Let's solve this step by step.
### Step 1: Calculate the Mean ([tex]\(\bar{x}\)[/tex])
The given [tex]\( x \)[/tex] values are:
[tex]\[ 6, 15, 13, 2, 11 \][/tex]
The mean ([tex]\(\bar{x}\)[/tex]) is calculated as follows:
[tex]\[
\bar{x} = \frac{\sum x_i}{n}
\][/tex]
Where [tex]\( n \)[/tex] is the number of values.
[tex]\[
\bar{x} = \frac{6 + 15 + 13 + 2 + 11}{5} = \frac{47}{5} = 9.4
\][/tex]
### Step 2: Calculate [tex]\((x - \bar{x})\)[/tex] for each [tex]\(x\)[/tex] value
[tex]\[
x - \bar{x} =
\begin{cases}
6 - 9.4 = -3.4 \\
15 - 9.4 = 5.6 \\
13 - 9.4 = 3.6 \\
2 - 9.4 = -7.4 \\
11 - 9.4 = 1.6 \\
\end{cases}
\][/tex]
### Step 3: Calculate [tex]\((x - \bar{x})^2\)[/tex] for each [tex]\(x\)[/tex] value
[tex]\[
(x - \bar{x})^2 =
\begin{cases}
(-3.4)^2 = 11.56 \\
5.6^2 = 31.36 \\
3.6^2 = 12.96 \\
(-7.4)^2 = 54.76 \\
1.6^2 = 2.56 \\
\end{cases}
\][/tex]
### Step 4: Calculate [tex]\(\sum (x - \bar{x})^2\)[/tex]
[tex]\[
\sum (x - \bar{x})^2 = 11.56 + 31.36 + 12.96 + 54.76 + 2.56 = 113.2
\][/tex]
### Step 5: Calculate the Sample Standard Deviation [tex]\(s\)[/tex]
The standard deviation for a sample is calculated as follows:
[tex]\[
s = \sqrt{\frac{\sum (x - \bar{x})^2}{n - 1}}
\][/tex]
Where [tex]\( n - 1 \)[/tex] is the degrees of freedom, in this case, [tex]\( 5 - 1 = 4 \)[/tex].
[tex]\[
s = \sqrt{\frac{113.2}{4}} = \sqrt{28.3} \approx 5.319774431308154
\][/tex]
### Step 6: Round to Two Decimal Places
The rounded standard deviation [tex]\( s \)[/tex] is:
[tex]\[
s \approx 5.32
\][/tex]
### Summary of Results
[tex]\[
\bar{x} = 9.4 \\
(x - \bar{x}) = [-3.4, 5.6, 3.6, -7.4, 1.6] \\
(x - \bar{x})^2 = [11.56, 31.36, 12.96, 54.76, 2.56] \\
\sum (x - \bar{x})^2 = 113.2 \\
s = 5.32
\][/tex]
### Final Answer:
The sample standard deviation is approximately 5.32.
