Question

The algebraic sum of the deviations from arithmetic mean is always:
A. zero
B. One
C. equal
D. unequal

Answer 1

Hint: This question is a general statement. So, here we have taken an example to show the value of the algebraic sum of the deviations from arithmetic mean for your better understanding.

Complete step-by-step answer:

The sum of algebraic sum of the deviations from the arithmetic mean is always zero.

Now we will see this by considering an example.

For example: 2,5,8,11,14 be the observations.

The arithmetic mean of the observations is \[\overline x = \dfrac{{2 + 5 + 8 + 11 + 14}}{5} = \dfrac{{40}}{5} = 8\]

The algebraic sum of the deviations from the arithmetic mean is \[\sum {\left( {{x_i} - \overline x } \right)} \]

\[

   \Rightarrow \sum {\left( {{x_i} - x} \right)} = \left( {2 - 8} \right) + \left( {5 - 8} \right) + \left(

{8 - 8} \right) + \left( {11 - 8} \right) + \left( {14 - 8} \right) \\

   \Rightarrow \sum {\left( {{x_i} - \overline x } \right) = - 6 - 3 + 0 + 3 + 6 = 0} \\

\]

Hence the algebraic sum of the deviations from arithmetic mean of the observations is equal to zero.

Therefore, the algebraic sum of the deviations from the arithmetic mean is always zero.

Thus, the correct option is A. Zero

Note: For the observations \[{x_1},{x_2},{x_3},.....................,{x_n}\] the arithmetic mean is given by \[\overline x = \dfrac{{{x_1} + {x_2} + {x_3} + ..................... + {x_n}}}{n}\]. The algebraic sum of the deviations from the arithmetic mean is given by \[\sum {\left( {{x_i} - \overline x } \right)} \].