Find median using linear interpolation method

To find the median using the linear interpolation method, we first need to understand the scenario where this method is typically used: **grouped data**. For grouped data, the median is estimated within a certain class interval. The linear interpolation method helps estimate where the median would lie within that interval.

### Steps to Find the Median using Linear Interpolation

Given a frequency distribution table with class intervals, follow these steps:

1. **Determine the Median Class**: 
   Find the cumulative frequency for each class. The median class is the class interval where the cumulative frequency exceeds \( \frac{N}{2} \), where \( N \) is the total number of observations.

2. **Identify Key Values**:
   - **\( L \)**: The lower boundary of the median class.
   - **\( N \)**: The total number of observations.
   - **\( F \)**: The cumulative frequency of the class before the median class.
   - **\( f_m \)**: The frequency of the median class.
   - **\( h \)**: The width of the median class interval.

3. **Apply the Linear Interpolation Formula**:
   The formula for the median \( M \) is:

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

   This formula linearly interpolates the position of the median within the median class.

### Example Calculation

Let's assume the following grouped frequency distribution:

| Class Interval | Frequency |
|----------------|-----------|
| 0 - 10         | 5         |
| 10 - 20        | 9         |
| 20 - 30        | 12        |
| 30 - 40        | 8         |
| 40 - 50        | 6         |

1. **Find the Total Frequency (N)**:
   \[
   N = 5 + 9 + 12 + 8 + 6 = 40
   \]

2. **Determine Cumulative Frequency**:

| Class Interval | Frequency | Cumulative Frequency |
|----------------|-----------|----------------------|
| 0 - 10         | 5         | 5                    |
| 10 - 20        | 9         | 14                   |
| 20 - 30        | 12        | 26                   |
| 30 - 40        | 8         | 34                   |
| 40 - 50        | 6         | 40                   |

3. **Find the Median Class**:
   \[
   \frac{N}{2} = \frac{40}{2} = 20
   \]
   The median class is **20 - 30** (where cumulative frequency first exceeds 20).

4. **Identify Key Values**:
   - \( L = 20 \) (lower boundary of the median class)
   - \( F = 14 \) (cumulative frequency of the class before the median class)
   - \( f_m = 12 \) (frequency of the median class)
   - \( h = 10 \) (class width)

5. **Apply the Formula**:
   \[
   M = 20 + \left( \frac{20 - 14}{12} \right) \times 10
   \]
   \[
   M = 20 + \left( \frac{6}{12} \right) \times 10
   \]
   \[
   M = 20 + 5 = 25
   \]

### Conclusion
The estimated median using the linear interpolation method is **25**.

Would you like more details or have any questions?

### Related Questions:
1. How do you find the mode for grouped data?
2. What is the difference between the mean and median in statistics?
3. How do you interpret the median in a real-world context?
4. What are other methods to estimate the median?
5. How does the choice of class intervals affect the median estimation?

**Tip:** When dealing with grouped data, always ensure class intervals are consistent (same width) for accurate interpolation.

Learn how to find the median using the linear interpolation method in statistics with a step-by-step explanation and example calculation. This problem involves interpreting grouped data and applying the linear interpolation formula to estimate the median class. Suitable for high school students studying statistics.

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