Question

in the data set below, what are the lower quartile, the me
7 5 7 2 3 2 5 1
lower quartile =
median =
upper quartile =

Explanation:

Step1: Sort the data

Sort the data set \(7,5,7,2,3,2,5,1\) in ascending order: \(1,2,2,3,5,5,7,7\).

Step2: Find the median

Since there are \(n = 8\) data points. The median is the average of the \(\frac{n}{2}\)th and \((\frac{n}{2}+1)\)th values. \(\frac{n}{2}=4\), \(\frac{n}{2}+1 = 5\). The \(4\)th value is \(3\) and the \(5\)th value is \(5\). Median \(M=\frac{3 + 5}{2}=4\).

Step3: Find the lower quartile

The lower half of the data is \(1,2,2,3\). Since there are \(n_1=4\) data points in the lower half. The lower quartile \(Q_1\) is the average of the \(\frac{n_1}{2}\)th and \((\frac{n_1}{2}+1)\)th values. \(\frac{n_1}{2}=2\), \(\frac{n_1}{2}+1 = 3\). The \(2\)nd value is \(2\) and the \(3\)rd value is \(2\). \(Q_1=\frac{2+2}{2}=2\).

Step4: Find the upper quartile

The upper half of the data is \(5,5,7,7\). Since there are \(n_2 = 4\) data points in the upper half. The upper quartile \(Q_3\) is the average of the \(\frac{n_2}{2}\)th and \((\frac{n_2}{2}+1)\)th values. \(\frac{n_2}{2}=2\), \(\frac{n_2}{2}+1=3\). The \(2\)nd value is \(5\) and the \(3\)rd value is \(7\). \(Q_3=\frac{5 + 7}{2}=6\).

Answer:

lower quartile \(=2\), median \(=4\), upper quartile \(=6\)