https://file.aiquickdraw.com/mathbot/file/1727391570002trdqb9q0.jpg

Let's go through the process of calculating the **mean of grouped scores** step by step using the image you provided.

### Step 1: Understand the table
The table contains the following columns:
- **Intervals**: These are the class intervals for the grouped data (e.g., 39–41, 36–38, etc.).
- **f**: This is the frequency, representing how many times scores within that interval appear.
- **d**: This is the deviation of the midpoint of each class from a selected reference point (this reference point is usually the midpoint of the class interval with the highest frequency).
- **fd**: This is the product of frequency (f) and deviation (d).

Here’s the breakdown of the table:

| Intervals | f  | d  | fd  |
|-----------|----|----|-----|
| 39–41     | 1  | 6  | 6   |
| 36–38     | 2  | 5  | 10  |
| 33–35     | 4  | 4  | 16  |
| 30–32     | 3  | 3  | 9   |
| 27–29     | 3  | 2  | 6   |
| 24–26     | 5  | 1  | 5   |
| 21–23     | 8  | 0  | 0   |
| 18–20     | 6  | -1 | -6  |
| 15–17     | 3  | -2 | -6  |
| 12–14     | 3  | -3 | -9  |
| 9–11      | 1  | -4 | -4  |

### Step 2: Summing values
- The **sum of frequencies (f)** is given as \( N = 40 \).
- The **sum of fd (∑fd)** is given as \( 30 \).

### Step 3: Identify the class midpoint
The Arithmetic Mean (AM) is usually calculated based on the midpoint of the class with the highest frequency. In this case, the class interval with the highest frequency is **21–23** with a frequency of 8. 
The midpoint of this class is calculated as:
\[
\text{Midpoint} = \frac{21 + 23}{2} = 22
\]
This 22 becomes our **assumed mean (A.M.)**.

### Step 4: Formula for calculating the mean of grouped data
The formula to calculate the mean (X̄) using the assumed mean method is:

\[
X̄ = A.M. + \left(\frac{∑fd}{N}\right) \times i
\]
Where:
- \( A.M. \) is the assumed mean (22 in this case),
- \( ∑fd \) is the sum of the product of frequencies and deviations (30 in this case),
- \( N \) is the total frequency (40 in this case),
- \( i \) is the class interval size. From the intervals given, the size is consistent at 3 (e.g., 39–41, 36–38).

### Step 5: Substitute the values
Now, plug in the values into the formula:

\[
X̄ = 22 + \left( \frac{30}{40} \right) \times 3
\]

### Step 6: Simplify the expression
First, calculate the fraction:

\[
\frac{30}{40} = 0.75
\]

Then, multiply by the class interval \( i = 3 \):

\[
0.75 \times 3 = 2.25
\]

Now, add this result to the assumed mean:

\[
X̄ = 22 + 2.25 = 24.25
\]

### Final Answer: The mean of the grouped scores is **24.25**.

---

### Do you want more details or have any questions?

Here are 5 related questions to explore:
1. How does the assumed mean method help in simplifying the calculations for grouped data?
2. What are other methods to find the mean of grouped data besides the assumed mean method?
3. How is the deviation calculated for each class interval?
4. What is the significance of class interval width in calculating the mean?
5. How would the result change if the assumed mean was different?

#### Tip:
Always double-check the class intervals to ensure consistency in width (i), as variations can affect the accuracy of the mean calculation.

Learn how to calculate the mean of grouped scores using the Assumed Mean Method. This problem covers statistical concepts such as grouped data, frequency distribution, and the assumed mean formula. Suitable for students in Grades 10-12.

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