Question
Answers

For any collection of n items, $ \sum {\left( {x - \overline x } \right)}$
A.$\sum x$
B.$\overline x$
C.$\text{n}\overline x$
D.$0$

Answer
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Hint:If the values of variables $x$ are ${x_1},{x_2},{x_3},.....,{x_n}$, where $'n'$ is the total number of values, then
Arithmetic mean $\left( {\overline x } \right)$
$\begin{gathered}
   = \dfrac{{{x_1} + {x_2} + {x_3} + ..... + {x_n}}}{n} = \dfrac{1}{n}\sum\limits_{i = 1}^{i = n} {{x_i}} \\
    \\
\end{gathered} $
The symbol $\sum\limits_{i = 1}^{i = n} {{x_i}} $, denotes the sum ${x_1} + {x_2} + {x_3} + ..... + {x_n}.$
The arithmetic mean of a set of observations is equal to their sum divided by the total number of observations.

Complete step-by-step answer:
Let the total number of observations are ‘n’ and given, the mean of observation be $'x'$.
Then,
$\begin{gathered}
  \dfrac{{{x_1} + {x_2} + {x_3} + ..... + {x_n}}}{n} = x \\
   \Rightarrow {x_1} + {x_2} + {x_3} + ..... + {x_n} = nx........(i) \\
\end{gathered} $
Then,

$\begin{gathered}
  \sum {\left( {x - \overline x } \right)} = \left[ {\left( {{x_1} - x} \right) + \left( {{x_2} - x} \right) + \left( {{x_3} - x} \right) + ...... + \left( {{x_n} - x} \right)} \right] \\
  {\text{ = }}\left[ {\left( {{x_1} + {x_1} + {x_1} + ...... + {x_1}} \right) - \left( {x + x + x + .....n{\text{ times}}} \right)} \right] \\
\end{gathered} $
Since, from $\left( i \right)$ above, we have
${x_1} + {x_2} + {x_3} + ..... + {x_n} = nx$ and $x + x + x + ....... + x = nx$
Therefore,
$\sum {\left( {x - \overline x } \right)} = nx - nx = 0$

So, the correct answer is “Option D”.

Note:The arithmetic mean of a set of observations is equal to their sum divided by the total number of observations.
$\begin{gathered}
  \dfrac{{{x_1} + {x_2} + {x_3} + ..... + {x_n}}}{n} = x \\
   \Rightarrow {x_1} + {x_2} + {x_3} + ..... + {x_n} = nx \\
\end{gathered} $
And also,
$x + x + x + ....... + x = nx$.