Question - Calculating Interquartile Range (IQR) for a Data Set
mathinterquartile range calculationiqr formula and calculationdata set quartiles calculationmedian, q1, q3 calculation
Solution:
The interquartile range (IQR) is the difference between the first quartile (Q1) and the third quartile (Q3) in a set of data. It measures the middle 50% of the data or the spread of the middle half of the data.First, we need to find Q1 and Q3 for the given data set:\[ 1, 2, 4, 4, 4, 4, 4, 8, 8 \]Since there are 9 data points, the median (second quartile, Q2) will be the middle value, which is the fifth value when the data is ordered - in this case, 4.To find Q1, we take the lower half of the data not including the median:\[ 1, 2, 4, 4 \]The lower half has an even number of data points (4), so Q1 will be the average of the middle two numbers in this subset:\[ Q1 = \frac{(2 + 4)}{2} = \frac{6}{2} = 3 \]Next, find Q3 by taking the upper half of the data not including the median:\[ 4, 4, 8, 8 \]The upper half also has an even number of data points (4), so Q3 will be the average of the middle two numbers in this subset:\[ Q3 = \frac{(4 + 8)}{2} = \frac{12}{2} = 6 \]Now let's calculate the IQR:\[ IQR = Q3 - Q1 = 6 - 3 = 3 \]The interquartile range for the given data set is 3.
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