Question

The sum of the deviations of the variates from the arithmetic mean is always.

• A. $+1$
• B. $0$
• C. $-1$
• D. Real number

Answer 1

Answer: (B)

$0$

The sum of the deviations of a given set of observations from their arithmetic mean is always zero. It is due to the property that the arithmetic mean is characterised as the centre of gravity. i.e. sum of positive deviation from the mean is equal to the sum of negative deviations.

For example:

$3,4,6,8,14$

$\overline{x}=\frac{3+4+6+8+14}{5}=7$

$x_i$ $x_i-\overline{x}$

$3$ $-4$

$4$ $-3$

$6$ $-1$

$8$ $1$

$14$ $7$

$\sum (x_i-\overline{x})=-8+8=0$

Hence, the sum of the deviations about mean is $0$.