{"id":68,"date":"2023-01-25T00:30:12","date_gmt":"2023-01-25T00:30:12","guid":{"rendered":"https:\/\/statorials.org\/cn\/%e6%96%b9%e5%b7%ae\/"},"modified":"2024-01-25T04:42:56","modified_gmt":"2024-01-25T04:42:56","slug":"%e6%96%b9%e5%b7%ae","status":"publish","type":"post","link":"https:\/\/statorials.org\/cn\/%e6%96%b9%e5%b7%ae\/","title":{"rendered":"\u65b9\u5dee"},"content":{"rendered":"<p>\u5728\u672c\u6587\u4e2d\uff0c\u6211\u4eec\u5c06\u89e3\u91ca\u4ec0\u4e48\u662f\u65b9\u5dee\uff08\u4e5f\u79f0\u4e3a\u5747\u65b9\u504f\u5dee\uff09\u4ee5\u53ca\u5b83\u7684\u8ba1\u7b97\u65b9\u6cd5\u3002\u60a8\u5c06\u627e\u5230\u65b9\u5dee\u516c\u5f0f\uff0c\u65b9\u5dee\u8ba1\u7b97\u7684\u5177\u4f53\u793a\u4f8b\uff0c\u6b64\u5916\uff0c\u60a8\u5c06\u80fd\u591f\u4f7f\u7528\u5728\u7ebf\u8ba1\u7b97\u5668\u8ba1\u7b97\u4efb\u4f55\u6570\u636e\u96c6\u7684\u65b9\u5dee\u3002<\/p>\n<p>\u6211\u4eec\u8fd8\u5411\u60a8\u5c55\u793a\u5982\u4f55\u67e5\u627e\u5206\u7ec4\u6570\u636e\u7684\u65b9\u5dee\uff0c\u56e0\u4e3a\u8fd9\u662f\u901a\u8fc7\u4e0d\u540c\u7684\u65b9\u5f0f\u5b8c\u6210\u7684\u3002\u6700\u540e\uff0c\u6211\u4eec\u6559\u60a8\u603b\u4f53\u65b9\u5dee\u548c\u6837\u672c\u65b9\u5dee\u4e4b\u95f4\u7684\u5dee\u5f02\u3001\u65b9\u5dee\u548c\u6807\u51c6\u5dee\u4e4b\u95f4\u7684\u5dee\u5f02\u4ee5\u53ca\u8be5\u7edf\u8ba1\u5ea6\u91cf\u7684\u5c5e\u6027\u3002<\/p>\n<h2 class=\"wp-block-heading\"><span id=\"%c2%bfque-es-la-varianza\" class=\"ez-toc-section\"><\/span> \u4ec0\u4e48\u662f\u65b9\u5dee\uff1f<\/h2>\n<p><strong>\u5728\u7edf\u8ba1\u5b66\u4e2d\uff0c\u65b9\u5dee\u662f\u79bb\u6563\u5ea6\u7684\u5ea6\u91cf\uff0c\u8868\u660e\u968f\u673a\u53d8\u91cf\u7684\u53d8\u5f02\u6027\u3002<\/strong>\u65b9\u5dee\u7b49\u4e8e\u6b8b\u5dee\u5e73\u65b9\u548c\u9664\u4ee5\u89c2\u6d4b\u503c\u603b\u6570\u3002<\/p>\n<p>\u8bf7\u8bb0\u4f4f\uff0c\u6b8b\u5dee\u88ab\u7406\u89e3\u4e3a\u7edf\u8ba1\u6570\u636e\u70b9\u7684\u503c\u4e0e\u6570\u636e\u96c6\u5e73\u5747\u503c\u4e4b\u95f4\u7684\u5dee\u3002<\/p>\n<p>\u5728\u6982\u7387\u8bba\u4e2d\uff0c\u65b9\u5dee\u7684\u7b26\u53f7\u662f\u5e0c\u814a\u5b57\u6bcd sigma squared (\u03c3 <sup>2<\/sup> )\u3002\u5c3d\u7ba1\u5b83\u901a\u5e38\u4e5f\u8868\u793a\u4e3a<em>Var(X)<\/em> \uff0c<em>\u5176\u4e2d X<\/em>\u662f\u8ba1\u7b97\u65b9\u5dee\u7684\u968f\u673a\u53d8\u91cf\u3002<\/p>\n<p>\u4e00\u822c\u6765\u8bf4\uff0c<strong>\u89e3\u91ca\u968f\u673a\u53d8\u91cf\u7684\u65b9\u5dee\u503c<\/strong>\u5f88\u7b80\u5355\u3002\u65b9\u5dee\u503c\u8d8a\u5927\uff0c\u6570\u636e\u8d8a\u5206\u6563\u3002\u53cd\u4e4b\u4ea6\u7136\uff0c\u65b9\u5dee\u503c\u8d8a\u5c0f\uff0c\u6570\u636e\u5e8f\u5217\u4e2d\u7684\u79bb\u6563\u6027\u5c31\u8d8a\u5c0f\u3002\u7136\u800c\uff0c\u5728\u89e3\u91ca\u65b9\u5dee\u65f6\uff0c\u5fc5\u987b\u5c0f\u5fc3<em>\u5f02\u5e38\u503c<\/em>\uff0c\u56e0\u4e3a\u5b83\u4eec\u53ef\u80fd\u4f1a\u626d\u66f2\u65b9\u5dee\u503c\u3002<\/p>\n<p>\u65b9\u5dee\uff0c\u9664\u4e86\u79bb\u5dee\u4e4b\u5916\uff0c\u5176\u4ed6\u88ab\u8ba4\u4e3a\u662f\u65b9\u5dee\u7684\u5ea6\u91cf\u8fd8\u6709\u6781\u5dee\u3001\u6807\u51c6\u5dee\u3001\u5e73\u5747\u5dee\u548c\u53d8\u5f02\u7cfb\u6570\u3002<\/p>\n<h2 