## Elementary Technical Mathematics

The standard deviation is $s=5.5$.
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 $s=5.5$.