To find the upper and lower outlier boundaries for a data set, we can use the Interquartile Range (IQR) method. Here’s how you can do it:

Step 1: Arrange the Data in Ascending Order

First, let's arrange the data in ascending order:

$3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 10, 10, 13, 14, 25$

Step 2: Find the Quartiles (Q1, Q3)

• Q1 (First Quartile): This is the median of the lower half of the data. For this data set, the lower half consists of the first 11 numbers: $3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 6$ The median (Q1) is 5.

• Q3 (Third Quartile): This is the median of the upper half of the data. The upper half consists of the last 11 numbers: $6, 6, 6, 7, 7, 8, 8, 10, 10, 13, 14$ The median (Q3) is 8.

Step 3: Calculate the Interquartile Range (IQR)

$\text{IQR} = Q3 - Q1 = 8 - 5 = 3$

Step 4: Calculate the Outlier Boundaries

• Lower Bound: $Q1 - 1.5 \times \text{IQR}$ $5 - 1.5 \times 3 = 5 - 4.5 = 0.5$

• Upper Bound: $Q3 + 1.5 \times \text{IQR}$ $8 + 1.5 \times 3 = 8 + 4.5 = 12.5$

Step 5: Identify Outliers

• Lower Outlier Boundary: 0.5
• Upper Outlier Boundary: 12.5

Any data points below 0.5 or above 12.5 are considered outliers.

Outliers in the Data Set

• Values above 12.5: 14, 25
• Values below 0.5: None

Final Answer

• Lower Outlier Boundary: 0.5
• Upper Outlier Boundary: 12.5

Would you like to go into more detail or have any questions?

Here are five related questions you might consider:

1. How do you calculate the median for an odd-sized data set?
2. What happens if the data set has no outliers?
3. Can outliers significantly affect the mean and standard deviation of a data set?
4. What are the differences between the IQR method and the Z-score method for detecting outliers?
5. How would you handle outliers in a data set for statistical analysis?

Tip: Outliers can often provide valuable insights into the variability or anomalies in your data, so don't always discard them without considering their significance.