To calculate the interquartile range (IQR) of the given data set, we first need to find the first quartile (Q1) and the third quartile (Q3). The IQR is then calculated as \( IQR = Q3 - Q1 \).
Here are the steps:
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Order the data (already ordered): \( 25, 27, 28, 28, 37, 41, 47, 65, 65, 65 \)
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Find the median (Q2): Since there are 10 data points (even number), the median is the average of the 5th and 6th values.
5th value: 37
6th value: 41
Median (Q2) = \(\frac{37 + 41}{2} = \frac{78}{2} = 39\) -
Find Q1 (the median of the lower half): The lower half of the data is \( 25, 27, 28, 28, 37 \). The median of this half (5 numbers) is the 3rd value.
Q1 = 28
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Find Q3 (the median of the upper half): The upper half of the data is \( 41, 47, 65, 65, 65 \). The median of this half (5 numbers) is the 3rd value.
Q3 = 65
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Calculate the IQR: \( IQR = Q3 - Q1 = 65 - 28 = 37 \)
Thus, the interquartile range (IQR) is 37.