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.
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.
This accurately represents the minimum, lower quartile, median, upper quartile, and maximum values of the data set.
To summarize the given data using a box plot, here are the steps:
Step 1: Arrange the data in ascending order:
8, 8, 10, 11, 12, 12, 13, 14, 16, 16, 18, 18, 18, 20, 24
Step 2: Find the median:
The median is the middle value of the ordered data set. In this case, the median is 14.
Step 3: Determine the lower and upper quartiles:
To find the lower quartile, take the median of the lower half of the data set. In this case, the lower quartile is 11.
To find the upper quartile, take the median of the upper half of the data set. In this case, the upper quartile is 18.
Step 4: Calculate the interquartile range (IQR):
The IQR is the difference between the upper quartile and the lower quartile. In this case, the IQR is 18 - 11 = 7.
Step 5: Determine the minimum and maximum values:
The minimum value is the smallest value in the data set, which is 8.
The maximum value is the largest value in the data set, which is 24.
Now we can check which of the box plot options 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.
Based on the calculations made in steps 2-5, the correct box plot is option C.
To determine the box plot that correctly summarizes the data, we need to understand how a box plot represents a set of data.
A box plot, also known as a box-and-whisker plot, displays the distribution of a dataset. It consists of five key values: minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum.
Here's how to create a box plot:
1. Sort the data in ascending order.
In this case, we have 15 data points:
8, 8, 10, 11, 12, 12, 13, 14, 16, 16, 18, 18, 18, 20, 24
2. Find the median (Q2).
The median splits the data into two equal halves. Since we have an odd number of data points, the median is the middle value. In this case, the median is 14.
3. Find the first quartile (Q1).
Q1 is the median of the lower half of the data. To find Q1, divide the data into two halves at the median and find the median of the lower half. The lower half of our data is:
8, 8, 10, 11, 12, 12, 13
The median of this lower half is 11.
4. Find the third quartile (Q3).
Q3 is the median of the upper half of the data. To find Q3, divide the data into two halves at the median and find the median of the upper half. The upper half of our data is:
16, 16, 18, 18, 18, 20, 24
The median of this upper half is 18.
5. Calculate the minimum and maximum values.
The minimum value is the smallest data point, which is 8.
The maximum value is the largest data point, which is 24.
6. Plot the box plot.
The box part of the box plot represents the interquartile range (IQR), which is the range from Q1 to Q3. The line inside the box represents the median (Q2). The whiskers extend from the box to the minimum and maximum values.
Now let's compare the options based on this information.
A. The end of the first whisker is at 8 (correct), the box starts at 12 (incorrect - should start at 11), the line inside the box is at 16 (incorrect - should be at 14), the box ends at 18 (correct), and the end of the second whisker is at 24 (correct).
B. The end of the first whisker is at 8 (correct), the box starts at 11 (correct), the line inside the box is at 14 (correct), the box ends at 18 (correct), and the end of the second whisker is at 24 (correct).
C. The end of the first whisker is at 8 (correct), the box starts at 11 (correct), the line inside the box is at 16 (correct), the box ends at 18 (correct), and the end of the second whisker is at 24 (correct).
D. The end of the first whisker is at 8 (correct), the box starts at 11 (correct), the line inside the box is at 14 (correct), the box ends at 20 (incorrect - should end at 18), and the end of the second whisker is at 24 (correct).
Based on the comparison, the box plot that correctly summarizes the data is option B, where 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.