Question

in the data set below, what is the interquartile range? 1 6 3 1 1 5

Explanation:

Step1: Sort the data

Sort the data set \(1,6,3,1,1,5\) in ascending order: \(1,1,1,3,5,6\).

Step2: Find the median

Since there are \(n = 6\) data points, the median is the average of the \(\frac{n}{2}\) - th and \((\frac{n}{2}+ 1)\) - th values. The \(\frac{6}{2}=3\) - rd value is \(1\) and the \(4\) - th value is \(3\). The median \(Q_2=\frac{1 + 3}{2}=2\).

Step3: Find \(Q_1\) and \(Q_3\)

The lower half of the data is \(1,1,1\). The median of the lower half (\(Q_1\)) is the \(\frac{3 + 1}{2}=2\) - nd value, so \(Q_1 = 1\).
The upper half of the data is \(3,5,6\). The median of the upper half (\(Q_3\)) is the \(\frac{3+1}{2}=2\) - nd value of the upper half, so \(Q_3=5\).

Step4: Calculate the inter - quartile range (IQR)

The formula for the inter - quartile range is \(IQR=Q_3 - Q_1\). Substitute \(Q_1 = 1\) and \(Q_3 = 5\) into the formula: \(IQR=5 - 1=4\).

Answer:

\(4\)