What are the default expressions of skewness and kurtosis used in Scipy?

Question

Scipy's (to date, version 0.19.1) Statistical Functions module (aka scipy.stats) contains the functions of scipy.stats.skew and scipy.stats.kurtosis to compute skewness and kurtosis of a data set (3rd and 4th statistical moments, respectively). Moreover, scipy.stats.describe calls these functions.

The definitions of skewness and kurtosis may vary; hence, no consensus on them in the literature. Then, which mathematical expressions are used in Scipy to define skewness and kurtosis in the two aforementioned functions with their default settings?

Answer 1

Both scipy.stats.skew and scipy.stats.kurtosis call the function of scipy.stats.moment, which computes the following for the k-th central moment of a data sample:

<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.0/MathJax.js?config=TeX-AMS_HTML-full"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({"HTML-CSS": { preferredFont: "TeX", availableFonts:["STIX","TeX"], linebreaks: { automatic:true }, EqnChunk:(MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { inlineMath: [ ["$", "$"], ["\\\\(","\\\\)"] ], displayMath: [ ["$$","$$"], ["\\[", "\\]"] ], processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { noUndefined: { attributes: { mathcolor: "red", mathbackground: "#FFEEEE", mathsize: "90%" } }, Macros: { href: "{}" } }, messageStyle: "none" });   </script>

$$m_k = \frac{1}{n} \sum_{i = 1}^n (x_i - \bar{x})^k$$

Accordingly, scipy.stats.skew with default settings (e.g. bias=True) computes:

<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.0/MathJax.js?config=TeX-AMS_HTML-full"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({"HTML-CSS": { preferredFont: "TeX", availableFonts:["STIX","TeX"], linebreaks: { automatic:true }, EqnChunk:(MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { inlineMath: [ ["$", "$"], ["\\\\(","\\\\)"] ], displayMath: [ ["$$","$$"], ["\\[", "\\]"] ], processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { noUndefined: { attributes: { mathcolor: "red", mathbackground: "#FFEEEE", mathsize: "90%" } }, Macros: { href: "{}" } }, messageStyle: "none" });   </script>

$$ S = \frac{m_3}{(m_2)^{1.5}} $$

scipy.stats.kurtosis with default settings:

<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.0/MathJax.js?config=TeX-AMS_HTML-full"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({"HTML-CSS": { preferredFont: "TeX", availableFonts:["STIX","TeX"], linebreaks: { automatic:true }, EqnChunk:(MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { inlineMath: [ ["$", "$"], ["\\\\(","\\\\)"] ], displayMath: [ ["$$","$$"], ["\\[", "\\]"] ], processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { noUndefined: { attributes: { mathcolor: "red", mathbackground: "#FFEEEE", mathsize: "90%" } }, Macros: { href: "{}" } }, messageStyle: "none" });   </script>

$$ K = \frac{m_4}{(m_2)^{2}} $$