If the mean of n observations x 1, x 2, ldots ldots x n is overline x , then the sum of deviations of observations from mean is :-

Question

If the mean of n observations x 1, x 2, ldots ldots x n is overline x , then the sum of deviations of observations from mean is :- (A) 0 (B) n barx (C) ( barx/n) (D) None of these

Tardigrade Admin · Accepted Answer

∵ ( x 1+ x 2+ ldots+ x n / n )= overline x ......(i) ∴ Req. sum =( x 1- overline x )+( x 2- overline x )+ ldots .+( x n - overline x ) =( x 1+ x 2+ ldots+ x n )- n overline x = n overline x - n overline x =0 (from eq. (i))

Q. If the mean of $n$ observations $x_{1}, x_{2}, \dots\dots x_{n}$ is $\overline{x}$ , then the sum of deviations of observations from mean is :-

0

$n \overset{x}{ˉ}$

$\frac{x ˉ}{n}$

None of these

$∵ \frac{x _{1} + x _{2} + \dots + x _{n}}{n} = \overline{x} ......$ (i)
$∴$ Req. sum $= (x_{1} - \overline{x}) + (x_{2} - \overline{x}) + \dots . + (x_{n} - \overline{x})$
$= (x_{1} + x_{2} + \dots + x_{n}) - n \overline{x}$
$= n \overline{x} - n \overline{x} = 0$ (from eq. (i))

Q. If the mean of n observations x1​,x2​,……xn​ is x, then the sum of deviations of observations from mean is :-

0

nxˉ

nxˉ​

None of these

∵nx1​+x2​+…+xn​​=x......(i) ∴ Req. sum =(x1​−x)+(x2​−x)+….+(xn​−x) =(x1​+x2​+…+xn​)−nx =nx−nx=0 (from eq. (i))

Q. If the mean of $n$ observations $x_{1}, x_{2}, \dots\dots x_{n}$ is $\overline{x}$ , then the sum of deviations of observations from mean is :-

$n \overset{x}{ˉ}$

$\frac{x ˉ}{n}$

$∵ \frac{x _{1} + x _{2} + \dots + x _{n}}{n} = \overline{x} ......$ (i)
$∴$ Req. sum $= (x_{1} - \overline{x}) + (x_{2} - \overline{x}) + \dots . + (x_{n} - \overline{x})$
$= (x_{1} + x_{2} + \dots + x_{n}) - n \overline{x}$
$= n \overline{x} - n \overline{x} = 0$ (from eq. (i))