To calculate the 5-number summary of the dataset, we need to determine the following:
- Minimum
- First Quartile (Q1)
- Median (Q2)
- Third Quartile (Q3)
- Maximum
Given dataset: \[ 5, 6, 6, 7, 7, 8, 8, 8, 9, 9, 14, 16 \]
Step 1: Find the Minimum and Maximum.
- Minimum = 5
- Maximum = 16
Step 2: Find the Median (Q2). To find the median, we need to find the middle value. The dataset has 12 numbers (even), so the median is the average of the 6th and 7th values.
The 6th value is 8, and the 7th value is 8. \[ \text{Median} = \frac{8 + 8}{2} = 8 \]
Step 3: Find Q1 (First Quartile). Q1 is the median of the first half of the data (the lower 50%). The first half is: \[ 5, 6, 6, 7, 7, 8 \] This has 6 numbers. Thus, the median (Q1) is the average of the 3rd and 4th values: The 3rd value is 6 and the 4th value is 7. \[ Q1 = \frac{6 + 7}{2} = 6.5 \]
Step 4: Find Q3 (Third Quartile). Q3 is the median of the second half of the data (the upper 50%). The second half is: \[ 8, 8, 9, 9, 14, 16 \] This half has 6 numbers. Thus, the median (Q3) is the average of the 3rd and 4th values: The 3rd value is 9 and the 4th value is 9. \[ Q3 = \frac{9 + 9}{2} = 9 \]
Step 5: Summarize the Five-number summary.
- Minimum: 5
- Q1: 6.5
- Median (Q2): 8
- Q3: 9
- Maximum: 16
Thus, the 5-number summary is: 5, 6.5, 8, 9, 16.
Answer to the question: The third quartile (Q3) is 9.