arise from variability in the data or may indicate measurement errors. In this scenario, the value of 180 is considered an outlier among the wait times, and its presence can distort the mean while having a minimal effect on the median. Understanding how to identify and handle outliers is crucial in statistical analysis.