https://file.aiquickdraw.com/mathbot/file/1727052447450w65eck6h.jpg

Let's go step by step to solve the questions based on the data provided in the image.

### 1. **Finding the Range:**
The range is calculated as the difference between the maximum and minimum values of the dataset.

- **Maximum value** = 11.5
- **Minimum value** = 9

\[
\text{Range} = 11.5 - 9 = 2.5 \text{ years}
\]

### 2. **Finding the Standard Deviation:**
To find the standard deviation, we first calculate the mean (average) and then the variance.

- First, let's list all the values:  
  \(9.5, 11.5, 11, 10, 11, 10.5, 9, 11, 9.5, 10.5, 10, 11, 10, 9.5, 9.5, 10, 9.5, 10, 9, 9.5\)

The number of data points \( n = 20 \).

1. **Calculate the mean (average):**

\[
\text{Mean} = \frac{\text{Sum of all values}}{n}
\]

\[
\text{Mean} = \frac{(9.5 + 11.5 + 11 + 10 + 11 + 10.5 + 9 + 11 + 9.5 + 10.5 + 10 + 11 + 10 + 9.5 + 9.5 + 10 + 9.5 + 10 + 9 + 9.5)}{20}
\]

2. **Find the variance and then the square root to get the standard deviation.**

I'll calculate that next.

### 3. **Five-number summary:**
The five-number summary includes:
- **Minimum**: 9
- **Q1 (First quartile)**: 9.5
- **Median (Q2)**: 10
- **Q3 (Third quartile)**: 10.625 (rounded)
- **Maximum**: 11.5

Let me calculate the standard deviation now.### Standard Deviation:
- The **mean** is approximately \( 10.1 \) years.
- The **variance** is approximately \( 0.51 \).
- The **standard deviation** is approximately \( 0.7 \) years (rounded to the nearest tenth).

### Summary of Results:
1. **Range**: 2.5 years
2. **Standard Deviation**: 0.7 years
3. **Five-number summary**:
   - Minimum: 9
   - Q1: 9.5
   - Median: 10
   - Q3: 10.625
   - Maximum: 11.5

Do you need further details on these calculations, or any other clarifications?

### Related Questions:
1. How is the quartile (Q1, Q3) calculated from a dataset?
2. What is the difference between population and sample standard deviation?
3. How does skewness in a dataset affect the range and standard deviation?
4. Can outliers in a dataset significantly impact the five-number summary?
5. What are other measures of data dispersion apart from standard deviation?

### Tip:
For large datasets, tools like Excel or statistical software can simplify calculations for range, standard deviation, and quartiles.

Find the range, standard deviation, and five-number summary for the ages of 20 fifth-graders.

Learn how to calculate the range, standard deviation, and five-number summary for a dataset of 20 students' ages. This problem explores basic statistics concepts suitable for Grades 6-8.

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