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To find the mean of the students' scores from the given table, we’ll use the concept of a grouped frequency distribution. Here are the steps:

1. **Identify the Midpoint (Class Mark) for Each Score Range**: The midpoint for each range is calculated by averaging the lower and upper limits.
2. **Multiply Each Midpoint by its Frequency** to get the total for each range.
3. **Sum of the Products**: Add up all the products obtained from step 2.
4. **Sum of Frequencies**: Add up all the frequencies.
5. **Calculate the Mean** using the formula:

\[
\text{Mean} = \frac{\sum ( \text{Midpoint} \times \text{Frequency})}{\sum \text{Frequency}}
\]

### Calculations

| Score Range | Frequency (\(f\)) | Midpoint (\(x\)) | \(f \times x\) |
|-------------|--------------------|------------------|----------------|
| 76 – 80     | 3                 | 78              | \(3 \times 78 = 234\) |
| 71 – 75     | 4                 | 73              | \(4 \times 73 = 292\) |
| 66 – 70     | 8                 | 68              | \(8 \times 68 = 544\) |
| 61 – 65     | 10                | 63              | \(10 \times 63 = 630\) |
| 56 – 60     | 12                | 58              | \(12 \times 58 = 696\) |
| 51 – 55     | 6                 | 53              | \(6 \times 53 = 318\) |

### Summing Up
- Sum of \(f \times x\): \(234 + 292 + 544 + 630 + 696 + 318 = 2714\)
- Sum of Frequencies (\(f\)): \(3 + 4 + 8 + 10 + 12 + 6 = 43\)

### Mean Calculation
\[
\text{Mean} = \frac{2714}{43} \approx 63.14
\]

The mean score of the students is approximately **63.14**.

Would you like more details or have any questions?

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### Related Questions

1. How would you calculate the median score from this grouped data?
2. What does the mean score tell us about the students' performance?
3. How can we calculate the mode for grouped data like this?
4. What is the significance of choosing the midpoint for each range in finding the mean?
5. How would the mean change if the frequencies for the score ranges were altered?

### Tip
In grouped frequency distributions, the midpoint approximation gives a close estimate of the central tendency but may not be exact since individual scores within the ranges are not known.

Find the mean of students' scores based on the grouped frequency distribution table.

Learn how to find the mean of students' scores using a grouped frequency distribution table. This problem involves identifying midpoints, multiplying by frequencies, and applying the mean formula. Suitable for Grades 7-9 students and those studying statistics.

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