Question

Calculate mean, variance and standard deviation of the following distribution :

$\begin{array}{ccccccc}\text{Class :}& 1-10& 10-20& 20-30& 30-40& 40-50& 50-60\\ \text{Frequency:}& 11& 29& 18& 4& 5& 3\end{array}$

Answer 1

Let A, the assumed mean, be 25.5 Here h = 10

$\begin{array}{cccccc}\text{Class}& x_i& y_i\frac{x_i-25.5}{10}& f_i& f_i{y}_i& f_i{y}_i^2\\ 1-10& 5.5& -2& 11& -22& 44\\ 10-20& 15.5& -1& 29& -29& 29\\ 20-30& 25.5& 0& 18& 0& 0\\ 30-40& 35.5& 0& 4& 4& 4\\ 40-50& 45.5& 2& 5& 10& 20\\ 50-60& 55.5& 3& 3& 9& 27\\ & & & 70& -28& 124\end{array}$

$x^{\prime }=\frac{\sum f_i{y}_i}{\sum f_i}=-0.4$

Mean = $\overline{x}=25.5+(-10)\left(0.4\right)=21.5$

Variance =$=\left({\sigma }^2\right){\left[\frac{h}{N}\sqrt{N\sum f_i{y}_i^2-(\sum f_i{y}_i{)}^2}\right]}^2$

= $\frac{10\times 10}{70\times 70}\left[70\right(124)-(-28{)}^2]$

$=\frac{70\left(124\right)}{7\times 7}-\frac{28\times 28}{7\times 7}=\frac{1240}{7}-16=161$

S.D. $\left(\sigma \right)=\sqrt{161}=12.7$