Suppose you have two data plots: Plot 1 is symmetric, and Plot 2 is the same as Plot 1 but with one large outlier added to the right. How do the $\mathrm{mean}$ and $\mathrm{median}$ change from Plot 1 to Plot 2?

• A. The $\mathrm{mean}$ increases more than the $\mathrm{median}$.
• B. The $\mathrm{median}$ increases more than the $\mathrm{mean}$.
• C. Both the $\mathrm{mean}$ and $\mathrm{median}$ increase by the same amount.
• D. Neither the $\mathrm{mean}$ nor the $\mathrm{median}$ changes.

Answer: (A)

Verified step by step guidance

1. Step 1: Understand the definitions of mean and median. The mean is the average of all data points, calculated as \(\text{mean} = \frac{\sum x_i}{n}\), where \(x_i\) are the data points and \(n\) is the number of points. The median is the middle value when the data are ordered from smallest to largest.
2. Step 2: Consider the effect of adding a large outlier to the right (a very large value) on the mean. Since the mean sums all values and divides by \(n\), a large outlier will increase the total sum significantly, thus increasing the mean.
3. Step 3: Consider the effect of adding the same large outlier on the median. The median depends on the middle value(s) of the ordered data. Adding one large outlier to the right will shift the data set but may not change the middle value much, especially if the data set is large.
4. Step 4: Compare the changes in mean and median. Because the mean incorporates all values, including the outlier, it will increase more than the median, which is more resistant to extreme values.
5. Step 5: Conclude that when a large outlier is added to the right of a symmetric data set, the mean increases more than the median.