Question

in the data set below, what is the interquartile range? 25 59 74 76 78

Explanation:

Step1: Find the median

The data set is \(25,59,74,76,78\). The median (second quartile \(Q_2\)) is the middle value. Since there are \(n = 5\) values, the median is the third value. So \(Q_2=74\).

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

The lower half of the data is \(25,59\). The median of the lower half (\(Q_1\)) is \(\frac{25 + 59}{2}=\frac{84}{2}=42\).
The upper half of the data is \(76,78\). The median of the upper half (\(Q_3\)) is \(\frac{76+78}{2}=\frac{154}{2} = 77\).

Step3: Calculate the inter - quartile range (IQR)

The formula for the inter - quartile range is \(IQR=Q_3 - Q_1\). Substitute \(Q_1 = 42\) and \(Q_3=77\) into the formula. So \(IQR=77 - 42=35\).

Answer:

\(35\)