1. **Problem Statement:** We need to understand the definition of the mean and identify a data set whose mean is not equal to any value in the set. 2. **Definition of Mean:** The mean (or average) of a data set is calculated by summing all the data entries and then dividing by the number of entries. 3. **Formula for Mean:** $$\text{Mean} = \frac{\text{Sum of all data entries}}{\text{Number of data entries}}$$ 4. **Explanation of Options:** - Option A describes an outlier, not the mean. - Option B correctly defines the mean. - Option C describes the mode. - Option D describes the median. 5. **Check each data set to find the mean and see if it equals any data entry:** - For data set $4, 4, 6, 6$: $$\text{Mean} = \frac{4 + 4 + 6 + 6}{4} = \frac{20}{4} = 5$$ 5 is not in the set. - For data set $4, 6, 8, 10, 12$: $$\text{Mean} = \frac{4 + 6 + 8 + 10 + 12}{5} = \frac{40}{5} = 8$$ 8 is in the set. - For data set $17, 18, 19, 20, 21$: $$\text{Mean} = \frac{17 + 18 + 19 + 20 + 21}{5} = \frac{95}{5} = 19$$ 19 is in the set. - For data set $4, 4, 4, 5, 6, 6, 6$: $$\text{Mean} = \frac{4 + 4 + 4 + 5 + 6 + 6 + 6}{7} = \frac{35}{7} = 5$$ 5 is in the set. 6. **Conclusion:** The data set $4, 4, 6, 6$ has a mean of 5, which is not a value in the set. **Final answers:** - Definition of mean: Option B - Data set whose mean is not in the set: $4, 4, 6, 6$