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Traditionally (see Johnson et al., 2005, for instance), the (population) standardised skewness and kurtosis can be denoted as

$\sqrt{\beta_1}= \frac{\mu_3}{\sigma^3}\;\;$ and $\;\;\beta_2= \frac{\mu_4}{\sigma^4}$,

respectively, where $\mu_r$ stands for the $r$-th central moment and $\sigma$ denotes the standard deviation of the random variable being considered.

My question is two-fold:

(1) I am curious. Where do those $\beta_1$ and $\beta_2$ actually come from? How/why do they appear? It is just a notation question. How are they originated?

(2) I am confused. Does $\sqrt{\beta_1}$ actually mean $\pm\sqrt{\beta_1}$? I mean, the standadised skewness index $\mu_3/\sigma^3$ can be negative, if I am not wrong. So, why the square root on $\beta_1$? What kind of notation is that? So, this is also a notation question, but in this case I am not sure whether I have correctly understood the meaning of $\sqrt{\beta_1}$.


REFERENCES

Johnson, N. L., Kemp, A. W., & Kotz, S. (2005). Univariate Discrete Distributions (3rd ed.). https://doi.org/10.1002/0471715816

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    $\begingroup$ Both these questions are discussed to some small extent as asides in other answers and comments but it may be worth having a question to address them directly $\endgroup$ – Glen_b Oct 23 '19 at 3:38
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This goes back to Pearson (1895) [1]. He used $\beta_1$ to refer to the square of the moment-based skewness coefficient, and $\beta_2$ to refer to the corresponding kurtosis. (I don't think they appear in his 1893 paper.)

Specifically on p351 he defines

$\beta_1 = \frac{\mu_3^2}{\mu_2^3}$ and $\beta_2=\frac{\mu_4}{\mu_2^2}$.

Why exactly he chose to do it that way, I am unsure, but it did simplify a couple of things he was talking about (slightly -- both at that point and elsewhere) and he may simply have not thought beyond that immediate need when he wrote it.

Most of the real data he dealt with was mildly right skew; it may not have been seen as especially important to worry about the negative skewness case when he was initially writing about it.

This definition in the 1895 paper led to $\sqrt{\beta_1}$ being used to refer to the skewness coefficient $\frac{\mu_3}{\mu_2^{3/2}}$ (i.e. to be whatever the original signed quantity was, not the principal root of $\beta_1$). This is confusing, sometimes even to people very well aware of the convention. It leads to a consequent error in a paper by Doane on histogram bins for example, where the meaning of $\sqrt{\beta_1}$ changes in the paper from the original signed skewness to the principal root of $\beta_1$ without warning.

Its fairly common these days to use $\gamma_1$ for the skewness, and then to write $\beta_1=\gamma_1^2$ if need be, which avoids the issue.

[1]: Pearson, Karl (1895). "Contributions to the mathematical theory of evolution, II: Skew variation in homogeneous material".
Philosophical Transactions of the Royal Society. 186: 343–414. pdf

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