To determine the first quartile (Q1), median (Q2), third quartile (Q3), and identify the outliers for the dataset, let's go through each calculation.

Step-by-Step Solution

1. Organize the Data

The data, in ascending order, is: $39, 44, 45, 46, 51, 53, 54, 55, 56, 58, 59, 61, 63, 64, 64, 65, 65, 67, 76, 84, 90, 91, 92$

2. Quartiles Calculation

• Median (Q2): This is the middle value. Since we have 23 data points, the median will be the 12th value.

  $Q2 = 61$

• First Quartile (Q1): This is the median of the lower half (first 11 values).

  $Q1 = 54$

• Third Quartile (Q3): This is the median of the upper half (last 11 values).

  $Q3 = 65$

3. Interquartile Range (IQR)

$IQR = Q3 - Q1 = 65 - 54 = 11$

4. Outlier Boundaries

• Lower Outlier Boundary: $Q1 - 1.5 \times IQR$

  $54 - (1.5 \times 11) = 54 - 16.5 = 37.5$

• Upper Outlier Boundary: $Q3 + 1.5 \times IQR$

  $65 + (1.5 \times 11) = 65 + 16.5 = 81.5$

5. Identifying Outliers

• Outliers are any values below 37.5 or above 81.5.

  In this dataset, the values above 81.5 are: 84, 90, 91, 92

Summary

• First Quartile (Q1): 54
• Median (Q2): 61
• Third Quartile (Q3): 65
• Lower Outlier Boundary: 37.5
• Upper Outlier Boundary: 81.5
• Outliers: 84, 90, 91, 92

Would you like more details on these calculations, or have any questions?

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Follow-up Questions:

1. How is the interquartile range used in detecting outliers?
2. What are other methods to detect outliers beyond IQR?
3. How does sample size affect the quartiles in a dataset?
4. What if the dataset were normally distributed? How would quartiles and outliers change?
5. Can quartiles be calculated for non-numeric data types?

Tip: Quartiles divide data into four equal parts, and the IQR-based method is a robust way to detect outliers in skewed distributions.