Question

in the data set below, what are the lower quartile, the median, and the upper quartile?
96 86 84 49 92 24 49 86
lower quartile =
median =
upper quartile =

Explanation:

Step1: Sort the data

First, sort the data set \(24,49,49,52,84,85,85,96\).

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 median \(M=\frac{52 + 84}{2}=68\).

Step3: Find the lower quartile

The lower half of the data is \(24,49,49,52\). Since \(n_1=4\) (number of 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. \(\frac{n_1}{2}=2\), \(\frac{n_1}{2}+1=3\). \(Q_1=\frac{49+49}{2}=49\).

Step4: Find the upper quartile

The upper half of the data is \(84,85,85,96\). Since \(n_2 = 4\) (number of 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. \(\frac{n_2}{2}=2\), \(\frac{n_2}{2}+1=3\). \(Q_3=\frac{85 + 85}{2}=85\).

Answer:

lower quartile \(=49\), median \(=68\), upper quartile \(=85\)