To find the median of a frequency distribution, we first need to follow these steps:

Steps to Calculate the Median:

1. Identify the cumulative frequency to locate the median class.
2. Determine the total number of observations.
3. Find the median class.
4. Use the formula for the median for grouped data.

Step 1: Create a cumulative frequency table.

Class Interval	Frequency (f)	Cumulative Frequency (CF)
10-20	12	12
20-30	30	12 + 30 = 42
30-40	34	42 + 34 = 76
40-50	65	76 + 65 = 141
50-60	45	141 + 45 = 186
60-70	5	186 + 5 = 191
70-80	18	191 + 18 = 209

Step 2: Calculate the total number of observations.

The total number of observations, $N = 209$.

Step 3: Find the median class.

The median is located at the $\frac{N}{2}$-th observation. So, $\frac{209}{2} = 104.5$. The cumulative frequency just greater than 104.5 is 141, which corresponds to the class interval 40-50. This is the median class.

Step 4: Apply the median formula for grouped data.

The formula for the median is:

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

Where:

• $L = 40$ (lower boundary of the median class),
• $\frac{N}{2} = 104.5$,
• $CF = 76$ (cumulative frequency of the class preceding the median class),
• $f = 65$ (frequency of the median class),
• $h = 10$ (class width).

Now, substitute the values into the formula:

$\text{Median} = 40 + \left(\frac{104.5 - 76}{65}\right) \times 10$ $\text{Median} = 40 + \left(\frac{28.5}{65}\right) \times 10$ $\text{Median} = 40 + (0.4385) \times 10$ $\text{Median} = 40 + 4.385$ $\text{Median} \approx 44.39$

Final Answer:

The median of the data is approximately 44.39.

Would you like more details on this process or have any questions?

Here are 5 questions related to the concept of finding the median:

1. How is the cumulative frequency table constructed?
2. What is the difference between grouped and ungrouped data when finding the median?
3. Can the median class change if the frequency distribution is adjusted slightly?
4. Why do we use the lower boundary of the median class in the formula?
5. How would the median be affected if we increased the class width?

Tip: Always ensure you identify the median class correctly by checking the cumulative frequencies carefully.