Find the mean for the data items in the given frequency distribution.

\begin{tabular}{|l|l|l|l|l|l|l|l|l|}
\hline
Score, [tex]$x$[/tex] & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
\hline
Frequency, [tex]$f$[/tex] & 1 & 2 & 3 & 5 & 6 & 4 & 6 & 3 \\
\hline
\end{tabular}

The mean is [tex]$\square$[/tex] (Round to 3 decimal places as needed.)



Answer :

To find the mean of the given frequency distribution, we can follow these steps:

1. List the Scores and Their Corresponding Frequencies:
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline \text{Score}, x & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline \text{Frequency}, f & 1 & 2 & 3 & 5 & 6 & 4 & 6 & 3 \\ \hline \end{array} \][/tex]

2. Find the Total Number of Data Items:
The total number of data items is calculated by summing the frequencies.
[tex]\[ \text{Total data items} = 1 + 2 + 3 + 5 + 6 + 4 + 6 + 3 = 30 \][/tex]

3. Calculate the Sum of (Score * Frequency):
We will multiply each score by its corresponding frequency and then sum these products.
[tex]\[ \begin{array}{|c|c|} \hline \text{Score}, x & \text{Frequency}, f & x \times f \\ \hline 1 & 1 & 1 \times 1 = 1 \\ \hline 2 & 2 & 2 \times 2 = 4 \\ \hline 3 & 3 & 3 \times 3 = 9 \\ \hline 4 & 5 & 4 \times 5 = 20 \\ \hline 5 & 6 & 5 \times 6 = 30 \\ \hline 6 & 4 & 6 \times 4 = 24 \\ \hline 7 & 6 & 7 \times 6 = 42 \\ \hline 8 & 3 & 8 \times 3 = 24 \\ \hline \end{array} \][/tex]
[tex]\[ \text{Sum of } (x \times f) = 1 + 4 + 9 + 20 + 30 + 24 + 42 + 24 = 154 \][/tex]

4. Calculate the Mean:
The mean is given by the formula:
[tex]\[ \text{Mean} = \frac{\sum (x \times f)}{\sum f} \][/tex]
Plugging in the values we have:
[tex]\[ \text{Mean} = \frac{154}{30} \approx 5.133 \][/tex]

Thus, the mean, rounded to three decimal places, is:
[tex]\[ \boxed{5.133} \][/tex]