Answer
The standard deviation is $s=5.5$.
Work Step by Step
The mean of the $n$ numbers is,
$\text{mean}=\frac{\text{Sum of }n\text{ numbers}}{n}$
And,
$\begin{align}
& \text{mean}=\frac{\text{165}\text{.4}}{6} \\
& =27.56 \\
& =27.6
\end{align}$
Find the difference between each piece of data and the mean.
$\begin{align}
& 20.2-27.6=-7.4 \\
& 22.6-27.6=-5 \\
& 27.3-27.6=-0.3 \\
& 29.6-27.6=2 \\
\end{align}$
And,
$\begin{align}
& 30.6-27.6=3 \\
& 35.1-27.6=7.5 \\
\end{align}$
Square each difference and find the sum of the squares,
$\begin{align}
& {{\left( -7.4 \right)}^{2}}=54.76 \\
& {{\left( -5 \right)}^{2}}=25 \\
& {{\left( -0.3 \right)}^{2}}=0.09 \\
& {{\left( 2 \right)}^{2}}=4
\end{align}$
And,
$\begin{align}
& {{\left( 3 \right)}^{2}}=9 \\
& {{\left( 7.5 \right)}^{2}}=56.25
\end{align}$
Sum of the square of the differences is $149.1$
Therefore the standard deviation is,
$\begin{align}
& s=\sqrt{\frac{\text{Sum of (measurement - mean}{{\text{)}}^{2}}}{n-1}} \\
& s=\sqrt{\frac{149.1}{6-1}} \\
& s=\sqrt{\frac{149.1}{5}} \\
\end{align}$
Further simplify,
$\begin{align}
& s=\sqrt{29.82} \\
& s=5.46 \\
& s=5.5 \\
\end{align}$
Hence the standard deviation is $s=5.5$.
