Question

in the data set below, what is the interquartile range? 28 28 29 33 42 48

Explanation:

Step1: Find the median

The data set is \(28, 28, 29, 33, 42, 48\). Since \(n = 6\) (even number of data points), the median is the average of the \(\frac{n}{2}\)th and \((\frac{n}{2}+ 1)\)th values. \(\frac{6}{2}=3\) and \(\frac{6}{2}+1 = 4\). The median \(Q_2=\frac{29 + 33}{2}=31\).

Step2: Find \(Q_1\) (lower quartile)

The lower half of the data is \(28, 28, 29\). Since \(n = 3\) (odd number of data points), \(Q_1\) is the \(\frac{n + 1}{2}\)th value. \(\frac{3+1}{2}=2\)nd value. So \(Q_1 = 28\).

Step3: Find \(Q_3\) (upper quartile)

The upper half of the data is \(33, 42, 48\). Since \(n = 3\) (odd number of data points), \(Q_3\) is the \(\frac{n + 1}{2}\)th value. \(\frac{3+1}{2}=2\)nd value. So \(Q_3=42\).

Step4: Calculate the inter - quartile range (IQR)

The formula for IQR is \(IQR=Q_3 - Q_1\). Substitute \(Q_1 = 28\) and \(Q_3 = 42\) into the formula. \(IQR=42-28\).

Answer:

\(14\)