To determine if the data is skewed right, left, or symmetric in a box-and-whisker plot, we need to first calculate the key statistics:

1. Order the Data: Arrange the data in ascending order.
2. Find the Median (Q2): The middle value when the data is sorted.
3. Find the First Quartile (Q1): The median of the lower half of the data (excluding the median).
4. Find the Third Quartile (Q3): The median of the upper half of the data (excluding the median).
5. Determine the Interquartile Range (IQR): $Q3 - Q1$.
6. Look at the Whiskers: The whiskers extend to the minimum and maximum data values that are not outliers (1.5 * IQR above Q3 or below Q1).

Step 1: Order the Data

The data in ascending order is: 24, 25, 26, 26, 27, 27, 27, 28, 28, 28, 29, 29, 29, 29, 29, 30, 30, 30, 30, 31, 31, 31, 31, 32, 32, 33, 34

Step 2: Find the Median (Q2)

There are 27 values in the dataset, so the median is the middle value (the 14th number): Median (Q2) = 29

Step 3: Find the First Quartile (Q1)

The lower half of the data is: 24, 25, 26, 26, 27, 27, 27, 28, 28, 28, 29, 29, 29 The median of this lower half (Q1) is the middle number: Q1 = 27

Step 4: Find the Third Quartile (Q3)

The upper half of the data is: 29, 29, 30, 30, 30, 30, 31, 31, 31, 31, 32, 32, 33, 34 The median of this upper half (Q3) is the middle number: Q3 = 31

Step 5: Interquartile Range (IQR)

IQR = Q3 - Q1 = 31 - 27 = 4

Step 6: Look at the Whiskers

• The lower whisker extends from Q1 to the smallest data point greater than or equal to $Q1 - 1.5 \times IQR = 27 - 6 = 21$, so the lower whisker ends at 24 (the smallest value within this range).
• The upper whisker extends from Q3 to the largest data point less than or equal to $Q3 + 1.5 \times IQR = 31 + 6 = 37$, so the upper whisker ends at 34 (the largest value within this range).

Conclusion: Skewness

Since the median (29) is closer to Q1 (27) and the upper whisker extends a bit further than the lower whisker, the plot would likely be slightly skewed to the right. This means there is a longer tail on the higher end of the data.

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Would you like more details or an explanation of how to create the box-and-whisker plot? Here are some related questions for further learning:

1. How do outliers affect a box-and-whisker plot?
2. What is the role of quartiles in understanding data distribution?
3. How do you calculate the interquartile range and why is it useful?
4. How can you tell if data is normally distributed based on a box plot?
5. What other types of plots can show data skewness besides box-and-whisker plots?

Tip: When interpreting a box plot, the direction of the skew is indicated by the relative positions of the median to the quartiles and the length of the whiskers.