Question

in the data set below, what is the interquartile range? 39 65 62 25 10 65 81 81 64

Explanation:

Step1: Sort the data set

Sort the data set in ascending order: \(10, 25, 39, 62, 64, 65, 65, 81, 81\)

Step2: Find the median (\(Q_2\))

Since there are \(n = 9\) data points, the median is the \(\frac{n + 1}{2}=\frac{9+1}{2}=5^{th}\) value. So \(Q_2=64\)

Step3: Find \(Q_1\)

The lower half of the data is \(10, 25, 39, 62\). The median of the lower half (for \(n = 4\) data points, median is the average of the \(2^{nd}\) and \(3^{rd}\) values). \(Q_1=\frac{25 + 39}{2}=\frac{64}{2}=32\)

Step4: Find \(Q_3\)

The upper half of the data is \(65, 65, 81, 81\). The median of the upper half (for \(n = 4\) data points, median is the average of the \(2^{nd}\) and \(3^{rd}\) values). \(Q_3=\frac{65+81}{2}=\frac{146}{2}=73\)

Step5: Calculate the inter - quartile range (IQR)

The formula for IQR is \(IQR=Q_3 - Q_1\). So \(IQR = 73-32=41\)

Answer:

\(41\)