Question

1. consider the following set of data: 5.6, 6.4, 4.0, 1.6, 3.2, 10.6, 6.4, 3.0, 3.0, 3.0 6a complete the table. lower extreme, lower quartile (q₁), median, upper quartile (q₃), upper extreme

Explanation:

Step1: Order the data

First, we order the data set from smallest to largest: \(1.6, 3.0, 3.0, 3.0, 3.2, 4.0, 5.6, 6.4, 6.4, 10.6\)

Step2: Find Lower extreme

The lower extreme is the smallest value in the data set. From the ordered data, the smallest value is \(1.6\).

Step3: Find Median

The median is the middle value of the ordered data set. Since there are \(n = 10\) (even number of data points), the median is the average of the \(\frac{n}{2}\)th and \((\frac{n}{2}+ 1)\)th values. \(\frac{n}{2}=\frac{10}{2}=5\)th value and \((\frac{n}{2}+ 1)=6\)th value. The 5th value is \(3.2\) and the 6th value is \(4.0\). So median \(=\frac{3.2 + 4.0}{2}=\frac{7.2}{2}=3.6\)? Wait, no, wait. Wait, the ordered data: let's re - check the ordering. The data points are \(1.6, 3.0, 3.0, 3.0, 3.2, 4.0, 5.6, 6.4, 6.4, 10.6\). The 5th term is \(3.2\) and the 6th term is \(4.0\). So median \(=\frac{3.2 + 4.0}{2}=3.6\)? Wait, no, maybe I made a mistake. Wait, the number of data points \(n = 10\). The median position is between the 5th and 6th term. So median \(=\frac{x_{5}+x_{6}}{2}\), where \(x_{5}=3.2\) and \(x_{6}=4.0\), so median \(=\frac{3.2 + 4.0}{2}=3.6\).

Step4: Find Lower quartile (\(Q_1\))

The lower quartile \(Q_1\) is the median of the lower half of the data. The lower half of the data (for \(n = 10\), the first 5 data points) is \(1.6, 3.0, 3.0, 3.0, 3.2\). The median of this set (since there are 5 data points, the 3rd term) is \(3.0\).

Step5: Find Upper quartile (\(Q_3\))

The upper quartile \(Q_3\) is the median of the upper half of the data. The upper half of the data (the last 5 data points) is \(4.0, 5.6, 6.4, 6.4, 10.6\). The median of this set (the 3rd term) is \(6.4\).

Step6: Find Upper extreme

The upper extreme is the largest value in the data set. From the ordered data, the largest value is \(10.6\).

Answer:

• Lower extreme: \(1.6\)
• Lower quartile (\(Q_1\)): \(3.0\)
• Median: \(3.6\)
• Upper quartile (\(Q_3\)): \(6.4\)
• Upper extreme: \(10.6\)