To find the median using the interpolation method from a grouped frequency distribution, we'll need to follow these steps:

1. Identify the median class: This is the class where the cumulative frequency (CF) reaches or just exceeds half of the total frequency ($N/2$).

2. Calculate the cumulative frequencies (if not already provided).

3. Apply the interpolation formula:

   $\text{Median} = L + \left(\frac{\frac{N}{2} - F}{f}\right) \times h$

   Where:

   • $L$ = Lower boundary of the median class
   • $N$ = Total frequency
   • $F$ = Cumulative frequency of the class before the median class
   • $f$ = Frequency of the median class
   • $h$ = Class width

I'll extract the relevant data from the image you provided and perform the calculation.The image you provided contains a frequency distribution table. Let's summarize the data first:

Class Interval	Frequency (f)
5 - 10	6
10 - 15	9
15 - 20	12
20 - 25	15
25 - 30	9
30 - 35	8
35 - 40	5

Step 1: Calculate the Cumulative Frequency (CF)

We need to calculate the cumulative frequency to identify the median class.

Class Interval	Frequency (f)	Cumulative Frequency (CF)
5 - 10	6	6
10 - 15	9	15
15 - 20	12	27
20 - 25	15	42
25 - 30	9	51
30 - 35	8	59
35 - 40	5	64

Step 2: Determine the Median Class

The median class is found by identifying the class where the cumulative frequency just exceeds $N/2$.

Total Frequency ($N$) = 64

$\frac{N}{2} = \frac{64}{2} = 32$

The median class is the one where the cumulative frequency first exceeds 32, which is the class interval 20 - 25.

Step 3: Apply the Interpolation Formula

Now, use the interpolation formula to find the median.

Given:

• $L = 20$ (lower boundary of the median class)
• $N = 64$
• $F = 27$ (cumulative frequency of the class before the median class)
• $f = 15$ (frequency of the median class)
• $h = 5$ (class width)

$\text{Median} = 20 + \left(\frac{32 - 27}{15}\right) \times 5$ $\text{Median} = 20 + \left(\frac{5}{15}\right) \times 5$ $\text{Median} = 20 + \left(\frac{5}{3}\right) \times 1$ $\text{Median} = 20 + 1.67 = 21.67$

Final Answer

The median of the given data using the interpolation method is 21.67.

Would you like more details or have any further questions?

Related Questions

1. What is the mean of the given data set?
2. How do you calculate the mode for this data set?
3. Can you explain how the cumulative frequency is calculated?
4. How would the median change if the frequency of the class interval 15 - 20 was different?
5. How does the concept of quartiles relate to the median in grouped data?

Tip

When calculating the median for grouped data, always ensure the class intervals are continuous (i.e., no gaps between intervals). If not, adjustments should be made before applying the formula.