Question

an compare measures of spread
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1. ten members of the library’s book club that meets twice a month were also asked how

many books they read in the past year. this data is displayed in the box plot below.

box plot image with x-axis from 20 to 35

a. compare/contrast the median of the two data sets.

b. compare/contrast the range of the two data sets.

c. compare/contrast the interquartile range of the two data sets.

Explanation:

Step1: Analyze Median

The median is the middle value (or average of two middle values) in a data set, represented by the line inside the box in a box - plot. For the given box - plot of the book club data, we need to assume there is another data set (not fully shown here, but typically in such problems, we compare with a previous data set). Let's assume we know the median of the other data set. If the median line in this box - plot is at 28 (from the x - axis marking), and if the other data set has a different median, say, if the other data set's median was, for example, 25, then we can say the median of this book club data set (28) is higher than the other. But since the problem mentions "two data sets" and only one box - plot is shown, we assume the first data set (maybe from a previous part) and this one. The key is that the median is the middle line in the box. So we identify the median of this data set from the box - plot (the vertical line in the box) and compare it to the other data set's median.

Step2: Analyze Range

The range of a data set is calculated as \( \text{Range}=\text{Maximum value}-\text{Minimum value} \). From the box - plot, the minimum value is 20 (the left - most dot) and the maximum value is 35 (the right - most dot), so the range for this data set is \( 35 - 20=15 \). For the other data set, we would calculate its range (max - min) and then compare. If the other data set had a range of, say, 10, then this data set has a larger range.

Step3: Analyze Interquartile Range (IQR)

The interquartile range is calculated as \( \text{IQR}=Q_3 - Q_1 \), where \( Q_3 \) is the third quartile (right end of the box) and \( Q_1 \) is the first quartile (left end of the box). From the box - plot, \( Q_1 = 27 \) (left end of the box) and \( Q_3=31 \) (right end of the box), so \( \text{IQR}=31 - 27 = 4 \). For the other data set, we calculate its \( \text{IQR}=Q_3^{\prime}-Q_1^{\prime} \) and then compare. If the other data set had an IQR of 5, then this data set has a smaller IQR.

(Note: Since the problem seems to have a missing part (the other data set's box - plot or description), the above is a general method. If we assume that this is a follow - up problem where the first data set was, for example, with min = 22, max = 30, median = 25, \( Q_1 = 23 \), \( Q_3 = 28 \)):

Part a:

If the other data set has a median of 25 and this data set has a median of 28, then the median of the book club data set (28) is greater than the median of the other data set (25).

Part b:

If the other data set has min = 22 and max = 30, its range is \( 30 - 22 = 8 \). The range of this data set is \( 35 - 20=15 \). So the range of the book club data set (15) is greater than the range of the other data set (8).

Part c:

If the other data set has \( Q_1 = 23 \) and \( Q_3 = 28 \), its IQR is \( 28 - 23 = 5 \). The IQR of this data set is \( 31 - 27 = 4 \). So the interquartile range of this data set (4) is less than the interquartile range of the other data set (5).

Answer:

(Assuming the above - mentioned comparison with a typical other data set):

a. The median of the book club data set (28) is greater than the median of the other data set (e.g., 25).

b. The range of the book club data set (\( 35 - 20 = 15 \)) is greater than the range of the other data set (e.g., \( 30 - 22 = 8 \)).

c. The interquartile range of the book club data set (\( 31 - 27 = 4 \)) is less than the interquartile range of the other data set (e.g., \( 28 - 23 = 5 \)).

(Note: The actual answer depends on the other data set's values. If more information about the other data set is provided, the answer can be adjusted accordingly.)