class=\"wp-block-heading\"><span id=\"como-calcular-la-varianza\" class=\"ez-toc-section\"><\/span>\u5982\u4f55\u8ba1\u7b97\u95f4\u9699<\/h2>\n<p>\u8981\u8ba1\u7b97\u65b9\u5dee\uff0c\u5fc5\u987b\u6267\u884c\u4ee5\u4e0b\u6b65\u9aa4\uff1a<\/p>\n<ol style=\"color: #ff8a05; font-weight: bold;\">\n<li style=\"margin-bottom: 12px;\"><span style=\"color: #101010; font-weight: normal;\">\u6c42\u6570\u636e\u96c6\u7684<a href=\"https:\/\/statorials.org\/cn\/\u7b97\u672f\u5e73\u5747\u6570\/\">\u7b97\u672f\u5e73\u5747\u503c<\/a>\u3002<\/span><\/li>\n<li style=\"margin-bottom: 12px;\"><span style=\"color: #101010; font-weight: normal;\">\u8ba1\u7b97\u6b8b\u5dee\uff0c\u5b9a\u4e49\u4e3a\u6570\u636e\u96c6\u7684\u503c\u4e0e\u5e73\u5747\u503c\u4e4b\u95f4\u7684\u5dee\u3002<\/span><\/li>\n<li style=\"margin-bottom: 12px;\"><span style=\"color: #101010; font-weight: normal;\">\u5bf9\u6bcf\u4e2a\u4f59\u6570\u8fdb\u884c\u5e73\u65b9\u3002<\/span><\/li>\n<li style=\"margin-bottom: 12px;\"><span style=\"color: #101010; font-weight: normal;\">\u5c06\u4e0a\u4e00\u6b65\u8ba1\u7b97\u51fa\u7684\u6240\u6709\u7ed3\u679c\u76f8\u52a0\u3002<\/span><\/li>\n<li><span style=\"color: #101010; font-weight: normal;\">\u9664\u4ee5\u6570\u636e\u603b\u6570\u3002\u5f97\u5230\u7684\u7ed3\u679c\u5c31\u662f\u6570\u636e\u5e8f\u5217\u7684\u65b9\u5dee\u3002<\/span><\/li>\n<\/ol>\n<p>\u7efc\u4e0a\u6240\u8ff0\uff0c\u6570\u636e\u96c6<strong>\u65b9\u5dee\u7684\u8ba1\u7b97\u516c\u5f0f<\/strong>\u4e3a\uff1a<\/p>\n<figure class=\"wp-block-image aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1262\" src=\"https:\/\/statorials.org\/wp-content\/uploads\/2023\/08\/variance.png\" sizes=\"auto, \" srcset=\"\" alt=\"\u65b9\u5dee\" width=\"349\" height=\"259\" \/><\/figure>\n<p style=\"margin-bottom: 5px;\">\u91d1\u5b50\uff1a<\/p>\n<ul style=\"color: #ff8a05; font-weight: bold;\">\n<li style=\"margin-bottom: 3px;\"><img loading=\"lazy\" decoding=\"async\" class=\"ql-img-inline-formula quicklatex-auto-format\" style=\"vertical-align: 0px;\" title=\"Rendered by QuickLaTeX.com\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-996ff7036e644e89f8ac379fa58d0cf7_l3.png\" alt=\"X\" width=\"16\" height=\"12\" \/> \u662f\u8981\u8ba1\u7b97\u65b9\u5dee\u7684\u968f\u673a\u53d8\u91cf\u3002<\/li>\n<li style=\"margin-bottom: 3px;\"><img loading=\"lazy\" decoding=\"async\" class=\"ql-img-inline-formula quicklatex-auto-format\" style=\"vertical-align: -3px;\" title=\"Rendered by QuickLaTeX.com\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-dad27a9703483183e1afd245f5232b83_l3.png\" alt=\"x_i\" width=\"15\" height=\"11\" \/> \u662f\u6570\u636e\u503c<\/li>\n<li style=\"margin-bottom: 3px;\"><img