Question

Fill In the Blanks

If $\overline{\mathrm{X}}$ represents the mean of n observations x₁,x₂,...,x_n, then the value of $\sum _{i=1}^n\left(x_i-\overline{\mathrm{X}}\right)$ is _____________.

Answer 1

It is given that, the mean of n observations x₁, x₂,..., x_n is $\overline{X}$.

$\therefore \overline{X}=\frac{x_1+x_2+...+x_n}{n}$

$\Rightarrow x_1+x_2+...+x_n=n\overline{X}$ .....(1)

Now,

$\sum _{i=1}^n\left(x_i-\overline{X}\right)$

$=\left(x_1-\overline{X}\right)+\left(x_2-\overline{X}\right)+...+\left(x_n-\overline{X}\right)$

$=x_1+x_2+...+x_n-\left(\underset{n\mathrm{times}}{\underbrace{\overline{X}+\overline{X}+...+\overline{X}}}\right)$

$=n\overline{X}-n\overline{X}$ [From (1)]

$=0$

Thus, the value of $\sum _{i=1}^n\left(x_i-\overline{X}\right)$ is 0.

If $\overline{\mathrm{X}}$ represents the mean of n observations x₁,x₂,...,x_n, then the value of $\sum _{i=1}^n\left(x_i-\overline{\mathrm{X}}\right)$ is ____0____.