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Using Sturge’s Rule, construct a frequency distribution with a cumulative frequency and upper boundary.

To create a frequency distribution using Sturges' Rule, we first need to determine the number of classes for the data. Sturges' Rule provides a formula for determining this:

\[
k = 1 + 3.322 \log_{10}(n)
\]

where:
- \( k \) is the number of classes,
- \( n \) is the total number of data points.

From the image, we know that the data contains 33 participants. Let's compute the number of classes and proceed with constructing the frequency distribution.Using Sturges' Rule, the number of classes \( k \) is 7. Now, let's move on to constructing the frequency distribution. 

### Step-by-step Process:
1. **Range**: Calculate the range of the data, which is the difference between the maximum and minimum values.
2. **Class Width**: Divide the range by the number of classes and round up to a reasonable number.
3. **Class Boundaries**: Define the class intervals based on the class width.
4. **Frequency Distribution**: Count how many data points fall within each class interval.
5. **Cumulative Frequency**: Compute the cumulative frequency for each class.

Let's first calculate the range, class width, and define the class intervals. I will then compute the frequency distribution.The class width is 9, and the class intervals based on this are:

1. \( 11 - 19 \)
2. \( 20 - 28 \)
3. \( 29 - 37 \)
4. \( 38 - 46 \)
5. \( 47 - 55 \)
6. \( 56 - 64 \)
7. \( 65 - 73 \)

Next, I'll compute the frequency distribution, cumulative frequency, and upper boundaries for each class interval.Here is the frequency distribution along with cumulative frequency and upper boundaries based on Sturges' Rule:

| Class Interval | Frequency | Cumulative Frequency | Upper Boundary |
|----------------|-----------|----------------------|----------------|
| 11 - 19        | 3         | 3                    | 19             |
| 20 - 28        | 3         | 6                    | 28             |
| 29 - 37        | 9         | 15                   | 37             |
| 38 - 46        | 7         | 22                   | 46             |
| 47 - 55        | 4         | 26                   | 55             |
| 56 - 64        | 5         | 31                   | 64             |
| 65 - 73        | 1         | 32                   | 73             |

This distribution summarizes the data from the quiz scores into class intervals, showing the frequency of scores within each range, the cumulative frequency, and the upper boundary for each class.

Do you need further details or have any questions? 

Here are five related questions for further exploration:
1. How would the distribution change with a different number of class intervals?
2. What are the potential advantages of using Sturges' Rule in frequency distribution?
3. How can the distribution be used to analyze performance in the quiz?
4. How would you compute the mean and standard deviation based on this grouped data?
5. How does this distribution compare with a histogram of the raw data?

**Tip**: Always check whether your class intervals fully encompass the data set to avoid missing any data points.

Learn how to construct a frequency distribution using Sturges' Rule for a quiz score dataset of 33 participants. The step-by-step guide includes calculating class intervals, cumulative frequency, and upper boundaries. Suitable for students in Grades 10-12.

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