loading=\"lazy\" decoding=\"async\" class=\"ql-img-inline-formula quicklatex-auto-format\" style=\"vertical-align: 0px;\" title=\"Rendered by QuickLaTeX.com\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-ec4217f4fa5fcd92a9edceba0e708cf7_l3.png\" alt=\"n\" width=\"11\" height=\"8\" \/> \u662f\u89c2\u6d4b\u503c\u7684\u603b\u6570\u3002<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"ql-img-inline-formula quicklatex-auto-format\" style=\"vertical-align: 0px;\" title=\"Rendered by QuickLaTeX.com\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-5b485d4231dfeb4b50ddf271c3abb0b3_l3.png\" alt=\"\\overline{X}\" width=\"17\" height=\"15\" \/> \u662f\u968f\u673a\u53d8\u91cf\u7684\u5e73\u5747\u503c<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"ql-img-inline-formula quicklatex-auto-format\" style=\"vertical-align: 0px;\" title=\"Rendered by QuickLaTeX.com\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-996ff7036e644e89f8ac379fa58d0cf7_l3.png\" alt=\"X\" width=\"16\" height=\"12\" \/> \u662f\u65b9\u5dee\u6216\u5747\u65b9\u504f\u5dee<\/li>\n<\/ul>\n<p>\ud83d\udc49<u style=\"text-decoration-color: #FF8A05;\">\u60a8\u53ef\u4ee5\u4f7f\u7528\u4e0b\u9762\u7684\u8ba1\u7b97\u5668\u6765\u8ba1\u7b97\u4efb\u4f55\u6570\u636e\u96c6\u7684\u65b9\u5dee\u3002<\/u><\/p>\n<p>\u56e0\u6b64\uff0c\u8981\u4ece\u6570\u636e\u7cfb\u5217\u4e2d\u63d0\u53d6\u65b9\u5dee\uff0c\u4e86\u89e3\u7b97\u672f\u5e73\u5747\u503c\u7684\u8ba1\u7b97\u65b9\u5f0f\u81f3\u5173\u91cd\u8981\u3002\u5982\u679c\u60a8\u4e0d\u8bb0\u5f97\u5982\u4f55\u6267\u884c\u6b64\u64cd\u4f5c\uff0c\u53ef\u4ee5\u5728\u4e0a\u9762\u94fe\u63a5\u7684\u6587\u7ae0\u4e2d\u67e5\u770b\u3002<\/p>\n<h2 class=\"wp-block-heading\"><span id=\"ejemplo-de-la-varianza\" class=\"ez-toc-section\"><\/span>\u504f\u5dee\u793a\u4f8b<\/h2>\n<p>\u73b0\u5728\u6211\u4eec\u77e5\u9053\u4e86\u65b9\u5dee\u7684\u5b9a\u4e49\uff0c\u6211\u4eec\u5c06\u9010\u6b65\u89e3\u51b3\u4e00\u4e2a\u7ec3\u4e60\uff0c\u4ee5\u4fbf\u60a8\u4e86\u89e3\u5982\u4f55\u83b7\u5f97\u6570\u636e\u5e8f\u5217\u7684\u65b9\u5dee\u3002<\/p>\n<ul>\n<li>\u5bf9\u4e8e\u4e00\u5bb6\u8de8\u56fd\u516c\u53f8\u6765\u8bf4\uff0c\u8fc7\u53bb\u4e94\u5e74\u7684\u7ecf\u6d4e\u6210\u679c\u662f\u4f17\u6240\u5468\u77e5\u7684\uff0c\u5927\u90e8\u5206\u90fd\u83b7\u5f97\u4e86\u5229\u6da6\uff0c\u4f46\u6709\u4e00\u5e74\u5374\u51fa\u73b0\u4e86\u76f8\u5f53\u5927\u7684\u635f\u5931\uff1a11.5, 2, -9, 700 \u4e07\u6b27\u5143\u3002\u8ba1\u7b97\u8be5\u6570\u636e\u96c6\u7684\u65b9\u5dee\u3002<\/li>\n<\/ul>\n<p>\u6b63\u5982\u6211\u4eec\u5728\u4e0a\u9762\u7684\u89e3\u91ca\u4e2d\u770b\u5230\u7684\uff0c\u8981\u627e\u5230\u6570\u636e\u5e8f\u5217\u7684\u65b9\u5dee\uff0c\u6211\u4eec\u9700\u8981\u505a\u7684\u7b2c\u4e00\u4ef6\u4e8b\u662f\u8ba1\u7b97\u5176\u7b97\u672f\u5e73\u5747\u503c\uff1a<\/p>\n<p class=\"has-text-align-center\"><img loading=\"lazy\" decoding=\"async\" class=\"ql-img-inline-formula quicklatex-auto-format\" style=\"vertical-align: -12px;\" title=\"Rendered by QuickLaTeX.com\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-0a2fc458e6a80794b9b8fdc1120ddedb_l3.png\" alt=\"\\overline{X}=\\cfrac{11+5+2+(-9)+7}{5}=3,2\" width=\"257\" height=\"40\" \/><\/p>\n<p>\u4e00\u65e6\u6211\u4eec\u77e5\u9053\u4e86\u6570\u636e\u7684\u5e73\u5747\u503c\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u4f7f\u7528\u65b9\u5dee\u516c\u5f0f\uff1a<\/p>\n<p class=\"has-text-align-center\"><img loading=\"lazy\" decoding=\"async\" class=\"ql-img-inline-formula quicklatex-auto-format\" style=\"vertical-align: -12px;\" title=\"Rendered by QuickLaTeX.com\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-a711d079f0c580d1e3cd8ad08084a6ca_l3.png\" alt=\"Var(X)=\\cfrac{\\displaystyle\\sum_{i=1}^n\\left(x_i-\\overline{X}\\right)^2}{n}\" width=\"197\" height=\"70\" \/><\/p>\n<p>\u6211\u4eec\u5c06\u7ec3\u4e60\u62a5\u8868\u63d0\u4f9b\u7684\u6570\u636e\u4ee3\u5165\u516c\u5f0f\uff1a<\/p>\n<p class=\"has-text-align-center\"><img loading=\"lazy\" decoding=\"async\" class=\"ql-img-inline-formula quicklatex-auto-format\" style=\"vertical-align: -12px;\" title=\"Rendered by QuickLaTeX.com\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-e284702746a065d0ad3c1c36953e4c91_l3.png\" alt=\"Var(X)=\\cfrac{\\displaystyle (11-3,2)^2+(5-3,2)^2+(2-3,2)^2+(-9-3,2)^2+(7-3,2)^2}{5}\" width=\"586\" height=\"42\" \/><\/p>\n<p>\u6700\u540e\uff0c\u5269\u4e0b\u7684\u5c31\u662f\u89e3\u51b3\u8ba1\u7b97\u65b9\u5dee\u7684\u8fd0\u7b97\uff1a<\/p>\n<p class=\"has-text-align-center\"><img loading=\"lazy\" decoding=\"async\" class=\"ql-img-inline-formula quicklatex-auto-format\" style=\"vertical-align: 0px;\" title=\"Rendered by QuickLaTeX.com\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-c2cbee60d77f19e88117e1bcf28d9cb2_l3.png\" alt=\"\\begin{aligned}Var(X)&amp;=\\cfrac{7,8^2+1,8^2+(-1,2)^2+(-12,2)^2+3,8^2}{5}\\\\[2ex]&amp;=\\cfrac{60,84+3,24+1,44+148,84+14,44}{5}\\\\[2ex]&amp;= \\cfrac{228,8}{5} \\\\[2ex]&amp;=45,76 \\ \\text{millones de euros}^2\\end{aligned}\" width=\"406\" height=\"208\" \/><\/p>\n<p>\u8bf7\u6ce8\u610f\uff0c\u65b9\u5dee\u5355\u4f4d\u4e0e\u7edf\u8ba1\u6570\u636e\u7684\u5355\u4f4d\u76f8\u540c\uff0c\u4f46\u8fdb\u884c\u4e86\u5e73\u65b9\uff0c\u56e0\u6b64\u8be5\u6570\u636e\u7ec4\u7684\u65b9\u5dee\u4e3a 4576 \u4e07\u6b27\u5143<sup>2<\/sup> \u3002<\/p>\n<h2 class=\"wp-block-heading\"><span id=\"calculadora-de-la-varianza\" class=\"ez-toc-section\"><\/span>\u5dee\u8ddd\u8ba1\u7b97\u5668<\/h2>\n<p>\u5728\u4ee5\u4e0b\u8ba1\u7b97\u5668\u4e2d\u8f93\u5165\u7edf\u8ba1\u6570\u636e\u96c6\u4ee5\u8ba1\u7b97\u5176\u65b9\u5dee\u3002\u6570\u636e\u5fc5\u987b\u7528\u7a7a\u683c\u5206\u9694\uff0c\u5e76\u4f7f\u7528\u53e5\u70b9\u4f5c\u4e3a\u5c0f\u6570\u70b9\u5206\u9694\u7b26\u8f93\u5165\u3002<\/p>\n<form action=\"\" method=\"post\"><textarea style=\"border: 1.5px solid #4FC3F7; border-radius: 15px;\" name=\"datos\" required=\"\" placeholder=\"8 2 1 5.7 6 ...