Question

If the sum of the deviations of a set of values $x_1,x_2,x_3,\dots ,x_n$ measured from 50 is -10 and the sum of deviations of the values from 46 is 70, then find the mean.

Answer 1

Given:
$x_1-50+x_2-50+\dots +x_n-50=-10$
$\Rightarrow (x_1+x_2+\dots +x_n)-50n=-10$
$\Rightarrow (x_1+x_2+\dots +x_n)=-10+50n---\left(i\right)$
Also given that,
$x_1-46+x_2-46+\dots +x_n-46=70$
$\Rightarrow (x_1+x_2+\dots +x_n)-46n=70$
$\Rightarrow x_1+x_2+\dots +x_n=70+46n---\left(ii\right)$
From Eqs. $\left(i\right)$ and $\left(ii\right)$
$-10+50n=70+46n\Rightarrow 4n=80\Rightarrow n=20$
From Eq. $\left(i\right)$
$x_1+x_2+\dots +x_n=990\Rightarrow Mean=\frac{x_1+x_2+\dots +x_n}{n}=\frac{990}{20}=49.5$