Question

If x₁, x₂,........x₁₈ are 18 observations such that $\sum _{i=1}^{18}(x_i-8)=9\mathrm{and}\sum _{i=1}^{18}{(x_i-8)}^2=45,$ then the standard deviation of these observations is _________________.

Answer 1

Given for 18 observations,
Given $\sum _{i=1}^{18}(x_i-8)=9,\sum _{i=1}^{18}{(x_i-8)}^2=45$
$\begin{array}{rcl}\mathrm{Since}\mathrm{variance}\mathrm{of}X& =& \frac{1}{n}\left(\sum _{i=1}^{18}{\left(x_i-8\right)}^2\right)-{\left(\sum _{i=1}^{18}\frac{\left(x_i-8\right)}{n}\right)}^2\\ & =& \frac{1}{18}\left(45\right)-1{\left(\frac{9}{18}\right)}^2\\ & =& \frac{5}{2}-{\left(\frac{1}{2}\right)}^2=\frac{5}{2}-\frac{1}{4}=\frac{10-1}{4}\\ & =& \frac{9}{4}\end{array}$
∴ Standard deviation = $\sqrt{\frac{9}{4}}=\frac{3}{2}=1.5$