Meaning of asymmetric third moment skewness Given the formula of the third moment that define the skewness:
$$skewness=E\Bigg[\bigg(\frac{x_i−\bar{X}}{σ}\bigg)^3 \Bigg] = \frac{\mu_3}{\sigma^3}
$$
I understand that this formula calculates the ratio of the spread of the data relatively to the mean and the standard deviation. and raising to the cube preserves the signs of the quantities.

so let's say I have a distribution and the result of this formula = 1.5, which means that it is positively skewed and the tail to the right is longer than the left. Is it correct to say that the tail to the right is 0.5 times longer than the left? if we want to describe the spread of 1 standard deviation for this distribution.

A precise interpretation of the skewness value 1.5 is given as follows:

*

*Construct the $z$-scores, $z_i =(x_i-\bar x)/s$.

*Calculate the values $s_i =z_i^3$.

*Draw the dot plot (like a histogram but instead dots) of the $s_i$.

*Since skewness is the mean of the $s_i$ values, and since a distribution's point of balance is the mean, the dot plot of the $s_i$ numbers balances at 1.5.

The above interpretation refers to sample skewness, but a nearly identical procedure gives you the precise interpretation of "population" skewness: drop the subscript, replace the mean and variance with their "population" counterparts, and draw the probability distribution graph of the resulting $s_i$. The point of balance of this graph is the "population" skewness.
