Question

Mean of 5 quantities is $\overline{x}$. If each quantity is decreased by 5 , then resulted mean is $\overline{X}$.
Realation between $\overline{x}$ and $\overline{X}$

Answer 1

Let the initial quantities be $x_1,x_2,x_3,x_4,x_5$
We have $\overline{x}=\frac{x_1+x_2+x_3+x_4+x_5}{5}\cdots \left(i\right)$
and
$\overline{X}=\frac{(x_1-5)+(x_2-5)+(x_3-5)+(x_4-5)+(x_5-5)}{5}=\frac{x_1+x_2+x_3+x_4+x_5-25}{5}=\frac{x_1+x_2+x_3+x_4+x_5}{5}-\frac{25}{5}=\frac{x_1+x_2+x_3+x_4+x_5}{5}-5=\overline{x}-5\Rightarrow \overline{X}=\overline{x}-5\left[\text{from (i)}\right]\Rightarrow \overline{X}<\overline{x}$
From converse property
$\overline{X}<\overline{x}\Rightarrow \overline{x}>\overline{X}$