To determine the mean scores from the given frequency distribution of scores, let's proceed step-by-step.
Here is the given frequency distribution:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Mark Interval} & \text{Frequency} \\
\hline
0-4 & 1 \\
5-9 & 5 \\
10-14 & 7 \\
15-19 & 13 \\
20-24 & 11 \\
25-29 & 9 \\
\hline
\end{array}
\][/tex]
### Step 1: Calculate Mid-Points of Each Interval
First, we need to find the mid-point for each class interval. The mid-point can be found by averaging the lower and upper bounds of each interval.
- For the interval [tex]\(0-4\)[/tex]:
[tex]\[
\frac{0+4}{2} = 2
\][/tex]
- For the interval [tex]\(5-9\)[/tex]:
[tex]\[
\frac{5+9}{2} = 7
\][/tex]
- For the interval [tex]\(10-14\)[/tex]:
[tex]\[
\frac{10+14}{2} = 12
\][/tex]
- For the interval [tex]\(15-19\)[/tex]:
[tex]\[
\frac{15+19}{2} = 17
\][/tex]
- For the interval [tex]\(20-24\)[/tex]:
[tex]\[
\frac{20+24}{2} = 22
\][/tex]
- For the interval [tex]\(25-29\)[/tex]:
[tex]\[
\frac{25+29}{2} = 27
\][/tex]
### Step 2: List Mid-Points and Frequencies
| Interval | Mid-point ([tex]\(x_i\)[/tex]) | Frequency ([tex]\(f_i\)[/tex]) |
|----------|---------------------|---------------------|
| 0-4 | 2 | 1 |
| 5-9 | 7 | 5 |
| 10-14 | 12 | 7 |
| 15-19 | 17 | 13 |
| 20-24 | 22 | 11 |
| 25-29 | 27 | 9 |
### Step 3: Calculate [tex]\( f_i \cdot x_i \)[/tex]
Next, calculate the product of each frequency and its corresponding mid-point.
| Interval | Mid-point ([tex]\(x_i\)[/tex]) | Frequency ([tex]\(f_i\)[/tex]) | [tex]\(f_i \cdot x_i\)[/tex] |
|----------|---------------------|---------------------|---------------------|
| 0-4 | 2 | 1 | 2 |
| 5-9 | 7 | 5 | 35 |
| 10-14 | 12 | 7 | 84 |
| 15-19 | 17 | 13 | 221 |
| 20-24 | 22 | 11 | 242 |
| 25-29 | 27 | 9 | 243 |
### Step 4: Find the Sum of Frequencies and Sum of [tex]\( f_i \cdot x_i \)[/tex]
Sum of frequencies ([tex]\(\sum f_i\)[/tex]):
[tex]\[
1 + 5 + 7 + 13 + 11 + 9 = 46
\][/tex]
Sum of the products ([tex]\(\sum f_i \cdot x_i\)[/tex]):
[tex]\[
2 + 35 + 84 + 221 + 242 + 243 = 827
\][/tex]
### Step 5: Calculate the Mean
Now, divide the sum of the products by the sum of the frequencies to find the mean:
[tex]\[
\text{Mean} = \frac{\sum f_i \cdot x_i}{\sum f_i} = \frac{827}{46} \approx 17.978
\][/tex]
So, the mean score is approximately [tex]\( 17.978 \)[/tex].
