To find the median of the given frequency distribution, we will follow a step-by-step process:
1. Identify the Total Frequency (N):
Find the total cumulative frequency from the table. The last entry in the 'c.f.' column gives the total frequency.
[tex]\[
N = 120
\][/tex]
2. Calculate the Median Position:
The median position is found using the formula:
[tex]\[
\text{Median Position} = \frac{N + 1}{2}
\][/tex]
Plugging in the value of [tex]\(N\)[/tex]:
[tex]\[
\text{Median Position} = \frac{120 + 1}{2} = \frac{121}{2} = 60.5
\][/tex]
3. Locate the Median Class:
The median class is the class where the cumulative frequency (c.f.) first exceeds the median position. We observe the cumulative frequencies to identify this class.
From the table, the cumulative frequency just before and after 60.5 are:
[tex]\[
\begin{array}{|c|c|c|}
\hline
x & f & c.f. \\
\hline
5 & 20 & 65 \\
6 & 25 & 90 \\
\end{array}
\][/tex]
Since 60.5 is between 45 and 65, the median class is the one where [tex]\( x = 6 \)[/tex].
4. Gather Median Class Information:
For the median class:
- Lower limit of the median class ([tex]\(L\)[/tex]): 5 (since we consider the class before 6 as the starting point)
- Cumulative frequency before the median class ([tex]\(c.f.\)[/tex] before): 65
- Frequency of the median class ([tex]\(f_m\)[/tex]): 25
- Class width ([tex]\(h\)[/tex]): Typically assumed to be 1 in such discrete data.
5. Apply the Median Formula:
The formula for the median in a frequency distribution is:
[tex]\[
\text{Median} = L + \left( \frac{\frac{N}{2} - c.f. \text{ before}}{f_m} \right) \times h
\][/tex]
6. Calculate the Median:
Substituting the values into the formula:
[tex]\[
\text{Median} = 5 + \left( \frac{60.5 - 65}{25} \right) \times 1
\][/tex]
Simplify inside the brackets first:
[tex]\[
\text{Median} = 5 + \left( \frac{-4.5}{25} \right)
\][/tex]
[tex]\[
\text{Median} = 5 + (-0.18)
\][/tex]
[tex]\[
\text{Median} = 4.82
\][/tex]
Therefore, the median of the given frequency distribution is approximately [tex]\(4.82\)[/tex].
