Alright, let's go through the solution step-by-step:
### Given Data
We have a frequency distribution of marks as follows:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}\hline
\text{Marks} & 5 & 6 & 7 & 8 & 9 \\ \hline
\text{Frequency} & 7 & 3 & 8 & 7 & 5 \\ \hline
\end{array}
\][/tex]
### 1. Mode
The mode is the value that occurs most frequently in the distribution. To find the mode, we look for the highest frequency.
- Marks: [tex]\[5, 6, 7, 8, 9\][/tex]
- Frequencies: [tex]\[7, 3, 8, 7, 5\][/tex]
Here, the highest frequency is 8, which corresponds to the mark 7. Therefore, the mode is:
[tex]\[ \text{Mode} = 7 \][/tex]
### 2. Range
The range is the difference between the maximum and minimum marks in the dataset.
- Maximum mark: [tex]\[9\][/tex]
- Minimum mark: [tex]\[5\][/tex]
The range is:
[tex]\[ \text{Range} = 9 - 5 = 4 \][/tex]
### 3. Median
The median is the middle value of the dataset when the data has been arranged in ascending order. Given the frequencies, let's first expand the dataset.
- Marks:
[tex]\[5 \text{ occurs } 7 \text{ times}\][/tex]
[tex]\[6 \text{ occurs } 3 \text{ times}\][/tex]
[tex]\[7 \text{ occurs } 8 \text{ times}\][/tex]
[tex]\[8 \text{ occurs } 7 \text{ times}\][/tex]
[tex]\[9 \text{ occurs } 5 \text{ times}\][/tex]
Expanding the dataset:
[tex]\[
\text{Expanded Marks} =
[5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9]
\][/tex]
There are 30 values in the expanded list. Since the number of values [tex]\( n \)[/tex] is even ([tex]\( n = 30 \)[/tex]), the median is the average of the [tex]\( \frac{n}{2} \)[/tex]th and [tex]\( \left( \frac{n}{2} + 1 \right) \)[/tex]th values.
The [tex]\( \frac{n}{2} = 15 \)[/tex]th value is 7, and the [tex]\( \frac{n}{2} + 1 = 16 \)[/tex]th value is also 7 in the sorted list.
Hence, the median is:
[tex]\[ \text{Median} = \frac{7 + 7}{2} = 7.0 \][/tex]
### Summary
Compiling all these results, we get:
- Mode: [tex]\( 7 \)[/tex]
- Range: [tex]\( 4 \)[/tex]
- Median: [tex]\( 7.0 \)[/tex]
Thus, we have the following final values:
[tex]\[ \text{Mode} = 7 \][/tex]
[tex]\[ \text{Range} = 4 \][/tex]
[tex]\[ \text{Median} = 7.0 \][/tex]
