Use the following grouped data set (and your calculator if you want to and know how) to find the quantity requested below.
\begin{tabular}{|c|c|}
\hline Classes & Frequency \\
\hline $0-< 10$ & 19 \\
$10-< 20$ & 23 \\
$20-< 30$ & 16 \\
$30-< 40$ & 12 \\
$40-< 50$ & 14 \\
$50-< 60$ & 24 \\
\hline & 108 \\
\hline
\end{tabular}
Calculate the sample standard deviation for the data set. Follow the rounding rule from the formula sheet.
Note: $\sum x^{2}=131,300$

Step 1 ：Given the grouped data set, we are asked to calculate the sample standard deviation. The formula for the sample standard deviation is \(s = \sqrt{\frac{\sum x^{2} - \frac{(\sum x)^{2}}{n}}{n-1}}\), where \(\sum x^{2}\) is the sum of the squares of the data values, \(\sum x\) is the sum of the data values, and \(n\) is the number of data values.

Step 2 ：We are given \(\sum x^{2} = 131,300\), but we don't have \(\sum x\) and \(n\). We can calculate these from the frequency table. The midpoint of each class is the representative value for the data values in that class. We can multiply the frequency of each class by its midpoint to get the total for that class, and then sum these totals to get \(\sum x\). The total frequency is \(n\).

Step 3 ：From the frequency table, we have the classes and their frequencies as follows: classes = [5, 15, 25, 35, 45, 55] and frequencies = [19, 23, 16, 12, 14, 24].

Step 4 ：By multiplying the frequency of each class by its midpoint, we get \(\sum x = 3210\). The total frequency is \(n = 108\).

Step 5 ：Substituting these values into the formula, we get \(s = \sqrt{\frac{131,300 - \frac{(3210)^{2}}{108}}{108-1}}\).

Step 6 ：Solving this, we get the sample standard deviation \(s\) to be approximately 18.31.

Step 7 ：Final Answer: The sample standard deviation for the data set is \(\boxed{18.31}\).