To calculate the 5-number summary of the dataset \(5, 6, 6, 7, 7, 8, 8, 8, 9, 9, 14, 16\), we need to determine the minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum.
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Minimum: The smallest number in the dataset is \(5\).
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Maximum: The largest number in the dataset is \(16\).
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Median (Q2): Since there are 12 numbers in the dataset (an even count), the median will be the average of the 6th and 7th numbers when arranged in ascending order:
- The 6th number is \(8\), and the 7th is also \(8\).
- \(Q2 = (8 + 8) / 2 = 8\).
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First Quartile (Q1): This is the median of the first half of the data (the first 6 numbers). The first half is \(5, 6, 6, 7, 7, 8\):
- The median of this subset is the average of the 3rd and 4th numbers: \(6\) and \(7\).
- \(Q1 = (6 + 7) / 2 = 6.5\).
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Third Quartile (Q3): This is the median of the second half of the data (the last 6 numbers). The second half is \(8, 8, 9, 9, 14, 16\):
- The median of this subset is the average of the 3rd and 4th numbers: \(9\) and \(9\).
- \(Q3 = (9 + 9) / 2 = 9\).
Now we can summarize the 5-number summary:
- Minimum: \(5\)
- Q1: \(6.5\)
- Median (Q2): \(8\)
- Q3: \(9\)
- Maximum: \(16\)
Among the choices given, the quartile 3 (Q3) is \(9\).
Final answer for quartile 3 (Q3): 9
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