1. **Problem Statement:** James has calculated the mean and median of a data set. The median is 147.5 and the mean is 209.6. We need to determine if the difference between these two values is significant and explain why it might have occurred. 2. **Understanding 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 data points and $n$ is the number of points. - The **median** is the middle value when data points are ordered from smallest to largest. 3. **Significance of Difference:** - A large difference between mean and median often indicates that the data is **skewed** or contains **outliers**. - Since the mean (209.6) is much larger than the median (147.5), this suggests the data is **right-skewed** (some very large values pull the mean upward). 4. **Why the Difference Occurred:** - The mean is sensitive to extreme values (outliers), while the median is more robust. - If the data set contains unusually high values, the mean increases significantly, but the median remains closer to the center of the data. 5. **Conclusion:** - The difference is significant because it indicates the data distribution is not symmetric. - The presence of outliers or skewness causes the mean to be higher than the median. This understanding helps in choosing appropriate measures of central tendency depending on data characteristics.