Question

during the last 15 years of his baseball career, andrew hit the following number of home runs each season.
35,24,32,36,40,32,40,38,36,33,11,20,19,22,8
state and label the values of the minimum, 1st quartile, median, 3rd quartile, and maximum.
using the line below, construct a box - and - whisker plot for this set of data.

Explanation:

Step1: Order the data

First, we order the data from least to greatest: \(8, 11, 19, 20, 22, 24, 32, 32, 33, 35, 36, 36, 38, 40, 40\)

Step2: Find the minimum and maximum

The minimum value is the smallest number in the ordered data, and the maximum value is the largest number.
Minimum: \(8\)
Maximum: \(40\)

Step3: Find the median (2nd quartile)

The median is the middle value of the ordered data. Since there are \(n = 15\) data points (an odd number), the median is the \(\frac{n + 1}{2}=\frac{15+ 1}{2}=8\)-th value.
Looking at the ordered data: \(8, 11, 19, 20, 22, 24, 32, \boldsymbol{32}, 33, 35, 36, 36, 38, 40, 40\)
Median (\(Q_2\)): \(32\)

Step4: Find the 1st quartile (\(Q_1\))

The 1st quartile is the median of the lower half of the data (excluding the median if \(n\) is odd). The lower half of the data is the first \(7\) values (since \(n = 15\), the lower half is from the 1st to the 7th value): \(8, 11, 19, 20, 22, 24, 32\)
The median of this lower half (which has \(n_1=7\) values) is the \(\frac{7 + 1}{2}=4\)-th value.
Looking at the lower half: \(8, 11, 19, \boldsymbol{20}, 22, 24, 32\)
1st Quartile (\(Q_1\)): \(20\)

Step5: Find the 3rd quartile (\(Q_3\))

The 3rd quartile is the median of the upper half of the data (excluding the median if \(n\) is odd). The upper half of the data is the last \(7\) values (from the 9th to the 15th value): \(33, 35, 36, 36, 38, 40, 40\)
The median of this upper half (which has \(n_2 = 7\) values) is the \(\frac{7+ 1}{2}=4\)-th value.
Looking at the upper half: \(33, 35, 36, \boldsymbol{36}, 38, 40, 40\)
3rd Quartile (\(Q_3\)): \(36\)

Answer:

• Minimum: \(8\)
• 1st Quartile (\(Q_1\)): \(20\)
• Median (\(Q_2\)): \(32\)
• 3rd Quartile (\(Q_3\)): \(36\)
• Maximum: \(40\)

To construct the box - and - whisker plot:

1. Draw a number line that covers the range from \(8\) to \(40\).
2. Plot the minimum value (\(8\)) and the maximum value (\(40\)) as the endpoints of the whiskers.
3. Draw a box from \(Q_1 = 20\) to \(Q_3=36\). Inside the box, draw a vertical line at the median \(Q_2 = 32\).