Find the variance and standard deviation for the following data:
$\begin{array}{cccccccc}{x}_i& 4& 8& 11& 17& 20& 24& 32\\ f_i& 3& 5& 9& 5& 4& 3& 1\end{array}$

Step 1: Finding mean
$\begin{array}{ccc}{x}_i& f_i& x_i{f}_i\\ 4& 3& 12\\ 8& 5& 40\\ 11& 9& 99\\ 17& 5& 85\\ 20& 4& 80\\ 24& 3& 72\\ 32& 1& 32\\ & \sum f_i=30& \sum x_i{f}_i=420\end{array}$
Mean $\overline{x}=\frac{\sum x_i{f}_i}{\sum f_i}$
$\Rightarrow \overline{x}=\frac{420}{30}$
$\Rightarrow \overline{x}=14$
Step 2: Finding variance and standard variance
$\begin{array}{ccccc}{x}_i& f_i& x_i-\overline{x}& (x_i-\overline{x}{)}^2& f_i(x_i-\overline{x}{)}^2\\ 4& 3& 4-14=-10& (-10{)}^2=100& 3\times 100=300\\ 8& 5& 8-14=-6& (-6{)}^2=36& 5\times 36=180\\ 11& 9& 11-14=-3& (-3{)}^2=9& 9\times 9=81\\ 17& 5& 17-14=3& (3{)}^2=9& 5\times 9=45\\ 20& 4& 20-14=6& (6{)}^2=36& 4\times 36=144\\ 24& 3& 24-14=10& (10{)}^2=100& 3\times 100=300\\ 32& 1& 32-14=18& (18{)}^2=324& 1\times 324=324\\ & \sum f_i=30& & \sum f_i(x_i-\overline{x}{)}^2=1374& \end{array}$
variance $\left({\sigma }^2\right)=\frac{\sum f_i(x_i-x{)}^2}{\sum f_i}$
$=\frac{1374}{30}=45.8$
Standard deviation $\left(\sigma \right)=\sqrt{45.8}=6.77$

Hence, the variance is 45.8 and the standard variance is $6.77$