Consider the boxplot below.
Box Plot with five Point Summary:  3,8,10,20,38
a.  What quarter has the smallest spread of data?
Third
Second
Fourth
First

b.  What is that spread?
c.  What quarter has the largest spread of data?
First
Third
Fourth
Second

d.  What is that spread?
e.  Find the Inter Quartile Range (IQR):
f.  Which interval has the most data in it?
12 - 36
20 - 38
10 - 20
3 - 10
g.  What value could represent the 53rd percentile?
5
11
53
20

To solve this problem, let’s first organize the five-number summary provided:  
- Minimum: **3**  
- Q1 (First Quartile): **8**  
- Median (Q2): **10**  
- Q3 (Third Quartile): **20**  
- Maximum: **38**  

### (a) What quarter has the smallest spread of data?
- **Q1 to Q2**: 8 to 10 → Spread = 10 - 8 = **2**  
- **Q2 to Q3**: 10 to 20 → Spread = 20 - 10 = **10**  
- **Q3 to Max**: 20 to 38 → Spread = 38 - 20 = **18**  
- **Min to Q1**: 3 to 8 → Spread = 8 - 3 = **5**

The smallest spread is between **Q1 and Q2** (Second quarter).  
**Answer:** Second quarter  

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### (b) What is that spread?
The spread of the second quarter (Q1 to Q2) is **2**.  
**Answer:** 2  

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### (c) What quarter has the largest spread of data?
From the calculations above:
- **Q3 to Max** (Third quarter) has the largest spread, which is **18**.  

**Answer:** Third quarter  

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### (d) What is that spread?
The spread of the third quarter is **18**.  
**Answer:** 18  

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### (e) Find the Interquartile Range (IQR):
The IQR is the difference between **Q3 and Q1**:
\[
IQR = Q3 - Q1 = 20 - 8 = 12
\]  
**Answer:** 12  

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### (f) Which interval has the most data in it?
- 12 - 36: **Not aligned with quartiles.**
- 20 - 38: Matches **Q3 to Max** (contains 25% of the data).
- 10 - 20: Matches **Q2 to Q3** (contains 25% of the data).
- 3 - 10: Matches **Min to Q2** (contains 50% of the data).

The interval **3 - 10** covers two quarters (Min to Q2), meaning **50%** of the data is contained in this range.  
**Answer:** 3 - 10  

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### (g) What value could represent the 53rd percentile?
- The **median (50th percentile)** is 10.
- The **53rd percentile** will be slightly above the median, closer to Q3.  
Thus, **11** is a reasonable estimate.

**Answer:** 11  

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Let me know if you have any further questions or need clarification!

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Here are **five related questions** to expand your understanding:
1. How do you interpret outliers in a box plot?
2. What does it mean if the IQR is very small?
3. Can two different datasets have the same five-number summary?
4. What other measures of spread exist besides the IQR?
5. How does skewness affect the shape of a box plot?

**Tip:** The IQR is often used to detect outliers by checking if any data point lies more than 1.5 times the IQR away from Q1 or Q3.

Explore a box plot with a five-number summary of 3, 8, 10, 20, 38. Analyze the spread of data across different quartiles, find the interquartile range (IQR), and determine the 53rd percentile. This statistical problem involves key concepts in quartile spread and percentile estimation, ideal for Grades 8-10.

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