To calculate the 5-number summary of the dataset \(15, 16, 16, 17, 17, 18, 18, 18, 19, 19, 29\), we need to determine the following:
- Minimum: The smallest number in the dataset.
- First Quartile (Q1): The median of the first half of the dataset.
- Median (Q2): The median of the entire dataset.
- Third Quartile (Q3): The median of the second half of the dataset.
- Maximum: The largest number in the dataset.
Step-by-Step Calculation:
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Minimum:
- The smallest number is \(15\).
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Maximum:
- The largest number is \(29\).
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Median (Q2):
- The dataset has \(11\) numbers. The median is the \(6^{th}\) number (since \(11\) is odd).
- The \(6^{th}\) number is \(18\).
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First Quartile (Q1):
- This is the median of the first half of the numbers: \(15, 16, 16, 17, 17\). Since there are \(5\) numbers (odd), the median is the \(3^{rd}\) number, which is \(16\).
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Third Quartile (Q3):
- This is the median of the second half of the numbers: \(18, 18, 19, 19, 29\). This also has \(5\) numbers (odd), so the median is the \(3^{rd}\) number, which is \(19\).
5-Number Summary:
- Minimum: \(15\)
- Q1: \(16\)
- Median (Q2): \(18\)
- Q3: \(19\)
- Maximum: \(29\)
The 5-number summary is (15, 16, 18, 19, 29).
Box Plot and Skewness:
When constructing the box plot for this data, we analyze the whiskers. The right whisker (from Q3 to the maximum) will be longer than the left whisker (from Q1 to the minimum) because the distance from Q3 (19) to the maximum (29) is greater than the distance from Q1 (16) to the minimum (15).
Answer Selection:
The response that best explains why the dataset is skewed is:
The right whisker is much longer than the left whisker.
1 answer