To calculate the 5-number summary of the dataset \( 15, 16, 16, 17, 17, 18, 18, 18, 19, 19, 29 \), we need to find the following:
- Minimum: The smallest number in the dataset.
- First Quartile (Q1): The median of the first half of the data.
- Median (Q2): The median of the entire dataset.
- Third Quartile (Q3): The median of the second half of the data.
- Maximum: The largest number in the dataset.
Step-by-step calculation:
1. Minimum:
The minimum value in the dataset is \( 15 \).
2. Maximum:
The maximum value in the dataset is \( 29 \).
3. Median (Q2):
To find the median, we determine the middle value. Since there are 11 numbers (odd), the median is the 6th number in the sorted list:
\( 15, 16, 16, 17, 17, 18, 18, 18, 19, 19, 29 \)
Thus, the median (Q2) is \( 18 \).
4. First Quartile (Q1):
To find Q1, we take the median of the first half of the data:
First half: \( 15, 16, 16, 17, 17 \) (5 values)
The median of this half is the 3rd number:
Thus, Q1 is \( 16 \).
5. Third Quartile (Q3):
To find Q3, we take the median of the second half of the data:
Second half: \( 18, 18, 19, 19, 29 \) (5 values)
The median of this half is the 3rd number:
Thus, Q3 is \( 19 \).
5-Number Summary:
- Minimum: \( 15 \)
- Q1: \( 16 \)
- Median (Q2): \( 18 \)
- Q3: \( 19 \)
- Maximum: \( 29 \)
So, the 5-number summary is: 15, 16, 18, 19, 29.
Box Plot Explanation:
When constructing the box plot, we notice that the right whisker (from Q3 to the maximum value) is much longer than the left whisker (from Q1 to Q1).
Thus, a reason why the dataset is skewed is:
The right whisker is much longer than the left whisker.