Question

what is the interquartile range of the set of data? 18, 22, 20, 23, 27, 18, 23, 27, 20 the iqr of the set of data is

Explanation:

Step1: Order the data

First, we order the data set from least to greatest. The data set is \(18, 22, 20, 23, 27, 18, 23, 27, 20\). After ordering, we get: \(18, 18, 20, 20, 22, 23, 23, 27, 27\).

Step2: Find the median (Q2)

The median (second quartile, \(Q_2\)) of a data set with \(n = 9\) (odd number of observations) is the middle value. The middle position is \(\frac{n + 1}{2}=\frac{9+ 1}{2}=5\)th value. Looking at the ordered data, the 5th value is \(22\), so \(Q_2=22\).

Step3: Find the lower quartile (Q1)

The lower quartile (\(Q_1\)) is the median of the lower half of the data. The lower half of the data (values below \(Q_2\)) is \(18, 18, 20, 20\). The number of values in the lower half is \(4\) (even number). The median of a set with \(n = 4\) is the average of the \(\frac{n}{2}\)th and \(\frac{n}{2}+ 1\)th values. So, the 2nd and 3rd values: \(\frac{18 + 20}{2}=\frac{38}{2}=19\). Thus, \(Q_1 = 19\).

Step4: Find the upper quartile (Q3)

The upper quartile (\(Q_3\)) is the median of the upper half of the data. The upper half of the data (values above \(Q_2\)) is \(23, 23, 27, 27\). The number of values in the upper half is \(4\) (even number). The median of a set with \(n = 4\) is the average of the \(\frac{n}{2}\)th and \(\frac{n}{2}+ 1\)th values. So, the 2nd and 3rd values: \(\frac{23+27}{2}=\frac{50}{2}=25\). Thus, \(Q_3 = 25\).

Step5: Calculate the interquartile range (IQR)

The interquartile range is calculated as \(IQR=Q_3 - Q_1\). Substituting the values of \(Q_3 = 25\) and \(Q_1=19\), we get \(IQR=25 - 19 = 6\).

Answer:

\(6\)