Given the following boxplot:
a) Which quarter has the smallest spread of data? What is that spread?
b) Which quarter has the largest spread of data? What is that spread?
c) Find the interquartile range (IQR).
d) Are there more data in the interval 5–10 or in the interval 10–13? How do
you know this?
e) Which interval has the fewest data in it? How do you know this?
i. 0–2
ii. 2–4
iii. 10–12
iv. 12–13
v. need more information

To answer the questions based on a boxplot, we need to interpret the quartiles, spreads, and intervals. While I don't have access to the visual image of the boxplot, I'll guide you through the typical steps to find these values assuming you can either describe or share the image:

### Steps for each question:

**a) Which quarter has the smallest spread of data? What is that spread?**  
- Each "quarter" in a boxplot corresponds to a quartile: the ranges from Q1 (25th percentile) to Q2 (median), and Q2 to Q3 (75th percentile), etc.
- The quarter with the smallest spread is the one where the distance between two quartiles is the shortest. By looking at the boxplot, observe which quarter has the smallest distance between adjacent quartiles.
- Once identified, subtract the lower quartile from the upper quartile to find the spread.

**b) Which quarter has the largest spread of data? What is that spread?**  
- This is the opposite of part (a). Look for the quarter where the spread is the largest, i.e., the widest distance between two quartiles. 
- Subtract the lower quartile from the upper quartile for the largest quarter to find the spread.

**c) Find the interquartile range (IQR).**  
- The IQR is the difference between the third quartile (Q3, or 75th percentile) and the first quartile (Q1, or 25th percentile). 
- IQR = Q3 - Q1. Look at the boxplot to find Q1 and Q3, then subtract them to find the IQR.

**d) Are there more data in the interval 5–10 or in the interval 10–13? How do you know this?**  
- In a boxplot, the width of the intervals represents the spread of data. However, the data density in each interval depends on how tightly packed the data points are. 
- If the interval from 5–10 is in a quarter that has a smaller range, it means that more data are packed into that quarter (as boxplots are evenly divided into quartiles).
- The interval with the smaller spread will contain more data because the boxplot divides the data into equal quartiles.

**e) Which interval has the fewest data in it? How do you know this?**
- We are asked to compare intervals (0–2, 2–4, 10–12, 12–13).
- In a boxplot, each quartile contains 25% of the data. Therefore, smaller spreads in the box indicate a denser concentration of data. 
- The interval with the largest spread will have fewer data, while a more compressed interval will have more data. Look for which interval stretches over the largest range.

Given this setup, if you can share the exact quartile information or describe the boxplot more, I can help you fill in the specific numerical values.

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### Expansion Questions:
1. How can we distinguish between skewness in the boxplot based on the positions of the quartiles?
2. Why is the IQR considered a more robust measure of spread compared to the range?
3. How would outliers in the boxplot affect the interpretation of the spread?
4. How does a boxplot visually represent the distribution of data compared to a histogram?
5. Can we infer anything about the symmetry of the data from the boxplot?

### Tip:
Always check for outliers in a boxplot, as they can provide additional insights into the spread and distribution of the data that aren't immediately visible from the quartiles alone.

Let me know if you need further details or would like to share an image!

Given the following boxplot: a) Which quarter has the smallest spread of data? What is that spread? b) Which quarter has the largest spread of data? What is that spread? c) Find the interquartile range (IQR). d) Are there more data in the interval 5–10 or in the interval 10–13? How do you know this? e) Which interval has the fewest data in it? How do you know this? i. 0–2 ii. 2–4 iii. 10–12 iv. 12–13 v. need more information

Learn how to analyze data using a boxplot by identifying the smallest and largest data spreads, calculating the interquartile range (IQR), and determining data distribution across intervals. This guide helps students interpret quartiles and data spread, suitable for Grades 9-12.

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