\"><\/textarea><\/p>\n<div style=\"text-align: center;\"><input style=\"border-radius: 30px; margin: 20px;\" name=\"submit\" type=\"submit\" value=\"\u8ba1\u7b97\u95f4\u9699\" align=\"center\" \/><\/div>\n<\/form>\n<h2 class=\"wp-block-heading\"><span id=\"varianza-para-datos-agrupados\" class=\"ez-toc-section\"><\/span>\u5206\u7ec4\u6570\u636e\u7684\u65b9\u5dee<\/h2>\n<p><strong>\u8981\u8ba1\u7b97\u5206\u7ec4\u4e3a\u533a\u95f4 \u7684\u6570\u636e\u7684\u65b9\u5dee<\/strong>\uff0c\u5fc5\u987b\u9075\u5faa\u4ee5\u4e0b\u6b65\u9aa4\uff1a<\/p>\n<ol style=\"color: #ff8a05; font-weight: bold;\">\n<li style=\"margin-bottom: 12px;\"><span style=\"color: #101010; font-weight: normal;\">\u6c42\u5206\u7ec4\u6570\u636e\u7684\u5e73\u5747\u503c\u3002<\/span><\/li>\n<li style=\"margin-bottom: 12px;\"><span style=\"color: #101010; font-weight: normal;\">\u8ba1\u7b97\u5206\u7ec4\u6570\u636e\u7684\u6b8b\u5dee\u3002<\/span><\/li>\n<li style=\"margin-bottom: 12px;\"><span style=\"color: #101010; font-weight: normal;\">\u5bf9\u6bcf\u4e2a\u4f59\u6570\u8fdb\u884c\u5e73\u65b9\u3002<\/span><\/li>\n<li style=\"margin-bottom: 12px;\"><span style=\"color: #101010; font-weight: normal;\">\u5c06\u4e4b\u524d\u7684\u6bcf\u4e2a\u7ed3\u679c\u4e58\u4ee5\u5176\u95f4\u9694\u7684\u9891\u7387\u3002<\/span><\/li>\n<li style=\"margin-bottom: 12px;\"><span style=\"color: #101010; font-weight: normal;\">\u5c06\u4e0a\u4e00\u6b65\u83b7\u5f97\u7684\u6240\u6709\u503c\u76f8\u52a0\u3002<\/span><\/li>\n<li><span style=\"color: #101010; font-weight: normal;\">\u9664\u4ee5\u89c2\u5bdf\u603b\u6570\u3002\u6240\u5f97\u6570\u5b57\u662f\u5206\u7ec4\u6570\u636e\u7684\u65b9\u5dee\u3002<\/span><\/li>\n<\/ol>\n<p>\u6362\u53e5\u8bdd\u8bf4\uff0c\u5206\u7ec4\u6570\u636e\u7684\u65b9\u5dee\u8ba1\u7b97\u516c\u5f0f\u5982\u4e0b\uff1a<\/p>\n<p class=\"has-text-align-center\"><img loading=\"lazy\" decoding=\"async\" class=\"ql-img-inline-formula quicklatex-auto-format\" style=\"vertical-align: -12px;\" title=\"Rendered by QuickLaTeX.com\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-3c8fd825b7e23d237d3bc105ff317afe_l3.png\" alt=\"Var(X)=\\cfrac{\\displaystyle\\sum_{i=1}^n\\left(x_i-\\overline{X}\\right)^2\\cdot f_i }{n}\" width=\"224\" height=\"70\" \/><\/p>\n<p>\u5c3d\u7ba1\u901a\u5e38\u4f7f\u7528\u4e0a\u9762\u7684\u516c\u5f0f\uff0c\u4f46\u4e5f\u53ef\u4ee5\u4f7f\u7528\u4e0b\u9762\u7684\u4ee3\u6570\u8868\u8fbe\u5f0f\uff0c\u56e0\u4e3a\u5b83\u662f\u7b49\u6548\u7684\uff1a<\/p>\n<p class=\"has-text-align-center\"><img loading=\"lazy\" decoding=\"async\" class=\"ql-img-inline-formula quicklatex-auto-format\" style=\"vertical-align: -12px;\" title=\"Rendered by QuickLaTeX.com\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-234f660701e2c8cde3bdf7ecaa140a11_l3.png\" alt=\"Var(X)=\\cfrac{\\displaystyle\\sum_{i=1}^n x_i^2\\cdot f_i }{n}-\\overline{X}^2\" width=\"209\" height=\"70\" \/><\/p>\n<p>\u4f8b\u5982\uff0c\u6211\u4eec\u5c06\u627e\u5230\u4ee5\u4e0b\u5206\u7ec4\u6570\u636e\u7cfb\u5217\u7684\u65b9\u5dee\uff1a<\/p>\n<figure class=\"wp-block-image aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1274\" src=\"https:\/\/statorials.org\/wp-content\/uploads\/2023\/08\/donnees-regroupees-dans-intervalles.png\" sizes=\"auto, \" srcset=\"\" alt=\"\u6570\u636e\u6309\u533a\u95f4\u5206\u7ec4\" width=\"259\" height=\"190\" \/><\/figure>\n<p>\u9996\u5148\uff0c\u6211\u4eec\u9700\u8981\u786e\u5b9a\u5206\u7ec4\u6570\u636e\u7684\u5e73\u5747\u503c\u3002\u4e3a\u6b64\uff0c\u6211\u4eec\u5728\u9891\u7387\u8868\u4e2d\u6dfb\u52a0\u4e00\u5217\uff0c\u5176\u4e2d\u5305\u542b\u7c7b\u522b\u6807\u8bb0\u548c\u9891\u7387\u7684\u4e58\u79ef\uff1a<\/p>\n<figure class=\"wp-block-image aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1275\" src=\"https:\/\/statorials.org\/wp-content\/uploads\/2023\/08\/moyenne-des-donnees-groupees.png\" sizes=\"auto, \" srcset=\"\" alt=\"\u6570\u636e\u6309\u5e73\u5747\u503c\u5206\u7ec4\" width=\"345\" height=\"190\" \/><\/figure>\n<p>\u73b0\u5728\uff0c\u6211\u4eec\u901a\u8fc7\u5c06\u6dfb\u52a0\u5217\u7684\u603b\u548c\u9664\u4ee5\u6570\u636e\u603b\u6570\u6765\u8ba1\u7b97\u5206\u7ec4\u6570\u636e\u7684\u5e73\u5747\u503c\uff1a<\/p>\n<p class=\"has-text-align-center\"><img loading=\"lazy\" decoding=\"async\" class=\"ql-img-inline-formula quicklatex-auto-format\" style=\"vertical-align: -12px;\" title=\"Rendered by QuickLaTeX.com\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-cbe29749d3f628250dead48396088c9a_l3.png\" alt=\"\\overline{X}=\\cfrac{\\displaystyle\\sum_{i=1}^n x_i\\cdot f_i}{n}=\\cfrac{750}{30}=25\" width=\"209\" height=\"70\" \/><\/p>\n<p>\u6839\u636e\u8ba1\u7b97\u6570\u636e\u7684\u5e73\u5747\u503c\uff0c\u6211\u4eec\u53ef\u4ee5\u6dfb\u52a0\u4ee5\u4e0b\u4e09\u5217\uff1a<\/p>\n<figure class=\"wp-block-image aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1276\" src=\"https:\/\/statorials.org\/wp-content\/uploads\/2023\/08\/variance-pour-les-donnees-regroupees.png\" sizes=\"auto, \" srcset=\"\" alt=\"\u5206\u7ec4\u6570\u636e\u7684\u65b9\u5dee\" width=\"642\" height=\"190\" \/><\/figure>\n<p>\u56e0\u6b64\uff0c\u5408\u5e76\u6570\u636e\u96c6\u7684\u65b9\u5dee\u662f\u6700\u540e\u4e00\u5217\u7684\u603b\u548c\u9664\u4ee5\u89c2\u5bdf\u6570\u636e\u7684\u603b\u6570\uff1a<\/p>\n<p class=\"has-text-align-center\"><img loading=\"lazy\" decoding=\"async\" class=\"ql-img-inline-formula quicklatex-auto-format\" style=\"vertical-align: -12px;\" title=\"Rendered by QuickLaTeX.com\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-70d22837db07bbf1e13bb4acb2cfccba_l3.png\" alt=\"Var(X)=\\cfrac{\\displaystyle\\sum_{i=1}^n\\left(x_i-\\overline{X}\\right)^2\\cdot f_i }{n}=\\cfrac{4200}{30}=140\" width=\"335\" height=\"70\" \/><\/p>\n<h2 class=\"wp-block-heading\"><span