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Let's go through each part of the question:

Given dataLet's go through each part of the question:

Given 

| Value | Frequency |
|-------|-----------|
| 623   | 13        |
| 532   | 6         |
| 599   | 11        |
| 582   | 14        |
| 585   | 7         |
| 609   | 3         |

### Part (a): Finding the Mean

To find the mean of the frequency distribution, use the formula:

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

1. **Calculate each term** \( \text{Value} \times \text{Frequency} \):

   - \( 623 \times 13 = 8099 \)
   - \( 532 \times 6 = 3192 \)
   - \( 599 \times 11 = 6589 \)
   - \( 582 \times 14 = 8148 \)
   - \( 585 \times 7 = 4095 \)
   - \( 609 \times 3 = 1827 \)

2. **Sum of \( \text{Value} \times \text{Frequency} \)**:
   \[
   8099 + 3192 + 6589 + 8148 + 4095 + 1827 = 31950
   \]

3. **Sum of Frequencies**:
   \[
   13 + 6 + 11 + 14 + 7 + 3 = 54
   \]

4. **Calculate the Mean**:
   \[
   \text{Mean} = \frac{31950}{54} \approx 591.7
   \]

So, the mean is approximately **591.7**.

### Part (b): Finding the Median

To find the median, locate the cumulative frequency that corresponds to the middle position in the dataset. Since the total frequency is 54, the median will be around the 27th and 28th values.

1. **Cumulative Frequencies**:
   - For 623: \( 13 \)
   - For 532: \( 13 + 6 = 19 \)
   - For 599: \( 19 + 11 = 30 \)
   
   The 27th and 28th values fall within the frequency interval for 599, meaning the median value is **599**.

### Part (c): Finding the Mode

The mode is the value with the highest frequency. 

- Checking the frequencies: 623 has 13, 532 has 6, 599 has 11, 582 has 14, 585 has 7, and 609 has 3.
- The highest frequency is **14**, which corresponds to the value **582**.

Therefore, the mode is **582**.

---

### Summary of Answers:
- **(a)** Mean: \( 591.7 \)
- **(b)** Median: \( 599 \)
- **(c)** Mode: \( 582 \)

Would you like more details on any part of the solution?

---

#### Further Practice Questions:
1. What is the range of the given values?
2. If the frequency of 609 increased by 2, how would the mean change?
3. Can there be multiple modes in a dataset? Why or why not?
4. How would removing the smallest value (532) affect the median?
5. Explain why the mean might be misleading if there is an outlier.

#### Tip:
When finding the median in frequency distributions, always check the cumulative frequency to identify the correct interval.

Consider the frequency distribution. Complete parts (a) through (c): (a) Find the mean of the frequency distribution, (b) Find the median of the frequency distribution, (c) Find the mode of the frequency distribution.

Learn how to find the mean, median, and mode of a given frequency distribution. This problem provides a step-by-step solution for calculating the central tendency measures of a dataset with values and frequencies. Suitable for students in Grades 10-12.

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