Question

If $x_1,x_2,....,x_n$ are values in a variable $x$ such that ${\sum }_{i=1}^n(x_i-2)=110$ and ${\sum }_{i=1}^n(x_i-5)=20.$
Find the value of $n$ and its mean.

Answer 1

$\sum x_i=x_1+x_2+...+x_n$

So, $\sum (x_i-2)=(x_1-2)+(x_2-2)+...+(x_n-2)$

Also, $\sum (x_i-2)=110$

$\Rightarrow$ $(x_1-2)+(x_2-2)+...+(x_n-2)=110$

$\Rightarrow$ $(x_1+x_2+..+x_n)-2n=110$

$\Rightarrow$ $\sum x_i-2n=110$ ---- ( 1 )

Similarly,

$\sum (x_i-5)=(x_1-5)+(x_2-5)+...+(x_n-5)$

Also, $\sum (x_i-5)=20$

$\Rightarrow$ $(x_1-5)+(x_2-5)+...+(x_n-5)=20$

$\Rightarrow$ $(x_1+x_2+...+x_n)-5n=20$

$\Rightarrow$ $\sum x_i-5n=20$ ---- ( 2 )

Subtracting ( 1 ) from ( 2 ) we get,

$\Rightarrow$ $\sum x_i-5n-(\sum x_i-2n)=20-110$

$\Rightarrow$ $-3n=-90$

$\therefore$ $n=30$

$\Rightarrow$ $\sum x_i-5\left(30\right)=20$

$\Rightarrow$ $\sum x_i=20+150$

$\therefore$ $\sum x_i=170$

$\Rightarrow$ $Mean=\frac{\sum x_i}{n}=\frac{170}{30}=5.67$