Graphic representation of kurtosis and skewness I'm curious to know if there is any graphic representation of the kurtosis and skewness. I am currently studying probability and statistics, I recognize that the average is at the center of the distribution, while the standard deviation represents a unit away from the center, on the other hand, the skewness allows me to know the symmetry and kurtosis the shape. How can I represent those numbers in the graph? That is, if I get from kurtosis 0.55, where should I measure that? On the other hand, if I get skewness 2.1, how do I place it? Where are these numbers in the bell drawing? If I want to draw it with pencil and paper, How can I do it?

 A: You can easily and precisely display both skewness and kurtosis on a histogram or density plot of certain transformations. 


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*First, transform the random variable or the data $X$ to $Z$-scores via $ Z = (X - \mu)/\sigma$. 

*For skewness, plot the distribution (discrete or continuous; note that the discrete case covers all empirical data) of $U = Z^3$. Since Skewness$(X)=E(U)$, the point on the horizontal axis where the distribution balances is equal to Skewness$(X)$.

*For kurtosis, plot the distribution (discrete or continuous; again note that the discrete case covers all empirical data) of $V = Z^4$. Since Kurtosis$(X)=E(V)$, the point on the horizontal axis where the distribution balances is equal to Kurtosis$(X)$.
Note that this representation also provides an easy way to visualize what higher and lower values of skewness and/or kurtosis tell you about a distribution (either empirical or theoretical). For example, suppose $X_1$ has kurtosis 4.3 and $X_2$ has kurtosis 5.8. What does this tell you about the difference between the distributions of $X_1$ and $X_2$? To answer, graph the distribution of $V_2 = \{(X_2 - \mu_2)/\sigma_2\}^4$, and place a fulcrum at 4.3 on the horizontal axis. The distribution "falls to the right" of the fulcrum, since 5.8 > 4.3.  Thus, either or both tails of the distribution of $X_2$ are heavier than those of $X_1$. 
Also, the difference in kurtosis values cannot be attributed to either greater mass near the center or to "peakedness" of the distribution whose kurtosis is 5.8. If either greater central mass or peakedness were the reason for the greater kurtosis, then the curve would "fall to the left" when balanced at 4.3, since both cases refer to increases of probability or leverage of data values near 0.
