Question
Taking some hypothetical data, show that sum of deviations taken from mean is zero.

Answer

The sum of the deviations, of all the values of x, from their arithmetic mean, is zero.Justification:
$\sum\text{f}_\text{i}(\text{x}_\text{i}-\overline{\text{x}})=\sum\text{f}_\text{i}\text{x}_\text{i}-\overline{\text{x}}\sum\text{f}_\text{i}=0$
$\overline{\text{x}}=\frac{\sum{\text{f}_\text{i}}\text{x}_\text{i}}{\sum\text{f}_\text{i}}$
$\therefore\ \sum\text{f}_\text{i}\text{x}_\text{i}=\overline{\text{x}}\sum\text{f}_\text{i}$
Since $\overline{\text{x}}$ is a constant.

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