A group of 15 students were assigned a novel to read during class. The data below represents the number of pages each student read.
8, 8, 10, 11, 12, 12, 13, 14, 16, 16, 18, 18, 18, 20, 24
Which of the following box plots correctly summarizes the data?
A.
The end of the first whisker is at 8, the box starts at 12, the line inside the box is at 16, the box ends at 18, and the end of the second whisker is at 24.
B.
The end of the first whisker is at 8, the box starts at 11, the line inside the box is at 14, the box ends at 18, and the end of the second whisker is at 24.
C.
The end of the first whisker is at 8, the box starts at 11, the line inside the box is at 16, the box ends at 18, and the end of the second whisker is at 24.
D.
The end of the first whisker is at 8, the box starts at 11, the line inside the box is at 14, the box ends at 20, and the end of the second whisker is at 24.
To construct a box plot, we need to find the minimum, first quartile, median, third quartile, and maximum.
The minimum is the smallest value, which is 8.
The first quartile is the median of the lower half of the data, which is 12.
The median is the middle value, or the average of the two middle values if there is an even number of data points. In this case, the median is 14.
The third quartile is the median of the upper half of the data, which is 18.
The maximum is the largest value, which is 24.
Therefore, the correct box plot is option B: The end of the first whisker is at 8, the box starts at 11, the line inside the box is at 14, the box ends at 18, and the end of the second whisker is at 24.
To summarize the given data, we need to create a box plot.
A box plot contains several components: the whiskers, the box, and the line inside the box.
The whiskers represent the minimum and maximum values of the dataset. The box represents the interquartile range, which contains the middle 50% of the data. The line inside the box represents the median.
Let's analyze the data to determine the correct box plot:
- The minimum value in the dataset is 8, so the end of the first whisker should be at 8.
- The maximum value in the dataset is 24, so the end of the second whisker should be at 24.
- The median is the middle value of the dataset when it is arranged in ascending order. Here are the data points arranged in ascending order: 8, 8, 10, 11, 12, 12, 13, 14, 16, 16, 18, 18, 18, 20, 24. The middle value is 14.
- The box should contain the interquartile range, which is the middle 50% of the data. In this case, the box should start at the value greater than the first quartile (Q1) and end at the value less than the third quartile (Q3).
To find Q1 and Q3, we need to find the values that divide the dataset into four equal parts.
- 15 students divided into four equal parts means each part contains 15/4 = 3.75 students. Since we can't have a fraction of a student, we'll consider the first quartile (Q1) as the value at position 4 (the fourth number when the data is arranged in ascending order) and the third quartile (Q3) as the value at position 12 (the twelfth number when the data is arranged in ascending order).
Arranging the data in ascending order:
8, 8, 10, 11, 12, 12, 13, 14, 16, 16, 18, 18, 18, 20, 24
The value at position 4 is 11, so Q1 = 11.
The value at position 12 is 18, so Q3 = 18.
Based on this analysis, the correct box plot is:
B.
The end of the first whisker is at 8, the box starts at 11, the line inside the box is at 14, the box ends at 18, and the end of the second whisker is at 24.
To summarize the given data using a box plot, we need to identify five key values: the minimum, the first quartile (Q1), the median (Q2 or middle quartile), the third quartile (Q3), and the maximum.
First, we can order the data from least to greatest: 8, 8, 10, 11, 12, 12, 13, 14, 16, 16, 18, 18, 18, 20, 24.
Now, let's find the five key values:
1. Minimum: The smallest value in the data set is 8.
2. Q1: The first quartile is the median of the lower half of the data. Since there are 15 data points, the lower half consists of the first 7 values. The median of this lower half is the average of the 4th and 5th values, which is (11 + 12) / 2 = 11.5.
3. Q2 (median): The median is the middle value of the data set. Since there are 15 data points, the median is the 8th value, which is 14.
4. Q3: The third quartile is the median of the upper half of the data. Since there are 15 data points, the upper half consists of the last 7 values. The median of this upper half is the average of the 3rd and 4th values, which is (18 + 18) / 2 = 18.
5. Maximum: The largest value in the data set is 24.
Now, let's match these values with the options provided:
A. This option has the first whisker ending at 8, the box starting at 12, the line inside the box at 16, the box ending at 18, and the second whisker ending at 24. This option correctly represents the data.
B. This option has the first whisker ending at 8, the box starting at 11, the line inside the box at 14, the box ending at 18, and the second whisker ending at 24. This option does not accurately represent the data since the first quartile is 11.5, not 11.
C. This option has the first whisker ending at 8, the box starting at 11, the line inside the box at 16, the box ending at 18, and the second whisker ending at 24. This option does not correctly represent the data since the third quartile is 18, not 16.
D. This option has the first whisker ending at 8, the box starting at 11, the line inside the box at 14, the box ending at 20, and the second whisker ending at 24. This option does not accurately represent the data since the third quartile is 18, not 20.
Therefore, the correct box plot that summarizes the data is option A.