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To solve the problem based on the given frequency distribution of box consignments, we need to compute the **mean**, **median**, and **mode** of the distribution. Here's a step-by-step guide:

### Given Frequency Distribution:

| Weight (Kg) | 15-19 | 20-24 | 25-29 | 30-34 | 35-39 | 40-44 | 45-49 | 50-54 | 55-59 | 60-64 | 65-69 |
|-------------|-------|-------|-------|-------|-------|-------|-------|-------|-------|-------|-------|
| No. of Boxes| 6     | 14    | 12    | 10    | 10    | 9     | 9     | 10    | 5     | 4     | 1     |

### 1. **Arithmetic Mean**:

The formula for the arithmetic mean is:

\[
\text{Mean} = \frac{\sum f_i x_i}{\sum f_i}
\]

Where:
- \( f_i \) is the frequency of each class.
- \( x_i \) is the mid-point of each class.

**Steps**:
- First, calculate the mid-point for each class.
- Multiply each mid-point by the frequency to get \( f_i x_i \).
- Then, sum all \( f_i x_i \).
- Finally, divide by the total frequency.

### 2. **Median**:

The median is the value that divides the data into two equal parts. The steps to find the median are:
1. Find the cumulative frequency.
2. Locate the median class, which is the class where the cumulative frequency exceeds half the total frequency.
3. Use the median formula:

\[
\text{Median} = L + \left( \frac{\frac{N}{2} - F}{f_m} \right) \times h
\]

Where:
- \( L \) is the lower boundary of the median class.
- \( N \) is the total frequency.
- \( F \) is the cumulative frequency of the class preceding the median class.
- \( f_m \) is the frequency of the median class.
- \( h \) is the class width.

### 3. **Mode**:

The mode is the class with the highest frequency. The formula to find the mode is:

\[
\text{Mode} = L + \left( \frac{f_m - f_{1}}{2f_m - f_1 - f_2} \right) \times h
\]

Where:
- \( L \) is the lower boundary of the modal class.
- \( f_m \) is the frequency of the modal class.
- \( f_1 \) is the frequency of the class preceding the modal class.
- \( f_2 \) is the frequency of the class following the modal class.
- \( h \) is the class width.

Let me now calculate these values.The computed values for the given frequency distribution are:

- **Mean**: 37.17
- **Median**: 36.2
- **Mode**: 23.2

These calculations follow standard statistical methods to find the central tendencies (mean, median, mode) from grouped data.

Would you like further details on the calculation process or any other related questions? 

Here are 5 related questions for practice:
1. How would the mean change if the number of boxes in the 40-44 kg category was increased?
2. What is the impact on the median if the class width of 25-29 is reduced?
3. How can we calculate the mean if the data provided were cumulative instead of grouped?
4. Why is the mode significantly lower than the mean and median in this dataset?
5. Can you explain the significance of the cumulative frequency in determining the median?

**Tip**: The mean is sensitive to extreme values, while the median is more robust in skewed distributions.

Given below is the frequency distribution of the consignment of boxes received by a trader. Compute the arithmetic mean, median and mode.

Learn how to calculate the mean, median, and mode of a frequency distribution. This example focuses on the weight distribution of boxes (15-69 kg) and provides a step-by-step solution, applying formulas for arithmetic mean, median, and mode. Suitable for students in Grades 9-12.

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