If $\overline{X}$ is the mean of the ten natural numbers $x_1, x_2, x_3, …, x_{10}$, show that $(x_1 - \overline{X}) + (x_2 - \overline{X}) + … + (x_{10} - \overline{X}) = 0$.
Given:
$\overline{X}$ is the mean of the ten natural numbers $x_1, x_2, x_3, …, x_{10}$.
To do:
We have to show that $(x_1 - \overline{X}) + (x_2 - \overline{X}) + … + (x_{10} - \overline{X}) = 0$.
Solution:
We know that,
Mean $\overline{X}=\frac{Sum\ of\ the\ observations}{Number\ of\ observations}$
Therefore,
Mean $\overline{\mathrm{X}}=\frac{x_{1}+x_{2}+x_{3}+x_{4}+\ldots+x_{10}}{10}$
$x_{1}+x_{2}+x_{3}+\ldots+x_{10}=10 \overline{\mathrm{X}}$.........(i)
$(x_{1}-\overline{\mathrm{X}})+(x_{2}-\overline{\mathrm{X}})+\ldots+(x_{10}-\overline{\mathrm{X}})=0$
LHS $=x_{1}-\overline{\mathrm{X}}+x_{2}-\overline{\mathrm{X}}+x_{3}-\overline{\mathrm{X}}+\ldots+x_{10}-\overline{\mathrm{X}}$
$=x_{1}+x_{2}+x_{3}+\ldots+x_{10}-10 \overline{\mathrm{X}}$
$=10 \overline{\mathrm{X}}-10 \overline{\mathrm{X}}$
$=0$
$=$ RHS
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