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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)

Responses

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

The whiskers are the same length.
The whiskers are the same length.

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.

1 answer

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 five values:

  1. Minimum: The smallest value in the dataset.
  2. First Quartile (Q1): The median of the first half of the dataset.
  3. Median (Q2): The middle value of the dataset.
  4. Third Quartile (Q3): The median of the second half of the dataset.
  5. Maximum: The largest value in the dataset.

Step 1: Minimum and Maximum

  • Minimum: 15
  • Maximum: 29

Step 2: Median (Q2)

Since there are 11 values, the median will be the sixth value in the ordered list.

  • Ordered dataset: \( 15, 16, 16, 17, 17, 18, 18, 18, 19, 19, 29 \)
  • Median (Q2): 18

Step 3: First Quartile (Q1)

To find Q1, we consider the first half of the dataset (the first five values):

  • First half: \( 15, 16, 16, 17, 17 \)
  • Q1 is the median of this half, which is the third value: 16

Step 4: Third Quartile (Q3)

To find Q3, we consider the second half of the dataset (the last five values):

  • Second half: \( 18, 18, 19, 19, 29 \)
  • Q3 is the median of this half, which is the third value: 19

5-Number Summary

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

Summary

The 5-number summary is:

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

Box Plot Interpretation

When constructing the box plot, we look at the lengths of the whiskers. The maximum (29) is much farther from the third quartile (19) than the minimum (15) is from the first quartile (16).

Thus, the correct reason explaining why the dataset is skewed can be determined as:

The right whisker is much longer than the left whisker.

This shows that there is a longer tail on the right side of the distribution, indicating positive (right) skewness.