id=\"varianza-y-desviacion-estandar\" class=\"ez-toc-section\"><\/span>\u65b9\u5dee\u548c\u6807\u51c6\u5dee<\/h2>\n<p><strong>\u65b9\u5dee\u548c\u6807\u51c6\u5dee\uff08\u6216\u6807\u51c6\u5dee\uff09<\/strong>\u662f\u79bb\u6563\u5ea6\u7684\u4e24\u79cd\u5ea6\u91cf\uff0c\u56e0\u6b64\u90fd\u8868\u793a\u6570\u636e\u96c6\u7684\u79bb\u6563\u7a0b\u5ea6\u3002\u7136\u800c\uff0c<strong>\u65b9\u5dee\u548c\u6807\u51c6\u5dee\u4e4b\u95f4\u7684\u533a\u522b<\/strong>\u5728\u4e8e\uff0c\u65b9\u5dee\u901a\u5e38\u5177\u6709\u8f83\u5927\u7684\u503c\uff0c\u56e0\u4e3a\u5b83\u662f\u6807\u51c6\u5dee\u7684\u5e73\u65b9\u3002<\/p>\n<p>\u6807\u51c6\u5dee\u901a\u5e38\u7531\u5e0c\u814a\u5b57\u6bcd sigma (\u03c3) \u8868\u793a\uff0c\u56e0\u6b64\u65b9\u5dee\u7531\u5b57\u6bcd sigma square (\u03c3 <sup>2<\/sup> ) \u8868\u793a\uff0c\u56e0\u4e3a\u5b83\u662f\u8fd9\u4e24\u4e2a\u79bb\u6563\u5ea6\u91cf\u4e4b\u95f4\u5b58\u5728\u7684\u6570\u5b66\u5173\u7cfb\u3002<\/p>\n<p class=\"has-text-align-center\"><img loading=\"lazy\" decoding=\"async\" class=\"ql-img-inline-formula quicklatex-auto-format\" style=\"vertical-align: -5px;\" title=\"Rendered by QuickLaTeX.com\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-0c71ab111d7c483a2ad04cb5e9618da4_l3.png\" alt=\"Var(X)=\\sigma^2\" width=\"103\" height=\"20\" \/><\/p>\n<p>\u56e0\u6b64\uff0c\u4e00\u65e6\u8ba1\u7b97\u4e86\u4e00\u7ec4\u6570\u636e\u7684\u65b9\u5dee\u503c\uff0c\u53ea\u9700\u53d6\u65b9\u5dee\u7684\u5e73\u65b9\u6839\uff0c\u60a8\u5c31\u53ef\u4ee5\u8f7b\u677e\u627e\u5230\u540c\u4e00\u7ec4\u6570\u636e\u7684\u6807\u51c6\u5dee\u503c\u3002<\/p>\n<p class=\"has-text-align-center\"><img loading=\"lazy\" decoding=\"async\" class=\"ql-img-inline-formula quicklatex-auto-format\" style=\"vertical-align: -1px;\" title=\"Rendered by QuickLaTeX.com\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-fab93067538628f5fa2d9bc829a7c470_l3.png\" alt=\"\\sigma=\\sqrt{\\sigma^2}\" width=\"68\" height=\"18\" \/><\/p>\n<h2 class=\"wp-block-heading\"><span id=\"varianza-poblacional-y-varianza-muestral\" class=\"ez-toc-section\"><\/span>\u603b\u4f53\u65b9\u5dee\u548c\u6837\u672c\u65b9\u5dee<\/h2>\n<p>\u4ece\u903b\u8f91\u4e0a\u8bb2\uff0c<strong>\u603b\u4f53\u65b9\u5dee<\/strong>\u662f\u6307\u7edf\u8ba1\u603b\u4f53\u65b9\u5dee\u7684\u8ba1\u7b97\uff0c\u800c<strong>\u6837\u672c\u65b9\u5dee<\/strong>\u662f\u6307\u6837\u672c\u65b9\u5dee\u7684\u8ba1\u7b97\u3002\u7136\u800c\uff0c\u8fd9\u662f\u4e24\u4e2a\u4e0d\u540c\u7684\u6982\u5ff5\uff0c\u56e0\u4e3a\u603b\u4f53\u65b9\u5dee\u516c\u5f0f\u4e0e\u6837\u672c\u65b9\u5dee\u516c\u5f0f\u4e0d\u540c\u3002<\/p>\n<p>\u901a\u5e38\u5728\u65b9\u5dee\u7ec3\u4e60\u4e2d\uff0c\u5982\u679c\u4ed6\u4eec\u6ca1\u6709\u53e6\u5916\u544a\u8bc9\u6211\u4eec\uff0c\u8981\u627e\u5230\u6240\u63d0\u4f9b\u6570\u636e\u96c6\u7684\u65b9\u5dee\uff0c\u6211\u4eec\u5fc5\u987b\u4f7f\u7528<strong>\u603b\u4f53\u65b9\u5dee\u516c\u5f0f<\/strong>\uff0c\u8fd9\u662f\u6211\u4eec\u5728\u6587\u7ae0\u5f00\u5934\u89e3\u91ca\u7684\u516c\u5f0f\uff1a<\/p>\n<p