To find the interquartile range (IQR) of the data, we need to follow these steps:
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Order the data: The data is already ordered:
\(8, 9, 9, 10, 12, 12, 13, 14, 15, 15, 15, 16, 18, 19, 20\) -
Find the first quartile (\(Q_1\)) and the third quartile (\(Q_3\)):
- The first quartile (\(Q_1\)) is the median of the first half of the data (the lower half).
- The third quartile (\(Q_3\)) is the median of the second half of the data (the upper half).
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Split the data into two halves:
- Lower half: \(8, 9, 9, 10, 12, 12, 13\) (7 data points)
- Upper half: \(14, 15, 15, 15, 16, 18, 19, 20\) (8 data points)
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Calculate \(Q_1\):
- The median (middle value) of the lower half is the 4th value:
- \(Q_1 = 10\)
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Calculate \(Q_3\):
- The median of the upper half: Since there are 8 points, the median will be the average of the 4th and 5th values:
- Values for the upper half are \(15, 15, 15, 16\) (the 4th value is 15, and the 5th value is 16).
- \(Q_3 = \frac{15 + 16}{2} = \frac{31}{2} = 15.5\)
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Calculate the interquartile range (IQR):
- \(IQR = Q_3 - Q_1 = 15.5 - 10 = 5.5\)
Thus, the interquartile range of the data is 5.5.