To find the mean of the given frequency distribution, follow these steps:
1. List the Scores and Frequencies:
- Scores ([tex]\(x\)[/tex]): 1, 2, 3, 4, 5, 6, 7, 8
- Frequencies ([tex]\(f\)[/tex]): 2, 6, 5, 5, 3, 3, 4, 3
2. Calculate the total number of observations:
The total number of observations is the sum of the frequencies.
[tex]\[
\text{Total number of observations} = \sum f = 2 + 6 + 5 + 5 + 3 + 3 + 4 + 3 = 31
\][/tex]
3. Calculate the weighted sum of the scores:
The weighted sum of the scores is the sum of the product of each score and its frequency.
[tex]\[
\text{Weighted sum of scores} = \sum (x \times f) = (1 \times 2) + (2 \times 6) + (3 \times 5) + (4 \times 5) + (5 \times 3) + (6 \times 3) + (7 \times 4) + (8 \times 3)
\][/tex]
Calculate each term:
[tex]\[
\begin{align*}
1 \times 2 &= 2 \\
2 \times 6 &= 12 \\
3 \times 5 &= 15 \\
4 \times 5 &= 20 \\
5 \times 3 &= 15 \\
6 \times 3 &= 18 \\
7 \times 4 &= 28 \\
8 \times 3 &= 24 \\
\end{align*}
\][/tex]
Now sum these products:
[tex]\[
\text{Weighted sum of scores} = 2 + 12 + 15 + 20 + 15 + 18 + 28 + 24 = 134
\][/tex]
4. Calculate the mean:
The mean ([tex]\(\bar{x}\)[/tex]) is given by the formula:
[tex]\[
\bar{x} = \frac{\sum (x \times f)}{\sum f}
\][/tex]
Substitute the calculated values into the formula:
[tex]\[
\bar{x} = \frac{134}{31} \approx 4.3226
\][/tex]
Thus, the mean of the given frequency distribution is approximately [tex]\(4.3226\)[/tex].
