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Calculate the five numbers summary of the skewed data set 15 16 16 17 18 18 19 29 when constructing a box plot on a number line which explains the reason why the data set is skewed

The median is it in exactly in the middle of the box the five numbers in the summary are exactly spread at equally spread out part the whiskers are the same length the right whisker is much longer than the left whisker

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To calculate the five-number summary of the given data set \(15, 16, 16, 17, 18, 18, 19, 29\), we'll follow these steps:

  1. Order the Data: The data is already ordered as \(15, 16, 16, 17, 18, 18, 19, 29\).

  2. Minimum: The smallest number in the data set is \( \text{Min} = 15 \).

  3. Maximum: The largest number in the data set is \( \text{Max} = 29 \).

  4. Median (Q2): To find the median, we can find the average of the two middle values (the 4th and 5th numbers) because there is an even number of data points. \[ \text{Median } (Q2) = \frac{17 + 18}{2} = \frac{35}{2} = 17.5 \]

  5. First Quartile (Q1): This is the median of the first half of the data (lower half): \(15, 16, 16, 17\). The median of these four numbers is the average of the 2nd and 3rd values. \[ Q1 = \frac{16 + 16}{2} = 16 \]

  6. Third Quartile (Q3): This is the median of the second half of the data (upper half): \(18, 18, 19, 29\). The median of these four numbers is the average of the 2nd and 3rd values. \[ Q3 = \frac{18 + 19}{2} = \frac{37}{2} = 18.5 \]

Now we can summarize the five-number summary:

  • Minimum: \(15\)
  • First Quartile (Q1): \(16\)
  • Median (Q2): \(17.5\)
  • Third Quartile (Q3): \(18.5\)
  • Maximum: \(29\)

Five-number summary:

  • Min: 15
  • Q1: 16
  • Median (Q2): 17.5
  • Q3: 18.5
  • Max: 29

Box Plot Explanation and Skewness:

When constructing a box plot based on these five numbers:

  • The box will stretch from \(Q1\) (16) to \(Q3\) (18.5). The line inside the box (the median) will be at \(17.5\), which is not exactly in the center of the box.

  • The whiskers extend out from the box to the minimum (15) on the left and the maximum (29) on the right. The left whisker will be from 16 (Q1) to 15 (min), and the right whisker will be from 18.5 (Q3) to 29 (max).

  • In this box plot, the right whisker is much longer than the left whisker, indicating that the data is positively skewed (right-skewed).

Conclusion:

The skewness of the data set arises from the presence of the outlier (29), which distorts the distribution of values. The median is closer to the first quartile than to the third quartile, which reinforces that the data set is skewed towards higher values.