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I'm studying the skewness of data distribution then the formula for measuring skewness is below:

$$\tilde\mu_3 = \frac{\sum_i^N(x_i-\bar X)^3}{(N-1)(\sigma^3)}$$

I would like to understand if it is possible to interpret the result of this formula geometrically as we usually do with the standard deviation or any kind of statistics, for example how can we quantify the asymmetric spread of the data on the left and the right side of the mean in order to roughly discover the shape of the distribution, something like that.
is this statistic can answer that question or do you have any suggestions?
thank you

tmo
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  • The mean and variance are easy to interpret geometrically because the geometries have a low dimensional shape. The skewness can be interpreted in a similar way, but it requires you to imagine a higher dimensional shape. – Sextus Empiricus Feb 19 '22 at 22:21
  • For instance here is an interpretation of variance as rotating the quantile function around the axis. https://stats.stackexchange.com/a/365385/164061 For skewness it works the same but you would have to do the rotation creating a 4d object. – Sextus Empiricus Feb 19 '22 at 22:23
  • Thanks, can you explain why it is 4d object? Or can you give a hint so I can look up on it? And does it tell us such as spread on the left is 2 times larger than the right? Thanks. – tmo Feb 19 '22 at 23:38
  • *"Say we wish to compute the first moment then $T(x) = x$. In the image below this corresponds to the area between F and the vertical line at $x=0$ (where the area on the left side may count as negative when $T(x)<0$). The second moment would correspond to the volume that the same area sweeps when it is rotated along the line at $x=0$ (with a factor $\pi$ difference)."* ... The rotation creates a sort of cone/vase shaped array of stacked disks (2d -balls) whose 3d volume is the variance. We can can continue this geometrical viewpoint to the third moment by considering stacked 3d-balls. – Sextus Empiricus Feb 20 '22 at 00:24

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