偏度

 
分布的偏度是指分布不对称的方向和程度,这种离散的偏差用偏度来衡量。根据偏度的正负可以分为正偏、负偏和无偏(对称)三种情况,对应三种偏度的示意图如下图所示。
 
Error creating node with http://docs.google.com/drawings/image?id=sN0sWyZWTcxFyhgEn1nGl3A&w=179&h=179&rev=23&ac=1. Unable to fetch: http://docs.google.com/drawings/image?id=sN0sWyZWTcxFyhgEn1nGl3A&w=179&h=179&rev=23&ac=1 --- Reason: Response code 404 (Not Found) --- Fetch details: { "attempt": 3, "method": "GET", "errorCode": "ERR_NON_2XX_3XX_RESPONSE", "responseStatusCode": 404, "responseStatusMessage": "Not Found", "requestHeaders": { "user-agent": "got (https://github.com/sindresorhus/got)", "accept-encoding": "gzip, deflate, br" }, "responseHeaders": { "cache-control": "no-cache, no-store, max-age=0, must-revalidate", "pragma": "no-cache", "expires": "Mon, 01 Jan 1990 00:00:00 GMT", "date": "Wed, 27 Mar 2024 06:08:42 GMT", "content-type": "text/html; charset=utf-8", "content-encoding": "gzip", "p3p": "CP=\"This is not a P3P policy! See g.co/p3phelp for more info.\"", "x-chromium-appcache-fallback-override": "disallow-fallback", "content-security-policy": "require-trusted-types-for 'script';report-uri https://csp.withgoogle.com/csp/docs-tt", "x-content-type-options": "nosniff", "x-xss-protection": "1; mode=block", "server": "GSE", "set-cookie": [ "NID=512=X5grZA-HJJ0FEOoVEQtxe3nfhapGFrsstuaMQd2nOKuKYF3va9w5HL7-L7b6MTWFiNNMfBRl9HixjayVfYVErJP41C8wu_aef-tI7jJsId3pTD1QMfSK0LNAU8Qnn4kXwK8xZZgwImw1XRshL_iTatTHSmCv7aagb03HojcG_mo; expires=Thu, 26-Sep-2024 06:08:42 GMT; path=/; domain=.google.com; HttpOnly" ], "alt-svc": "h3=\":443\"; ma=2592000,h3-29=\":443\"; ma=2592000", "connection": "close", "transfer-encoding": "chunked" } } --- Error creating node with http://docs.google.com/drawings/image?id=sYU0a1CyLAHj-k95K_nesMw&w=179&h=179&rev=8&ac=1. Unable to fetch: http://docs.google.com/drawings/image?id=sYU0a1CyLAHj-k95K_nesMw&w=179&h=179&rev=8&ac=1 --- Reason: Response code 404 (Not Found) --- Fetch details: { "attempt": 3, "method": "GET", "errorCode": "ERR_NON_2XX_3XX_RESPONSE", "responseStatusCode": 404, "responseStatusMessage": "Not Found", "requestHeaders": { "user-agent": "got (https://github.com/sindresorhus/got)", "accept-encoding": "gzip, deflate, br" }, "responseHeaders": { "cache-control": "no-cache, no-store, max-age=0, must-revalidate", "pragma": "no-cache", "expires": "Mon, 01 Jan 1990 00:00:00 GMT", "date": "Wed, 27 Mar 2024 06:09:00 GMT", "content-type": "text/html; charset=utf-8", "content-encoding": "gzip", "p3p": "CP=\"This is not a P3P policy! See g.co/p3phelp for more info.\"", "x-chromium-appcache-fallback-override": "disallow-fallback", "content-security-policy": "require-trusted-types-for 'script';report-uri https://csp.withgoogle.com/csp/docs-tt", "x-content-type-options": "nosniff", "x-xss-protection": "1; mode=block", "server": "GSE", "set-cookie": [ "NID=512=nhmoX6gVClBZZ1lw2eeZTtCBFKItLLpfJo4XQjUlOCuDQIO21p1N6jcIAXcS4KEP9Ecy7nj0AiRz5nxneqprMH3TR53Y-sT2hJ-jDWPNahD2fjfhaT-MzxSBtud2nLxeDOAqT70-faE30uNdWKMj_GoinAmlCBsBpQ5tcO0bidw; expires=Thu, 26-Sep-2024 06:09:00 GMT; path=/; domain=.google.com; HttpOnly" ], "alt-svc": "h3=\":443\"; ma=2592000,h3-29=\":443\"; ma=2592000", "connection": "close", "transfer-encoding": "chunked" } } --- Error creating node with http://docs.google.com/drawings/image?id=sUvedI5EoB2xMw1JFvwNh7g&w=179&h=179&rev=13&ac=1. Unable to fetch: http://docs.google.com/drawings/image?id=sUvedI5EoB2xMw1JFvwNh7g&w=179&h=179&rev=13&ac=1 --- Reason: Response code 404 (Not Found) --- Fetch details: { "attempt": 3, "method": "GET", "errorCode": "ERR_NON_2XX_3XX_RESPONSE", "responseStatusCode": 404, "responseStatusMessage": "Not Found", "requestHeaders": { "user-agent": "got (https://github.com/sindresorhus/got)", "accept-encoding": "gzip, deflate, br" }, "responseHeaders": { "cache-control": "no-cache, no-store, max-age=0, must-revalidate", "pragma": "no-cache", "expires": "Mon, 01 Jan 1990 00:00:00 GMT", "date": "Wed, 27 Mar 2024 06:09:07 GMT", "content-type": "text/html; charset=utf-8", "content-encoding": "gzip", "p3p": "CP=\"This is not a P3P policy! See g.co/p3phelp for more info.\"", "x-chromium-appcache-fallback-override": "disallow-fallback", "content-security-policy": "require-trusted-types-for 'script';report-uri https://csp.withgoogle.com/csp/docs-tt", "x-content-type-options": "nosniff", "x-xss-protection": "1; mode=block", "server": "GSE", "set-cookie": [ "NID=512=AyCS-Eg_DOIj845mro1o3EdyPH7JVmy2o1rCY_I_E9uWS6tDsMMXr7Yp4Bb5Rpb70med_7fR8hkAn4BAS0WdH1UkVWCEReAJfKZsdQApIn5C61LLgrLeAemVoUHvkXlQRODz7jPuprfgH_hpsh91-j3tBCQhG-zFwy2osCw9BKs; expires=Thu, 26-Sep-2024 06:09:06 GMT; path=/; domain=.google.com; HttpOnly" ], "alt-svc": "h3=\":443\"; ma=2592000,h3-29=\":443\"; ma=2592000", "connection": "close", "transfer-encoding": "chunked" } } ---
 
