Question

in the data set below, what are the lower quartile, the median, and the upper quartile? 47 70 75 95 96 97 lower quartile = median = upper quartile =

Explanation:

Step1: Arrange the data set

The data set is already in ascending order: \(47,70,75,95,96,97\)

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. \(\frac{n}{2}=3\), \(\frac{n}{2}+1 = 4\). The median \(M=\frac{75 + 95}{2}=\frac{170}{2}=85\)

Step3: Find the lower quartile

The lower half of the data is \(47,70,75\). Since \(n_1=3\) (odd number of data points in the lower half), the lower quartile \(Q_1\) is the \(\frac{n_1 + 1}{2}\)th value. \(\frac{3+1}{2}=2\)nd value. So \(Q_1 = 70\)

Step4: Find the upper quartile

The upper half of the data is \(95,96,97\). Since \(n_2=3\) (odd number of data points in the upper half), the upper quartile \(Q_3\) is the \(\frac{n_2+1}{2}\)th value. \(\frac{3 + 1}{2}=2\)nd value. So \(Q_3=96\)

Answer:

lower quartile \(=70\), median \(=85\), upper quartile \(=96\)