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

1. Construct a cumulative frequency table.
2. Determine the median class.
3. Use the median formula to calculate the median.

Step 1: Construct a cumulative frequency table

We first list the given intervals and their corresponding frequencies. We will then calculate the cumulative frequency (CF) by adding the frequencies progressively.

$x$ (Class Interval)	$f$ (Frequency)	Cumulative Frequency (CF)
20 - 30	3	3
30 - 40	5	8
40 - 50	20	28
50 - 60	10	38
60 - 70	5	43

Step 2: Find the median class

The total frequency (sum of all frequencies) is 43.

Now, we find the median class, which corresponds to the cumulative frequency that is just greater than or equal to half the total frequency, i.e., $\frac{43}{2} = 21.5$.

Looking at the cumulative frequency column, the first cumulative frequency greater than 21.5 is 28, which falls in the class interval 40 - 50.

Thus, the median class is 40 - 50.

Step 3: Use the median formula

The formula to calculate the median is:

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

Where:

• $L$ is the lower boundary of the median class (40 in this case).
• $N$ is the total frequency (43).
• $CF$ is the cumulative frequency of the class before the median class (8 for the class 30 - 40).
• $f$ is the frequency of the median class (20 for the class 40 - 50).
• $h$ is the class width (10, since the difference between 40 and 50 is 10).

Now, let's substitute the values into the formula:

$\text{Median} = 40 + \left( \frac{21.5 - 8}{20} \right) \times 10$

$\text{Median} = 40 + \left( \frac{13.5}{20} \right) \times 10$

$\text{Median} = 40 + (0.675 \times 10)$

$\text{Median} = 40 + 6.75$

$\text{Median} = 46.75$

Thus, the median is 46.75.

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Would you like further clarification on any part of the process?

Here are 5 related questions to expand your understanding:

1. What is the difference between the median and mean of a frequency distribution?
2. How do you determine the modal class of a frequency distribution?
3. How can you find the mean of a frequency distribution?
4. Why do we need cumulative frequency in finding the median of a grouped data set?
5. How do you handle open-ended intervals when calculating the median?

Tip: To avoid errors in constructing the cumulative frequency, always make sure that the cumulative frequencies are increasing and never decrease between rows.