Question

in the data set below, what is the interquartile range? 1 1 2 2 2 2 4 4 4 6 7 7

Explanation:

Step1: Find the median of the data set

The data set has \(n = 12\) values. The median is the average of the \(6^{th}\) and \(7^{th}\) values. The \(6^{th}\) value is \(2\) and the \(7^{th}\) value is \(4\). So the median \(Q_{2}=\frac{2 + 4}{2}=3\).

Step2: Find \(Q_{1}\) (lower - quartile)

The lower half of the data set is \(1,1,2,2,2,2\). The median of the lower half (since \(n = 6\)) is the average of the \(3^{rd}\) and \(4^{th}\) values. \(Q_{1}=\frac{2+2}{2}=2\).

Step3: Find \(Q_{3}\) (upper - quartile)

The upper half of the data set is \(4,4,6,7,7,7\). The median of the upper half (since \(n = 6\)) is the average of the \(3^{rd}\) and \(4^{th}\) values. \(Q_{3}=\frac{6 + 7}{2}=6.5\).

Step4: Calculate the inter - quartile range (IQR)

The formula for the inter - quartile range is \(IQR=Q_{3}-Q_{1}\). Substitute \(Q_{1}=2\) and \(Q_{3}=6.5\) into the formula. \(IQR = 6.5-2=4.5\).

Answer:

\(4.5\)