To determine the sample standard deviation for the given data set [tex]\(6, 5, 10, 11, 13\)[/tex], we follow these steps.
Given:
- Data points: [tex]\(6, 5, 10, 11, 13\)[/tex]
- Mean [tex]\(\bar{x} = 9\)[/tex]
First, we calculate the sum of squared differences from the mean using the formula [tex]\(\Sigma (x - \bar{x})^2\)[/tex].
Let's break it down using the table provided:
[tex]\[
\begin{array}{|c|c|c|}
\hline
x & x - \bar{x} & (x - \bar{x})^2 \\
\hline
6 & 6 - 9 = -3 & (-3)^2 = 9 \\
\hline
5 & 5 - 9 = -4 & (-4)^2 = 16 \\
\hline
10 & 10 - 9 = 1 & 1^2 = 1 \\
\hline
11 & 11 - 9 = 2 & 2^2 = 4 \\
\hline
13 & 13 - 9 = 4 & 4^2 = 16 \\
\hline
\end{array}
\][/tex]
Summing up the squares of the differences:
[tex]\[
\Sigma (x - \bar{x})^2 = 9 + 16 + 1 + 4 + 16 = 46
\][/tex]
Next, we use the formula for the sample standard deviation, [tex]\(s = \sqrt{\frac{1}{n-1} \Sigma (x - \bar{x})^2}\)[/tex], where [tex]\(n\)[/tex] is the number of data points.
Here [tex]\(n = 5\)[/tex], so [tex]\(n - 1 = 4\)[/tex].
Plugging in the values:
[tex]\[
s = \sqrt{\frac{46}{4}}
\][/tex]
Calculating the expression inside the square root:
[tex]\[
\frac{46}{4} = 11.5
\][/tex]
Now, taking the square root:
[tex]\[
s = \sqrt{11.5} \approx 3.391164991562634
\][/tex]
Finally, we round this result to the nearest tenth:
[tex]\[
s \approx 3.4
\][/tex]
Thus, the standard deviation for the given data set is [tex]\(\boxed{3.4}\)[/tex].
