0, 1, 2, 3, 5, 6, 9, 9, 11, 12, 13, 14, 15, 16, 19, 19, 20, 21, 25, 26, 29, 30, 32, 38, 40, 48, 49, 52, 67, 86. Find the lower and upper outlier boundaries.

To find the lower and upper outlier boundaries, we use the **interquartile range (IQR)** method. The steps are as follows:

1. **Order the data** (already done in the sequence):
   - 0, 1, 2, 3, 5, 6, 9, 9, 11, 12, 13, 14, 15, 16, 19, 19, 20, 21, 25, 26, 29, 30, 32, 38, 40, 48, 49, 52, 67, 86.

2. **Find the first quartile (Q1)**:
   - The first quartile, Q1, is the median of the lower half of the data (not including the overall median).
   - Lower half: 0, 1, 2, 3, 5, 6, 9, 9, 11, 12, 13, 14, 15, 16.
   - The median of this half is between 9 and 11, so Q1 = 10.

3. **Find the third quartile (Q3)**:
   - The third quartile, Q3, is the median of the upper half of the data (not including the overall median).
   - Upper half: 19, 19, 20, 21, 25, 26, 29, 30, 32, 38, 40, 48, 49, 52, 67, 86.
   - The median of this half is between 30 and 32, so Q3 = 31.

4. **Calculate the interquartile range (IQR)**:
   - IQR = Q3 - Q1 = 31 - 10 = 21.

5. **Calculate the lower and upper outlier boundaries**:
   - Lower boundary = Q1 - 1.5 * IQR = 10 - 1.5 * 21 = 10 - 31.5 = -21.5.
   - Upper boundary = Q3 + 1.5 * IQR = 31 + 1.5 * 21 = 31 + 31.5 = 62.5.

### Conclusion:
- **Lower outlier boundary** = -21.5
- **Upper outlier boundary** = 62.5

Any data points outside this range (-21.5, 62.5) are considered outliers. Therefore, the outliers in your data set are **67 and 86** (as they are greater than 62.5).

Would you like further details or clarification?

Here are some related questions to expand on this concept:
1. How do you interpret an IQR in terms of data spread?
2. What would happen if you used a different multiplier, like 2.0, instead of 1.5 for outliers?
3. How do you handle outliers in real-world data analysis?
4. Can outliers affect the mean or standard deviation significantly?
5. What is the purpose of the lower boundary, and why might it be negative in this case?

**Tip:** Outliers can often signal important anomalies in data, but sometimes they may indicate data entry errors or rare events. Always investigate them carefully!

Learn how to find lower and upper outlier boundaries using the IQR method with the dataset: 0, 1, 2, 3, 5, 6, 9, 9, 11, 12, 13, 14, 15, 16, 19, 19, 20, 21, 25, 26, 29, 30, 32, 38, 40, 48, 49, 52, 67, 86. This solution covers key concepts such as quartiles, interquartile range, and identifying outliers. Suitable for high school statistics students.

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