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The image you've uploaded contains a grouped frequency distribution table. The goal is to solve for \(Q_3\), the third quartile, of the grouped data.

Here’s the table extracted from the image:

| Class Interval | Frequency (f) | Cumulative Frequency (cf) |
|----------------|---------------|---------------------------|
| 81 - 90        | 2             | 40                        |
| 71 - 80        | 6             | 38                        |
| 61 - 70        | 15            | 32                        |
| 51 - 60        | 10            | 17                        |
| 41 - 50        | 7             | 7                         |

To find \(Q_3\), we use the formula for the third quartile in grouped dataThe image you've uploaded contains a grouped frequency distribution table. The goal is to solve for \(Q_3\), the third quartile, of the grouped data.

Here’s the table extracted from the image:

| Class Interval | Frequency (f) | Cumulative Frequency (cf) |
|----------------|---------------|---------------------------|
| 81 - 90        | 2             | 40                        |
| 71 - 80        | 6             | 38                        |
| 61 - 70        | 15            | 32                        |
| 51 - 60        | 10            | 17                        |
| 41 - 50        | 7             | 7                         |

To find \(Q_3\), we use the formula for the third quartile in grouped 

\[
Q_3 = L + \left(\frac{\frac{3N}{4} - F}{f}\right) \times h
\]

Where:
- \(L\) = lower boundary of the quartile class
- \(N\) = total frequency (which is 40 in this case)
- \(F\) = cumulative frequency of the class before the quartile class
- \(f\) = frequency of the quartile class
- \(h\) = class width

### Steps to Calculate \(Q_3\):

1. **Find \(\frac{3N}{4}\)**:
   \[
   \frac{3 \times 40}{4} = 30
   \]
   So, the third quartile corresponds to the cumulative frequency 30.

2. **Identify the Quartile Class**:
   The class interval where the cumulative frequency just exceeds 30 is 61 - 70.

3. **Determine the Quartile Class Parameters**:
   - \(L = 60.5\) (the lower boundary of 61 - 70)
   - \(F = 17\) (cumulative frequency before the quartile class)
   - \(f = 15\) (frequency of the quartile class)
   - \(h = 10\) (class width)

4. **Apply the Quartile Formula**:
   \[
   Q_3 = 60.5 + \left(\frac{30 - 17}{15}\right) \times 10
   \]
   Simplifying this:
   \[
   Q_3 = 60.5 + \left(\frac{13}{15}\right) \times 10
   \]
   \[
   Q_3 = 60.5 + \left(0.8667\right) \times 10
   \]
   \[
   Q_3 = 60.5 + 8.67 = 69.17
   \]

### Final Answer:
\(Q_3\) for the grouped data is approximately **69.17**.

Would you like more details on any specific part of the process? 

Here are 5 related questions for further exploration:
1. How would the calculation change if the class intervals were of different widths?
2. What is the significance of the quartile in data analysis?
3. How can we use the cumulative frequency graph to find the quartiles?
4. How do you determine which class interval the median lies in using grouped data?
5. What happens if the dataset has more or fewer intervals? How does it affect quartile calculation?

**Tip:** When solving for quartiles or percentiles, always check the cumulative frequency to identify the correct class interval before applying the formula.

Learn how to calculate the third quartile (Q3) for grouped data using the formula Q3 = L + ((3N/4 - F) / f) × h. This detailed solution explains each step of the process, suitable for high school students in Grades 9-12.

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