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.