5-Number Summary: Siblings Data Range, IQR & Quartiles

Answer

The requested statistics are: A) Range = $10$, B) IQR = $4$, C) between $1$ and $5$ siblings, and D) above $1$ sibling.

Explanation

The image shows a blank number line intended for a box-and-whisker plot, representing a data set of 15 student sibling counts. To solve this, we must first organize the data in ascending order.

1. Sort the Data The raw data is $\{1, 5, 2, 8, 7, 0, 1, 3, 1, 10, 4, 5, 5, 6, 4\}$. Once sorted from least to greatest: $\{0, 1, 1, 1, 2, 3, 4, 4, 5, 5, 5, 6, 7, 8, 10\}$

2. Find the 5-Number Summary

   • Minimum: $0$
   • First Quartile ($Q_1$): The median of the lower half (first 7 numbers) is the 4th value: $1$.
   • Median ($Q_2$): The middle value of the 15 numbers is the 8th value: $4$.
   • Third Quartile ($Q_3$): The median of the upper half (last 7 numbers) is the 12th value: $6$. (Actually, wait—let's re-verify: the 12th number is $6$). Let's check: $Q_1$ is at index 4 ($1$), Median is index 8 ($4$), $Q_3$ is index 12 ($6$).
   • Maximum: $10$

3. Calculate Range $Range = Maximum - Minimum = 10 - 0 = 10$ The range represents the total spread of our data from the smallest to the largest value.

4. Calculate Interquartile Range (IQR) $IQR = Q_3 - Q_1 = 6 - 2 = 4$ Wait, looking closely at the sorted list: $\{0, 1, 1, 1, 2, 3, 4, \mathbf{4}, 5, 5, 5, 6, 7, 8, 10\}$. The lower half is $\{0, 1, 1, \mathbf{1}, 2, 3, 4\}$, so $Q_1 = 1$. The upper half is $\{5, 5, 5, \mathbf{6}, 7, 8, 10\}$, so $Q_3 = 6$. $IQR = 6 - 1 = 5$. Let me re-calculate: $6 - 1 = 5$.

5. Answer Questions C and D

   • C (Middle 50%): This is represented by the range between $Q_1$ and $Q_3$, which is $1$ to $6$.
   • D (Above what number does 75% exist): 75% of the data lies above the first quartile ($Q_1$), which is $1$.

Final Answer

A) Range: $\boxed{10}$ B) IQR: $\boxed{5}$ C) Middle 50%: $\boxed{1 \text{ to } 6}$ D) 75% above: $\boxed{1}$

Common Mistakes

• Forgetting to sort: Always sort the list from smallest to largest first; otherwise, the medians and quartiles will be incorrect.
• Inclusive vs. Exclusive: When finding quartiles in an odd-numbered dataset, students often get confused about whether to include the median in the halves. By convention, you exclude the median itself.