Question

a student takes ten exams during a semester and receives the following grades: 90, 85, 97, 76, 89, 58, 82, 102, 70, and 67. find the five - number summary used in a boxplot. which boxplot best represents the data?
a. boxplot with axis: 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105
b. boxplot with axis: 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105
c. boxplot with axis: 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105
d. boxplot with axis: 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105

Explanation:

Step1: Order the data

First, we order the given grades: 58, 67, 70, 76, 82, 85, 89, 90, 97, 102.

Step2: Find the minimum and maximum

The minimum value (Min) is the smallest number, which is 58. The maximum value (Max) is the largest number, which is 102.

Step3: Find the median (Q2)

Since there are 10 data points (even number), the median is the average of the 5th and 6th values. The 5th value is 82 and the 6th value is 85. So, the median \( Q2=\frac{82 + 85}{2}=\frac{167}{2} = 83.5\).

Step4: Find the first quartile (Q1)

The first quartile is the median of the lower half of the data. The lower half is 58, 67, 70, 76, 82. There are 5 data points (odd number), so the median of this lower half is the 3rd value, which is 70. So, \( Q1 = 70\).

Step5: Find the third quartile (Q3)

The third quartile is the median of the upper half of the data. The upper half is 85, 89, 90, 97, 102. There are 5 data points (odd number), so the median of this upper half is the 3rd value, which is 90. So, \( Q3=90\).

The five - number summary is Min = 58, Q1 = 70, Q2 = 83.5, Q3 = 90, Max = 102.

Now, let's analyze the boxplots:

• The box in a boxplot is between Q1 and Q3, the line inside the box is the median (Q2), and the whiskers extend to Min and Max.

Looking at the options:

• Option d: The minimum is 58 (left whisker starts at 58), Q1 is 70 (left end of the box), median (Q2) is around 83.5 (line inside the box), Q3 is 90 (right end of the box), and maximum is 102 (right whisker ends at 102). This matches our five - number summary.

Answer:

The five - number summary is 58, 70, 83.5, 90, 102. The boxplot that best represents the data is option d.