To compare the frequencies above and below the median class, we will follow a step-by-step approach to identify key details such as the total frequency, the position of the median class, and the accumulated frequencies. Let's go through the solution methodically.
### Step 1: Identify given data
We have the following data:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
\text{Marks} & 0-10 & 0-20 & 0-30 & 0-40 & 0-50 & 0-60 \\
\hline
\text{Frequency} & 10 & 18 & 24 & 36 & 41 & 50 \\
\hline
\end{array}
\][/tex]
### Step 2: Total Frequency
The total frequency is given by the highest cumulative frequency:
[tex]\[ \text{Total Frequency} = 50 \][/tex]
### Step 3: Determine the Median Class
To find the median class:
1. Locate the middle position in the cumulative frequency distribution.
2. Based on the frequencies: 10, 18, 24, 36, 41, 50, the cumulative frequencies are as follows:
- [tex]\(0-10\)[/tex]: 10
- [tex]\(0-20\)[/tex]: 18
- [tex]\(0-30\)[/tex]: 24
- [tex]\(0-40\)[/tex]: 36
- [tex]\(0-50\)[/tex]: 41
- [tex]\(0-60\)[/tex]: 50
The median class can be identified by locating the position where the frequency count equals half of the total frequency plus one. This is because the cumulative frequency total is 50.
[tex]\[ \text{Median class position} = \left\lfloor \frac{total \ frequency}{2} \right\rfloor = \left\lfloor \frac{50}{2} \right\rfloor = 25 \][/tex]
The 25th frequency count falls within the class boundaries of [tex]\(0-40\)[/tex].
### Step 4: Frequencies Below and Above the Median Class
Let's split the frequencies around the median class:
- Below the median class:
The cumulative frequency just below 25th position is up to the class [tex]\(0-30\)[/tex]. So,
[tex]\[ \text{Frequency below median class} = 24 \][/tex]
- Above the median class:
The remaining frequencies above the 25th position will add up as:
[tex]\[ \text{Total Frequency} - \text{Frequency below Median Class} = 50 - 24 = 26 \][/tex]
### Summarize the Comparison
- Frequency below median class: 24
- Frequency above median class: 26
Based on our summary:
1. The cumulative frequency below the median class boundary (0-30) is 24.
2. The cumulative frequency above the median class boundary is 26.
### Conclusion
The result shows there are slightly more frequencies above the median class position compared to below, indicating the distribution is not perfectly symmetrical.
Thus, the frequencies below and above the median class are 24 and 26, respectively.
