Calculate the 5-number summary of the skewed dataset 15, 16, 16, 17, 17, 18, 18, 18, 19, 19, 29. When constructing the box plot on a number line, which explains a reason why the dataset is skewed?(1 point)

Question

Responses

The right whisker is much longer than the left whisker.
The right whisker is much longer than the left whisker.

The 5 numbers in the summary are equally spread apart.
The 5 numbers in the summary are equally spread apart.

The median is exactly in the middle of the box.
The median is exactly in the middle of the box.

The whiskers are the same length.

Answer 1

To calculate the 5-number summary of the dataset \(15, 16, 16, 17, 17, 18, 18, 18, 19, 19, 29\), we first need to identify the following five statistics:

1. Minimum: The smallest number in the dataset.
2. First Quartile (Q1): The median of the first half of the data (not including the median if the number of data points is odd).
3. Median (Q2): The middle number when the data is ordered.
4. Third Quartile (Q3): The median of the second half of the data (not including the median if the number of data points is odd).
5. Maximum: The largest number in the dataset.

Step-by-step calculation:

1. Minimum:

   • The smallest number is 15.

2. Median (Q2):

   • There are 11 numbers, so the median is the 6th number (since \( (11 + 1)/2 = 6 \)).
   • The ordered dataset is \(15, 16, 16, 17, 17, 18, 18, 18, 19, 19, 29\).
   • The 6th number is 18.

3. First Quartile (Q1):

   • The first half of the data is \(15, 16, 16, 17, 17\) (5 numbers).
   • The median of these numbers is the 3rd number, which is 16.

4. Third Quartile (Q3):

   • The second half of the data is \(18, 18, 19, 19, 29\) (5 numbers).
   • The median of these numbers is the 3rd number, which is 19.

5. Maximum:

   • The largest number is 29.

5-number summary:

• Minimum: 15
• Q1: 16
• Median (Q2): 18
• Q3: 19
• Maximum: 29

Thus, the 5-number summary is: 15, 16, 18, 19, 29.

Explanation of skewness:

When we construct a box plot using this data, we will observe the distribution:

• The minimum is 15 and the maximum is 29.
• The third quartile (Q3) is 19 and the first quartile (Q1) is 16.
• The median (Q2) is 18.

In the box plot representation, the whisker extending from Q1 (16) to the minimum (15) is shorter than the whisker extending from Q3 (19) to the maximum (29).

Therefore, the best explanation for the skewness of the dataset is:

• The right whisker is much longer than the left whisker.