Question

If $\overline{x}$ is the mean of ten natural numbers $x_1,x_2,x_3,...,x_{10}$, show that:

$\left(x_1-\overline{x}\right)+\left(x_2-\overline{x}\right)+...+\left(x_{10}-\overline{x}\right)=0$

Answer 1

Step 1: Find the value of the sum of ten natural numbers

Given,

$\overline{x}$ is the mean of ten natural numbers $x_1,x_2,x_3,...,x_{10}$

Here, the number of observations $\left(n\right)$ $=10$

we know that, $\mathrm{Mean}=\frac{\mathrm{Sum}\mathrm{of}\mathrm{terms}}{\mathrm{Number}\mathrm{of}\mathrm{terms}}$

$\therefore \overline{x}=\frac{x_1+x_2+x_3+...+x_{10}}{10}$

$\Rightarrow x_1+x_2+x_3+...+x_{10}=10\overline{x}$

Step 2: Show that $\left(x_1-\overline{x}\right)+\left(x_2-\overline{x}\right)+...+\left(x_{10}-\overline{x}\right)=0$

Consider, $LHS$

$\left(x_1-\overline{x}\right)+\left(x_2-\overline{x}\right)+...+\left(x_{10}-\overline{x}\right)$

$=x_1+x_2+x_3+...+x_{10}-\overline{x}-\overline{x}-\overline{x}-\overline{x}-\overline{x}-\overline{x}-\overline{x}-\overline{x}-\overline{x}-\overline{x}$

$=10\overline{x}-10\overline{x}$

$=0$

$=RHS$

$\therefore \left(x_1-\overline{x}\right)+\left(x_2-\overline{x}\right)+...+\left(x_{10}-\overline{x}\right)=0$

Hence, showed.