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.