In which data set is the mean greater than the median?

The relationship between the **mean** and the **median** of a data set provides valuable insights into the distribution's shape. Specifically, when the **mean is greater than the median**, the data set is typically **right-skewed (positively skewed)**. Here's a detailed explanation: ### **Understanding Skewness** 1. **Symmetrical Distribution:** - **Mean = Median = Mode** - Example: A normal distribution where data is evenly distributed around the central point. 2. **Right-Skewed (Positively Skewed) Distribution:** - **Mean > Median > Mode** - **Characteristics:** - Tail extends to the right (higher values). - A few significantly high values pull the mean upward. - Median remains less affected by extreme values. 3. **Left-Skewed (Negatively Skewed) Distribution:** - **Mean < Median < Mode** - **Characteristics:** - Tail extends to the left (lower values). - A few significantly low values pull the mean downward. - Median remains less affected by extreme values. ### **Example of a Right-Skewed Data Set** Consider the following data sets: - **Data Set A:** 2, 3, 5, 7, 9 - **Mean:** (2 + 3 + 5 + 7 + 9) / 5 = 5.2 - **Median:** 5 - **Relationship:** Mean > Median - **Data Set B:** 2, 3, 5, 7, 100 - **Mean:** (2 + 3 + 5 + 7 + 100) / 5 = 23.4 - **Median:** 5 - **Relationship:** Mean > Median - **Data Set C:** 2, 3, 5, 7, 9, 100 - **Mean:** (2 + 3 + 5 + 7 + 9 + 100) / 6 ≈ 21 - **Median:** (5 + 7) / 2 = 6 - **Relationship:** Mean > Median In all these examples, the presence of an unusually high value (like 100) skews the distribution to the right, resulting in the mean being greater than the median. ### **Real-World Applications** - **Income Levels:** Often right-skewed because a small number of individuals earn significantly higher incomes, raising the average (mean) above the median. - **Housing Prices:** Similar to income, a few high-priced properties can increase the mean price beyond the median price. ### **Identifying Skewness Without Calculation** - **Visual Inspection:** Plotting the data using histograms or box plots can help identify skewness. A longer tail on the right indicates a right-skewed distribution. - **Comparing Mean and Median:** Simply calculating both can provide clues about the distribution's skewness. ### **Conclusion** The **mean is greater than the median** in data sets that are **right-skewed (positively skewed)**. This typically occurs when there are outliers or a long tail on the higher end of the data, pulling the mean upwards while having a lesser effect on the median. If you have specific data sets you'd like to analyze, feel free to share them, and I can help determine which one has a mean greater than the median!