Question

1. consider the following set of data:

2.0, 12.8, 3.2, 4.0, 2.8, 6.6, 5.8, 6.6, 2.8, 2.8
11a 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: \(2.0, 2.8, 2.8, 2.8, 3.2, 4.0, 5.8, 6.6, 6.6, 12.8\)

Step2: Find Lower Extreme

The lower extreme is the smallest value in the data set. From the ordered data, the smallest value is \(2.0\).

Step3: Find Median

The median is the middle value of a data set with an even number of values, we take the average of the two middle numbers. There are \(n = 10\) values, so the median is the average of the \(5^{th}\) and \(6^{th}\) values. The \(5^{th}\) value is \(3.2\) and the \(6^{th}\) value is \(4.0\). The median is \(\frac{3.2 + 4.0}{2}=\frac{7.2}{2} = 3.6\)

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

The lower quartile is the median of the lower half of the data. The lower half of the data (first 5 values) is \(2.0, 2.8, 2.8, 2.8, 3.2\). The median of this set (which has \(n = 5\) values) is the \(3^{rd}\) value, which is \(2.8\)

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

The upper quartile is the median of the upper half of the data. The upper half of the data (last 5 values) is \(4.0, 5.8, 6.6, 6.6, 12.8\). The median of this set (which has \(n = 5\) values) is the \(3^{rd}\) value, which is \(6.6\)

Step6: Find Upper Extreme

The upper extreme is the largest value in the data set. From the ordered data, the largest value is \(12.8\)

Answer:

Lower extreme: \(2.0\)
Lower quartile (\(Q_1\)): \(2.8\)
Median: \(3.6\)
Upper quartile (\(Q_3\)): \(6.6\)
Upper extreme: \(12.8\)