很多方法可以把偏度数量化来衡量,常用的有偏斜度、矩偏度系数、四分位数偏度系数和Spearman偏度系数。
 
一、偏斜度
 
偏斜度能够反映以平均值为中心的分布的不对称程度,正偏斜度表示不对称部分的分布更趋向正值,负偏斜度表示不对称的分布更趋向负值。在Excel给出的函数中即采用偏斜度来衡量分布的偏度,对应统计函数库中的SKEW函数。
 
对应偏斜度的计算公式为
 
偏斜度 =  frac{n}{(n-1)(n-2)}Sigma(frac{x_{i}-bar{x} }{s})^3
 
例:运用 SKEW 求偏斜度
 
此例为某股票 18 个交易日的价格,试求股票价格的偏斜度。该示例文件可详见:偏斜度示例
 
如果通过以上链接无法在线查看,可以通过此链接下载Excel文件离线查看。
 

说明:SKEW,返回分布的偏斜度。
语法:SKEW(Number1, Number2, ...),其中Number1, Number2, ...为需要计算偏斜度的1~30个参数。
应注意的是,如果数据点个数少于3个,或者样本标准偏差为零,函数SKEW返回错误值#DIV/0!。

 

示例文件如下(如果Google电子表格可以打开的话可以查看,不过最近Google电子表格好像经常打不开):

可以看出,对应该股票价格的偏斜度为-0.9236。