To determine which average is the most sensitive to extreme values, we will analyze three types of averages: mean, median, and mode.
### Step-by-Step Solution:
1. **Understanding Extreme Values**:
- Extreme values, also known as outliers, are data points that are significantly higher or lower than the other values in a dataset. For example, in the dataset {1, 2, 3, 4, 5}, the extreme values are 1 (the minimum) and 5 (the maximum).
2. **Calculating the Mean**:
- The mean is calculated by adding all the values together and dividing by the number of values.
- For the dataset {1, 2, 3, 4, 5}:
\[
\text{Mean} = \frac{1 + 2 + 3 + 4 + 5}{5} = \frac{15}{5} = 3
\]
- Now, if we change one of the values, say we replace 2 with 10, the new dataset becomes {1, 10, 3, 4, 5}.
\[
\text{New Mean} = \frac{1 + 10 + 3 + 4 + 5}{5} = \frac{23}{5} = 4.6
\]
- The mean has changed significantly due to the extreme value (10).
3. **Calculating the Median**:
- The median is the middle value when the data is arranged in ascending order.
- For the dataset {1, 2, 3, 4, 5}, the median is 3 (the third value).
- If we change the value 2 to 10, the new dataset {1, 3, 4, 5, 10} still has a median of 3, as it is still the middle value.
4. **Calculating the Mode**:
- The mode is the value that appears most frequently in the dataset.
- In the dataset {1, 2, 3, 4, 5}, there is no repeating number, so there is no mode.
- If we change the dataset to {1, 2, 3, 3, 4, 5}, the mode is 3.
- If we change the value 2 to 10, the dataset becomes {1, 3, 3, 4, 5, 10}. The mode remains 3, as it is still the most frequently occurring number.
5. **Conclusion**:
- From the calculations above, we can see that the mean is significantly affected by extreme values, while the median and mode remain unchanged. Therefore, the average that is most sensitive to extreme values is the **mean**.
