Question

1. data set 6, 4, 23, 17, 12, 16, 21, 20

median = __ upper quartile = lower quartile = __
iqr = ____

Explanation:

Step1: Sort the data set

First, sort the data set \(4,6,12,16,17,20,21,23\).

Step2: Find the median

There are \(n = 8\) data - points. The median is the average of the \(\frac{n}{2}\)th and \((\frac{n}{2}+1)\)th ordered values. \(\frac{n}{2}=4\) and \(\frac{n}{2}+1 = 5\). The median \(Q_2=\frac{16 + 17}{2}=16.5\).

Step3: Find the lower half and lower quartile

The lower half of the data set is \(4,6,12,16\). 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 ordered values. \(\frac{n_1}{2}=2\) and \(\frac{n_1}{2}+1 = 3\). So \(Q_1=\frac{6 + 12}{2}=9\).

Step4: Find the upper half and upper quartile

The upper half of the data set is \(17,20,21,23\). 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 ordered values. \(\frac{n_2}{2}=2\) and \(\frac{n_2}{2}+1 = 3\). So \(Q_3=\frac{20+21}{2}=20.5\).

Step5: Calculate the inter - quartile range (IQR)

The inter - quartile range \(IQR=Q_3 - Q_1\). So \(IQR = 20.5-9 = 11.5\).

Answer:

Median = \(16.5\)
Upper Quartile = \(20.5\)
Lower Quartile = \(9\)
IQR = \(11.5\)