Question

independent practice

1. the data set 20, 36, 52, 56, 24, 16, 40, 4, 28 represents the number of books purchased by nine book club members in a year. construct a box plot for these data on the number line below.

Explanation:

Step1: Order the data set

First, we need to order the given data set from smallest to largest. The data set is \( 20, 36, 52, 56, 24, 16, 40, 4, 28 \). After ordering, we get \( 4, 16, 20, 24, 28, 36, 40, 52, 56 \).

Step2: Find the minimum and maximum

The minimum value (smallest number) in the ordered data set is \( 4 \), and the maximum value (largest number) is \( 56 \).

Step3: Find the median (Q2)

Since there are \( n = 9 \) data points (an odd number), the median is the middle value. The middle position is \( \frac{n + 1}{2}=\frac{9+ 1}{2}=5 \)-th value. Looking at the ordered data set \( 4, 16, 20, 24, 28, 36, 40, 52, 56 \), the 5 - th value is \( 28 \). So the median (Q2) is \( 28 \).

Step4: Find the first quartile (Q1)

The first quartile (Q1) is the median of the lower half of the data. The lower half of the data (excluding the median) is \( 4, 16, 20, 24 \). Since there are \( 4 \) data points (an even number), the median of this subset is the average of the two middle values. The two middle values are the 2 - nd and 3 - rd values: \( 16 \) and \( 20 \). So \( Q1=\frac{16 + 20}{2}=\frac{36}{2}=18 \).

Step5: Find the third quartile (Q3)

The third quartile (Q3) is the median of the upper half of the data. The upper half of the data (excluding the median) is \( 36, 40, 52, 56 \). Since there are \( 4 \) data points (an even number), the median of this subset is the average of the two middle values. The two middle values are the 2 - nd and 3 - rd values: \( 40 \) and \( 52 \). So \( Q3=\frac{40+52}{2}=\frac{92}{2} = 46 \).

Step6: Construct the box - plot

• The minimum value (\( 4 \)) is plotted at the left end of the whisker.
• The first quartile (\( Q1 = 18 \)) is the left end of the box.
• The median (\( Q2=28 \)) is the line inside the box.
• The third quartile (\( Q3 = 46 \)) is the right end of the box.
• The maximum value (\( 56 \)) is plotted at the right end of the whisker.

To plot on the given number line (with marks at \( 0,20,40,60 \)):

• The minimum (\( 4 \)) is between \( 0 \) and \( 20 \).
• \( Q1 = 18 \) is also between \( 0 \) and \( 20 \), closer to \( 20 \).
• The median (\( 28 \)) is between \( 20 \) and \( 40 \), closer to \( 20 \).
• \( Q3 = 46 \) is between \( 40 \) and \( 60 \), closer to \( 40 \).
• The maximum (\( 56 \)) is between \( 40 \) and \( 60 \), closer to \( 60 \).

Answer:

To construct the box - plot:

• Whisker left: \( 4 \)
• Box left (Q1): \( 18 \)
• Box middle (Median): \( 28 \)
• Box right (Q3): \( 46 \)
• Whisker right: \( 56 \)

(When drawing on the number line, mark the minimum at \( 4 \), Q1 at \( 18 \), median at \( 28 \), Q3 at \( 46 \), and maximum at \( 56 \) with the box between \( 18 \) and \( 46 \) and whiskers extending to \( 4 \) and \( 56 \).)