To find the mean of the data set, follow these steps:

1. Add all the numbers together: $4 + 5 + 6 + 6 + 6 + 7 + 7 + 7 + 7 + 8 = 63$

2. Count the number of data points: There are 10 numbers in the data set.

3. Divide the sum by the number of data points: $\text{Mean} = \frac{63}{10} = 6.3$

So, the mean of the data set is 6.3.

Would you like more details or have any questions?

Here are 5 related questions for further practice:

1. What is the median of the given data set?
2. How would you find the mode of this data set?
3. What is the range of the data?
4. How does adding a new data point of 9 affect the mean?
5. What is the impact of removing the smallest value from the data on the mean?

Tip: The mean is sensitive to every value in the data set, so any change in the data will affect the mean.