To calculate the 5-number summary of the dataset \(15, 16, 16, 17, 17, 18, 18, 18, 19, 19, 29\), we will identify the minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum values.
-
Minimum: The smallest value in the dataset is \(15\).
-
Maximum: The largest value in the dataset is \(29\).
-
Median (Q2): To find the median, we first count the number of values (which is \(11\), an odd number). The median is the value at position \((11 + 1) / 2 = 6\), which is the 6th value in the ordered list: \[ 15, 16, 16, 17, 17, \textbf{18}, 18, 18, 19, 19, 29 \] Therefore, the median is \(18\).
-
First Quartile (Q1): This is the median of the first half of the dataset, which includes the values \(15, 16, 16, 17, 17\) (the first 5 values). The median of these values is the 3rd value, which is \(16\).
-
Third Quartile (Q3): This is the median of the second half of the dataset, which includes the values \(18, 18, 19, 19, 29\) (the last 6 values). The median of these values is the 3rd value, which is \(19\).
So the 5-number summary is:
- Minimum: \(15\)
- Q1: \(16\)
- Median (Q2): \(18\)
- Q3: \(19\)
- Maximum: \(29\)
Now we summarize:
- 5-number summary: \(15, 16, 18, 19, 29\)
Explanation for Skewness
When we construct a box plot using this 5-number summary, we can analyze the whiskers:
- The left whisker will extend from \(Q1\) to the minimum (\(16\) to \(15\)).
- The right whisker will extend from \(Q3\) to the maximum (\(19\) to \(29\)).
Since the maximum value (\(29\)) is much further away from \(Q3\) than the minimum value (\(15\)) is from \(Q1\), this indicates that the right whisker is much longer than the left whisker.
Therefore, the correct explanation of why the dataset is skewed is: "The right whisker is much longer than the left whisker."