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?
There are multiple correct box plots that could summarize the data, as long as they are constructed correctly. However, we can assume that the "correct" box plot is the one that accurately represents the minimum, lower quartile, median, upper quartile, and maximum values of the data set.
The minimum value is 8 pages, the lower quartile is 12 pages, the median is 14 pages, the upper quartile is 18 pages, and the maximum value is 24 pages.
One possible correct box plot description could be as follows:
-The minimum is represented by a line segment at 8 pages, as the minimum value is 8.
-The lower quartile is represented by a box starting at 12 pages and ending at 13 pages, as the lower quartile is between the 12th and 13th values in the data set.
-The median is represented by a line segment at 14 pages, as the median is the value between the 7th and 8th values in the data set.
-The upper quartile is represented by a box starting at 16 pages and ending at 18 pages, as the upper quartile is between the 9th and 12th values in the data set.
-The maximum is represented by a line segment at 24 pages, as the maximum value is 24.
Therefore, any box plot that accurately represents these values is correct.
To create a box plot, we need to find the minimum, maximum, median, and quartiles of the data.
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 minimum and maximum:
Minimum: 8
Maximum: 24
Step 3: Find the median:
Since we have an odd number of data points, the median is the middle value, which is 14.
Step 4: Find the quartiles:
First quartile (Q1): The median of the lower half of the data.
Second quartile (Q2): The median.
Third quartile (Q3): The median of the upper half of the data.
Lower half of the data: 8, 8, 10, 11, 12, 12, 13
Upper half of the data: 16, 16, 18, 18, 18, 20, 24
First quartile (Q1): The median of the lower half, which is 11.
Third quartile (Q3): The median of the upper half, which is 18.
Now we have all the necessary values to create a box plot.
Among the given options, the box plot that correctly summarizes the data is the one with:
- Minimum: 8
- Q1: 11
- Median: 14
- Q3: 18
- Maximum: 24
To determine which box plot correctly summarizes the given data, we need to understand the components of a box plot and how it represents the data.
A box plot, also known as a box-and-whisker plot, provides a visual representation of the distribution of a dataset. It consists of five key values: the minimum value, the first quartile (Q1), the median (Q2), the third quartile (Q3), and the maximum value.
To construct a box plot, we perform the following steps:
1. Arrange the data points in increasing order.
2. Find the median, which is the middle value of the dataset. If the dataset has an odd number of values, the median is the middle value. If the dataset has an even number of values, the median is the average of the two middle values.
3. Find the first quartile, which is the median of the lower half of the dataset (the values below the median). This is the value that splits the lower 50% of the dataset.
4. Find the third quartile, which is the median of the upper half of the dataset (the values above the median). This is the value that splits the upper 50% of the dataset.
5. Calculate the interquartile range (IQR) by subtracting the first quartile from the third quartile (IQR = Q3 - Q1).
6. Determine the minimum and maximum values that fall within 1.5 times the IQR below Q1 and above Q3, respectively. These values are known as the "whiskers" of the box plot.
7. Plot the box plot using horizontal lines: a box from Q1 to Q3 with a line (median) inside, and vertical lines (whiskers) from Q1 to the minimum value and from Q3 to the maximum value. Any data points outside the whiskers are considered outliers and are plotted individually.
Now, let's apply these steps to the given data:
1. Arrange the data in increasing order:
8, 8, 10, 11, 12, 12, 13, 14, 16, 16, 18, 18, 18, 20, 24
2. Find the median:
Since the dataset has an odd number of values, the median is the middle value, which is 14.
3. Find the first quartile:
The lower half of the dataset is: 8, 8, 10, 11, 12, 12, 13.
The median of this lower half is 11.
4. Find the third quartile:
The upper half of the dataset is: 16, 16, 18, 18, 18, 20, 24.
The median of this upper half is 18.
5. Calculate the interquartile range (IQR):
IQR = Q3 - Q1 = 18 - 11 = 7
6. Determine the minimum and maximum values within 1.5 times the IQR:
The minimum is the smallest value within 1.5 times the IQR below Q1: 11 - (1.5 * 7) = 11 - 10.5 = 0.5
The maximum is the largest value within 1.5 times the IQR above Q3: 18 + (1.5 * 7) = 18 + 10.5 = 28.5
(Note: In this case, since the dataset does not contain any values below 0.5 or above 28.5, the whiskers will extend to the minimum and maximum values of the dataset.)
Now we can compare the given box plots to the information and construct the plot that matches our calculations.
Unfortunately, since you didn't provide the available box plots as options, I cannot determine the correct one from the given information. However, you can follow the steps mentioned above to find the correct box plot by identifying which one correctly represents the median, quartiles, and whiskers based on the calculated values from the given dataset.