To determine which measure is the greatest, let's calculate the following measures:

1. Median: The median is the middle value when the data is arranged in order. The data set is already arranged:
   $5, 6, 6, 8, 9, 10$
   Since there are 6 numbers, the median is the average of the two middle values (6 and 8):
   $\text{Median} = \frac{6 + 8}{2} = 7$

2. Range: The range is the difference between the highest and lowest values:
   $\text{Range} = 10 - 5 = 5$

3. Mean: The mean is the average of the numbers:
   $\text{Mean} = \frac{5 + 6 + 6 + 8 + 9 + 10}{6} = \frac{44}{6} \approx 7.33$

4. Mode: The mode is the most frequent value. In this case, the number 6 appears twice, so:
   $\text{Mode} = 6$

Comparison:

• Median = 7
• Range = 5
• Mean ≈ 7.33
• Mode = 6

Thus, the mean (option c) is the greatest.

Would you like more details or have any questions? Here are some related questions to deepen your understanding:

1. How is the mean different from the median in skewed distributions?
2. What happens to the range if the highest or lowest value changes?
3. Can a data set have more than one mode?
4. How do outliers affect the mean and median?
5. Why is the median often used in real-world data sets like income?

Tip: The mean is sensitive to extreme values (outliers), while the median gives a better central tendency for skewed data sets.