30, 31, 32, 33, 34, 35, 35, 37, 39, 40, 44, 46, 49, 49, 49, 51, 53, 56, 58, 58, 59, 59, 61, 62, 63, 64, 64, 64, 65, 66, 70, 70, 71, 72, 72, 76, 81, 82, 82, 86. The Q1 is 42 and the Q3 is 68. The median is 58.5. (c) Find the upper and lower outlier boundaries.

To find the upper and lower outlier boundaries, we can use the interquartile range (IQR) method. The formula for determining these boundaries is:

\[
\text{Lower boundary} = Q1 - 1.5 \times \text{IQR}
\]
\[
\text{Upper boundary} = Q3 + 1.5 \times \text{IQR}
\]

### Step 1: Calculate the IQR
The interquartile range (IQR) is the difference between the third quartile (Q3) and the first quartile (Q1).

\[
\text{IQR} = Q3 - Q1 = 68 - 42 = 26
\]

### Step 2: Find the lower boundary
\[
\text{Lower boundary} = Q1 - 1.5 \times \text{IQR} = 42 - 1.5 \times 26 = 42 - 39 = 3
\]

### Step 3: Find the upper boundary
\[
\text{Upper boundary} = Q3 + 1.5 \times \text{IQR} = 68 + 1.5 \times 26 = 68 + 39 = 107
\]

### Final Boundaries
- **Lower boundary:** 3
- **Upper boundary:** 107

Any data points outside this range (below 3 or above 107) would be considered outliers.

Would you like further details or have any questions?

Here are 5 related questions for deeper understanding:
1. How do we calculate quartiles from a data set?
2. What is the significance of the interquartile range (IQR)?
3. How do outliers affect data analysis?
4. What are some other methods to detect outliers besides the IQR method?
5. Why is 1.5 used as a multiplier when calculating outlier boundaries?

**Tip:** Quartiles divide data into four equal parts, and understanding them helps in assessing the spread and distribution of your data.

The query submitted by the user: 30, 31, 32, 33, 34, 35, 35, 37, 39, 40, 44, 46, 49, 49, 49, 51, 53, 56, 58, 58, 59, 59, 61, 62, 63, 64, 64, 64, 65, 66, 70, 70, 71, 72, 72, 76, 81, 82, 82, 86. The Q1 is 42 and the Q3 is 68. The median is 58.5. Find the upper and lower outlier boundaries.

Learn how to calculate outlier boundaries using the interquartile range method with a data set. The question involves determining the lower and upper boundaries using Q1 (42) and Q3 (68), and covers important concepts such as quartiles, IQR, and outliers. Suitable for students in Grades 10-12.

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