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Why is the skewness of the samples $[ 1,\,2,\,3,\,4,\, 5,\, 6,\, 7,\, 8 ,\, 9 ]$ zero?

Of course, it follows from definition, $\displaystyle E \left[\left(\frac{x_i - \text{mean}}{\text{deviation}}\right)^3\right]$.

But on the other hand a skewness of zero means a symmetrical distribution which is, obviously, not the case for the linear increasing values of the samples presented above?!

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The samples do match a symmetrical distribution - it's just not symmetrical with respect to $0$.

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I think measure of skewness is $$s_k={ \bar x-\text{Mo} \over s}={3( \bar x-\text{Me}) \over s}$$ where $\bar x=$ mean, $\text{Mo}=$ mode, $\text{Me}=$ median and $s=$ Standard deviation. With the given data we can conclude $s_k=0$, since mean=median.

Note that this distribution is not symmetric about $0$ but symmetric about $5$ which is obvious because the median of the given sample is $5$ not $0$.

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