Scipy 中使用的偏度和峰度的默认表达式是什么？
What are the default expressions of skewness and kurtosis used in Scipy?
Scipy（迄今为止，版本 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 调用这些函数。
偏度和峰度的定义可能不同；因此，文献中没有就它们达成共识。那么，Scipy中使用了哪些数学表达式来定义上述两个函数的偏度和峰度及其默认设置？
Both scipy.stats.skew and scipy.stats.kurtosis call the function of scipy.stats.moment，计算数据样本的第 k 个中心矩：
<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>
$$m_k = \frac{1}{n} \sum_{i = 1}^n (x_i - \bar{x})^k$$
因此，scipy.stats.skew 使用默认设置（例如 bias=True）计算：
<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 默认设置：
<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>
$$ K = \frac{m_4}{(m_2)^{2}} $$
Scipy（迄今为止，版本 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 调用这些函数。
偏度和峰度的定义可能不同；因此，文献中没有就它们达成共识。那么，Scipy中使用了哪些数学表达式来定义上述两个函数的偏度和峰度及其默认设置？
Both scipy.stats.skew and scipy.stats.kurtosis call the function of scipy.stats.moment，计算数据样本的第 k 个中心矩：
<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>
$$m_k = \frac{1}{n} \sum_{i = 1}^n (x_i - \bar{x})^k$$
因此，scipy.stats.skew 使用默认设置（例如 bias=True）计算：
<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 默认设置：
<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>
$$ K = \frac{m_4}{(m_2)^{2}} $$