Question

Find the variance and standard deviation for the following data:

$\begin{array}{cccccc}{x}_i& 10& 15& 18& 20& 25\\ f_i& 3& 2& 5& 8& 2\end{array}$

Answer 1

We may prepare the table given below:

$\therefore N={\sum }_{i=1}^5{f}_i=20,{\sum }_{i=1}^5{f}_i(x_i-\overline{x}{)}^2=340$

$\therefore$ variance, ${\sigma }^2=\frac{1}{N}.\sum f_i(x_i-\overline{x}{)}^2=(\frac{1}{20}\times 340)=17$

Standard deviation, $\sigma =\sqrt{17}=4.12$

Short Cut Method:

Let A be the assumed mean and let $d_i=(x_i-A)$

Then, ${\sigma }^2=\{\frac{\sum f_i{d}_i^2}{N}-{\left(\frac{\sum f_i{d}_i}{N}\right)}^2\}$