2, 3, 5, 9, 9, 11, 12, 12, 13, 13, 14, 15, 15, 16, 19, 19, 20, 21, 25, 26, 32, 32, 38, 40, 48, 49, 49, 52, 86, 97. The Q1 is 12 and the Q2 is 38. The median is 19. Find the lower and upper outlier boundaries.

To find the lower and upper outlier boundaries using the given data, we use the **Interquartile Range (IQR) method**.

The steps are as follows:

### 1. Calculate the IQR
The **Interquartile Range (IQR)** is the difference between the third quartile (Q3) and the first quartile (Q1):
\[
\text{IQR} = Q3 - Q1
\]

From the information you provided:
- \( Q1 = 12 \)
- \( Q2 = 38 \) (which is actually \( Q3 \), based on the context)

Thus:
\[
\text{IQR} = 38 - 12 = 26
\]

### 2. Calculate the outlier boundaries

#### Lower boundary:
The lower boundary is calculated using the formula:
\[
\text{Lower Boundary} = Q1 - 1.5 \times \text{IQR}
\]
\[
\text{Lower Boundary} = 12 - 1.5 \times 26 = 12 - 39 = -27
\]

#### Upper boundary:
The upper boundary is calculated using the formula:
\[
\text{Upper Boundary} = Q3 + 1.5 \times \text{IQR}
\]
\[
\text{Upper Boundary} = 38 + 1.5 \times 26 = 38 + 39 = 77
\]

### 3. Conclusion:
- The **lower outlier boundary** is \(-27\), meaning there are no lower outliers since all data points are positive.
- The **upper outlier boundary** is **77**, meaning any data points above 77 are considered outliers.

### Outliers:
From your data set, the only value greater than 77 is **86** and **97**, so these are outliers.

Would you like more details or have any questions? Here are 5 related questions you can explore:
1. How do quartiles divide a data set?
2. What happens if the IQR is very small?
3. How do extreme outliers affect statistical analysis?
4. How would you visualize this data using a box plot?
5. Can the IQR method detect all types of outliers?

**Tip:** The IQR method is a robust way to detect outliers because it is not influenced by extreme values like the mean and standard deviation methods.

The query submitted by the user: 2, 3, 5, 9, 9, 11, 12, 12, 13, 13, 14, 15, 15, 16, 19, 19, 20, 21, 25, 26, 32, 32, 38, 40, 48, 49, 49, 52, 86, 97. The Q1 is 12 and the Q2 is 38. The median is 19. Find the lower and upper outlier boundaries.

This page explains how to find outliers using the IQR method with the data set: 2, 3, 5, 9, 9, 11, 12, 12, 13, 13, 14, 15, 15, 16, 19, 19, 20, 21, 25, 26, 32, 32, 38, 40, 48, 49, 49, 52, 86, 97. It covers the calculation of Q1, Q3, and how to apply the formula to identify the lower and upper outlier boundaries.

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