Question

in the data set below, what is the interquartile range? 2 5 5 5 7 7 9 9

Explanation:

Step1: Find the median

The data set is \(2,5,5,5,7,7,9,9\). There are \(n = 8\) data points. The median is the average of the \(\frac{n}{2}\)th and \((\frac{n}{2}+1)\)th values. \(\frac{8}{2}=4\) and \(\frac{8}{2}+1 = 5\). The median \(Q_2=\frac{5 + 7}{2}=6\).

Step2: Find \(Q_1\) and \(Q_3\)

The lower half of the data is \(2,5,5,5\). There are \(n_1=4\) data points. \(Q_1\) is the average of the \(\frac{n_1}{2}\)th and \((\frac{n_1}{2}+1)\)th values. \(\frac{4}{2}=2\) and \(\frac{4}{2}+1 = 3\). \(Q_1=\frac{5+5}{2}=5\).
The upper half of the data is \(7,7,9,9\). There are \(n_2 = 4\) data points. \(Q_3\) is the average of the \(\frac{n_2}{2}\)th and \((\frac{n_2}{2}+1)\)th values. \(\frac{4}{2}=2\) and \(\frac{4}{2}+1=3\). \(Q_3=\frac{7 + 9}{2}=8\).

Step3: Calculate the inter - quartile range (IQR)

The formula for the inter - quartile range is \(IQR=Q_3 - Q_1\). Substitute \(Q_1 = 5\) and \(Q_3=8\) into the formula. \(IQR=8 - 5\).

Answer:

\(3\)