A class of 50 students takes a 5 -question quiz. The results are presented in the following frequency distribution:
\begin{tabular}{|c|c|}
\hline Number Correct, $x$ & Frequency. \\
\hline 0 & 8 \\
\hline 1 & 7 \\
\hline 2 & 7 \\
\hline 3 & 14 \\
\hline 4 & 8 \\
\hline 5 & 6 \\
\hline
\end{tabular}
1. Create a probability distribution.
2. Find the mean of the probability distribution.
3. Find the variance of the probability distribution.
4. Find the standard deviation of the probability distribution.

Step 1 ：First, we need to create a probability distribution. This is done by dividing the frequency of each score by the total number of students. This gives us the probability of each score.

Step 2 ：The probability distribution is as follows: \begin{tabular}{|c|c|} \hline Number Correct, $x$ & Probability \\ \hline 0 & 0.16 \\ \hline 1 & 0.14 \\ \hline 2 & 0.14 \\ \hline 3 & 0.28 \\ \hline 4 & 0.16 \\ \hline 5 & 0.12 \\ \hline \end{tabular}

Step 3 ：Next, we find the mean of the probability distribution. This is done by multiplying each score by its probability and summing these products. The mean of the probability distribution is \(\boxed{2.5}\).

Step 4 ：Then, we find the variance of the probability distribution. This is done by subtracting the mean from each score, squaring the result, multiplying by the probability of each score, and summing these products. The variance of the probability distribution is \(\boxed{2.53}\).

Step 5 ：Finally, we find the standard deviation of the probability distribution. This is done by taking the square root of the variance. The standard deviation of the probability distribution is \(\boxed{1.59}\).