Question

Find the standard deviation for the following data.

$\begin{array}{cccccc}{x}_i& 3& 8& 13& 18& 23\\ f_i& 7& 10& 15& 10& 6\end{array}$

Answer 1

Let us form the following table

$\begin{array}{ccccc}{x}_1& f_i& x_i{f}_i& x_i{}^2& f_i{x}_i{}^2\\ 3& 7& 21& 9& 63\\ 8& 10& 80& 64& 640\\ 13& 15& 195& 169& 2535\\ 18& 10& 180& 324& 3240\\ 23& 6& 138& 529& 3174\\ 48& 614& & & 9652\end{array}$

Now, by formula

$\sigma =\left(1/N\right)\sqrt{N\sum f_i{x}_i^2-(\sum f_i{x}_i{)}^2}$
$=\left(1/48\right)\sqrt{48\times 9652-(614{)}^2}$
$=\left(1/48\right)\sqrt{463296-376996}$
$=\frac{1}{48}\times 293.77$
$=6.12$

Therefore, standard deviation is 6.12