To construct a box-and-whisker plot (also known as a box plot) for this data sample of 12 values, follow these steps:

Data Set:

$35, 29, 44, 72, 34, 64, 41, 50, 54, 104, 39, 58$

Steps:

1. Sort the data in ascending order: $29, 34, 35, 39, 41, 44, 50, 54, 58, 64, 72, 104$

2. Find the minimum and maximum values (whiskers):

   • Minimum (Q0) = 29
   • Maximum (Q4) = 104

3. Determine the quartiles:

   • First Quartile (Q1): The median of the lower half of the data (excluding the median itself if the dataset is odd).

   • Second Quartile (Q2 or Median): The median of the entire dataset.

   • Third Quartile (Q3): The median of the upper half of the data (excluding the median itself if the dataset is odd).

   • Lower half: $29, 34, 35, 39, 41, 44$

     • Median (Q1) = $\frac{35 + 39}{2} = 37$

   • Upper half: $50, 54, 58, 64, 72, 104$

     • Median (Q3) = $\frac{58 + 64}{2} = 61$

4. Calculate the median (Q2):

   • The median (middle of the entire data set): $\frac{44 + 50}{2} = 47$

Five-Number Summary:

• Minimum = 29
• Q1 = 37
• Median (Q2) = 47
• Q3 = 61
• Maximum = 104

Box-and-Whisker Plot Components:

• The "box" extends from Q1 (37) to Q3 (61).
• The "whiskers" extend from the minimum value (29) to the maximum value (104).
• The median (47) is represented by a line inside the box.

Would you like further explanation on how to interpret the box-and-whisker plot?

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Relative Questions:

1. How do you interpret outliers in a box-and-whisker plot?
2. What does the interquartile range (IQR) tell us in this dataset?
3. How do whiskers in the box plot relate to variability?
4. Can a box plot show skewness in a dataset?
5. How would the box plot change if the dataset had more extreme values?

Tip: Box-and-whisker plots are great for visualizing data distribution, especially the spread and potential outliers.