Question

The sums of the deviations of a set of n values $x_1,x_2,....,x_n$measured from 15 and -3 are -90 and 54 respectively.Find the value of n and mean.

Answer 1

In first case, $(x_1-15)+(x_2-15)+(x_3-15)+...+(x_n-15)=-90\Rightarrow x_1+x_2+x_3+...+x_n-15\times n=-90\Rightarrow n\overline{x}-15n=-90...\left(i\right)$
Similarly, $n\overline{x}+3n=54...\left(ii\right)$

Subtracting, $-18n=-144\Rightarrow n=\frac{-144}{-18}=8$

Now, $n\overline{x}-15n=-90\Rightarrow 8\overline{x}-15\times 8=-90\Rightarrow 8\overline{x}=-90+120\Rightarrow 8\overline{x}=30\Rightarrow \overline{x}=\frac{30}{8}=\frac{15}{4}\Rightarrow \overline{x}=\frac{15}{4}Hence,n=8and\overline{x}=\frac{15}{4}$