To find the 5-number summary of the dataset \( 5, 6, 6, 7, 7, 8, 8, 8, 9, 9, 14, 16 \), we need to calculate the following:
- Minimum: The smallest number in the dataset.
- First Quartile (Q1): The median of the first half of the data.
- Median (Q2): The middle value of the dataset.
- Third Quartile (Q3): The median of the second half of the data.
- Maximum: The largest number in the dataset.
Step 1: Calculate Minimum and Maximum
- Minimum: \( 5 \)
- Maximum: \( 16 \)
Step 2: Calculate Median (Q2)
The dataset has \( 12 \) values (even). Thus, the median is the average of the 6th and 7th values.
Values: \( 5, 6, 6, 7, 7, 8, 8, 8, 9, 9, 14, 16 \)
The 6th and 7th values are both \( 8 \): \[ Q2 = \frac{8 + 8}{2} = 8 \]
Step 3: Calculate Q1
Q1 is the median of the first half of the dataset. The first half is:
\( 5, 6, 6, 7, 7, 8 \)
This has \( 6 \) values, so Q1 is the average of the 3rd and 4th values, which are \( 6 \) and \( 7 \): \[ Q1 = \frac{6 + 7}{2} = 6.5 \]
Step 4: Calculate Q3
Q3 is the median of the second half of the dataset. The second half is:
\( 8, 8, 9, 9, 14, 16 \)
Again, this has \( 6 \) values, so Q3 is the average of the 3rd and 4th values, which are \( 9 \) and \( 9 \): \[ Q3 = \frac{9 + 9}{2} = 9 \]
Conclusion
Now we have the 5-number summary:
- Minimum: \( 5 \)
- Q1: \( 6.5 \)
- Median (Q2): \( 8 \)
- Q3: \( 9 \)
- Maximum: \( 16 \)
Answer
Quartile 3 is 9. Thus, the correct response is:
- 9