To find the median of a frequency distribution, we need to follow a systematic process. Here's how we'll approach it:

Step 1: Organize the given data

We are given two columns: one for the values of the variable (X) and another for the corresponding frequencies (f).

X	f
0	7
1	14
2	18
3	36
4	51
5	54
6	52
7	18

Step 2: Calculate the cumulative frequency

The cumulative frequency (CF) is the running total of the frequencies.

X	f	Cumulative Frequency (CF)
0	7	7
1	14	7 + 14 = 21
2	18	21 + 18 = 39
3	36	39 + 36 = 75
4	51	75 + 51 = 126
5	54	126 + 54 = 180
6	52	180 + 52 = 232
7	18	232 + 18 = 250

Step 3: Find the total number of data points (N)

The total number of data points is the sum of the frequencies:

$N = 7 + 14 + 18 + 36 + 51 + 54 + 52 + 18 = 250$

Step 4: Determine the median class

The median is the value that divides the data into two equal halves. Since the total number of data points is 250, the median is located at:

$\frac{N}{2} = \frac{250}{2} = 125$

Now, we find the cumulative frequency that just exceeds 125. From the table, the cumulative frequency reaches 126 at $X = 4$, so the median class is for $X = 4$.

Step 5: Use the formula for the median

To calculate the median, we use the following formula for a frequency distribution:

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

Where:

• $L$ = lower boundary of the median class (for $X = 4$, $L = 3.5$)
• $N$ = total number of data points (250)
• $CF$ = cumulative frequency before the median class (for $X = 4$, $CF = 75$)
• $f$ = frequency of the median class (for $X = 4$, $f = 51$)
• $h$ = class width (the difference between successive $X$-values, here $h = 1$)

Step 6: Substituting the values into the formula:

$\text{Median} = 3.5 + \left(\frac{\frac{250}{2} - 75}{51}\right) \times 1$

$\text{Median} = 3.5 + \left(\frac{125 - 75}{51}\right)$

$\text{Median} = 3.5 + \left(\frac{50}{51}\right)$

$\text{Median} = 3.5 + 0.98 \approx 4.48$

Final Answer:

The median for the given frequency distribution is approximately 4.48.

Would you like further details or clarifications on any step?

Here are 5 related questions:

1. How do you calculate the mode of a frequency distribution?
2. How do you calculate the mean of a frequency distribution?
3. How do you find the range of a frequency distribution?
4. What is the difference between a cumulative frequency and a frequency distribution?
5. How do you interpret the cumulative frequency table?

Tip: Always double-check that your cumulative frequency matches the total frequency to avoid mistakes when calculating the median!