Question

Question 13
If $\overline{x}$ represents the mean of n observations $x_1,x_2,\dots \dots x_n$, then value of ${\sum }_{i-1}^n(x_i-\overline{x})$ is:

A) -1
B) 0
C) 1
D) n - 1

Answer 1

The answer is B.
Given, $\overline{x}=\frac{{\sum }_{i=1}^n{x}_i}{n}\Rightarrow {\sum }_{i=1}^n{x}_i=n\overline{x}\dots \dots \left(i\right)$
Now, ${\sum }_{i=1}^n(x-\overline{x})={\sum }_{i-1}{x}_i-{\sum }_{i-1}\overline{x}$
$=n\overline{x}-\overline{x}{\sum }_{i-1}^{n}1$ [from (i)]
$=n\overline{x}-\overline{x}n$
$=0$