Question

independent practice
using the line provided, construct a box - and - whisker plot for the 12 scores below.
26, 32, 19, 65, 57, 16, 28, 42, 40, 21, 38, 10
determine the number of scores that lie above the 3rd quartile.

Explanation:

Step1: Order the data

First, we order the 12 scores from least to greatest: \(10, 16, 19, 21, 26, 28, 32, 38, 40, 42, 57, 65\)

Step2: Find the median (second quartile, \(Q_2\))

Since there are \(n = 12\) data points (an even number), the median is the average of the 6th and 7th values. The 6th value is \(28\) and the 7th value is \(32\), so \(Q_2=\frac{28 + 32}{2}=30\)

Step3: Find the first quartile (\(Q_1\))

The first quartile is the median of the lower half of the data. The lower half consists of the first 6 values: \(10, 16, 19, 21, 26, 28\). The median of these 6 values (even number) is the average of the 3rd and 4th values. The 3rd value is \(19\) and the 4th value is \(21\), so \(Q_1=\frac{19+21}{2} = 20\)

Step4: Find the third quartile (\(Q_3\))

The third quartile is the median of the upper half of the data. The upper half consists of the last 6 values: \(32, 38, 40, 42, 57, 65\). The median of these 6 values (even number) is the average of the 3rd and 4th values. The 3rd value is \(40\) and the 4th value is \(42\), so \(Q_3=\frac{40 + 42}{2}=41\)

Step5: Determine the number of scores above \(Q_3\)

The third quartile (\(Q_3\)) divides the data such that 75% of the data is below it and 25% is above it. For \(n = 12\) data points, the number of scores above \(Q_3\) is \(n\times\frac{1}{4}\) (since 25% is \(\frac{1}{4}\)). So \(12\times\frac{1}{4}=3\)

Answer:

3