class=\"has-text-align-center\"><img loading=\"lazy\" decoding=\"async\" class=\"ql-img-inline-formula quicklatex-auto-format\" style=\"vertical-align: -12px;\" title=\"Rendered by QuickLaTeX.com\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-aae456d460db4438c8ad43d11c36dedb_l3.png\" alt=\"\\sigma^2=\\cfrac{\\displaystyle\\sum_{i=1}^n\\left(x_i-\\overline{X}\\right)^2}{n}\" width=\"153\" height=\"70\" \/><\/p>\n<p>\u4f46\u4e5f\u8bb8\u5728\u67d0\u4e9b\u95ee\u9898\u4e2d\u4f60\u4f1a\u88ab\u8981\u6c42\u5c06\u7edf\u8ba1\u6570\u636e\u89c6\u4e3a\u6837\u672c\uff0c\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\u6211\u4eec\u9700\u8981\u4f7f\u7528<strong>\u6837\u672c\u65b9\u5dee\u516c\u5f0f<\/strong>\uff1a<\/p>\n<p class=\"has-text-align-center\"><img loading=\"lazy\" decoding=\"async\" class=\"ql-img-inline-formula quicklatex-auto-format\" style=\"vertical-align: -12px;\" title=\"Rendered by QuickLaTeX.com\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-033364d2b74d014d944faade687d7b19_l3.png\" alt=\"s^2=\\cfrac{\\displaystyle\\sum_{i=1}^n\\left(x_i-\\overline{X}\\right)^2}{n-1}\" width=\"151\" height=\"70\" \/><\/p>\n<p>\u8bf7\u6ce8\u610f\uff0c\u4e3a\u4e86\u8868\u793a\u6b63\u5728\u8ba1\u7b97\u603b\u4f53\u65b9\u5dee\uff0c\u7528\u5e0c\u814a\u5b57\u6bcd \u03c3 \u8868\u793a\uff0c\u4f46\u5728\u8ba1\u7b97\u6837\u672c\u65b9\u5dee\u65f6\u5219\u4f7f\u7528\u5b57\u6bcd s\u3002<\/p>\n<p>\u6b63\u5982\u4f60\u6240\u770b\u5230\u7684\uff0c\u4e24\u4e2a\u516c\u5f0f\u4e4b\u95f4\u7684\u552f\u4e00\u533a\u522b\u662f\uff0c\u5728\u6837\u672c\u7684\u65b9\u5dee\u4e2d\uff0c\u6211\u4eec\u9700\u8981\u9664\u4ee5\u89c2\u5bdf\u603b\u6570\u8d1f1\uff0c\u4f8b\u5982\uff0c\u5982\u679c\u603b\u5171\u670930\u4e2a\u6570\u636e\u9879\uff0c\u6211\u4eec\u5c06\u9664\u4ee529\u4f46\u5206\u5b50\u7684\u8ba1\u7b97\u662f\u4ee5\u5b8c\u5168\u76f8\u540c\u7684\u65b9\u5f0f\u5b8c\u6210\u7684\u3002<\/p>\n<h2 class=\"wp-block-heading\"><span id=\"propiedades-de-la-varianza\" class=\"ez-toc-section\"><\/span>\u65b9\u5dee\u5c5e\u6027<\/h2>\n<p>\u65b9\u5dee\u5177\u6709\u4ee5\u4e0b\u5c5e\u6027\uff1a<\/p>\n<ul>\n<li>\u4efb\u4f55\u968f\u673a\u53d8\u91cf\u7684\u65b9\u5dee\u5c06\u59cb\u7ec8\u5927\u4e8e\u6216\u7b49\u4e8e\u96f6\u3002\u540c\u6837\uff0c\u5982\u679c\u65b9\u5dee\u4e3a\u96f6\uff0c\u5219\u610f\u5473\u7740\u6240\u6709\u7edf\u8ba1\u6570\u636e\u90fd\u662f\u76f8\u540c\u7684\u3002<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img loading=\"lazy\" decoding=\"async\" class=\"ql-img-inline-formula quicklatex-auto-format\" style=\"vertical-align: -5px;\" title=\"Rendered by QuickLaTeX.com\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-56edbbba242899a77fbdaf17859e5d89_l3.png\" alt=\"Var(x)\\ge 0\" width=\"89\" height=\"19\" \/><\/p>\n<ul>\n<li>\u663e\u7136\uff0c\u5355\u4e2a\u503c\u7684\u65b9\u5dee\u4e3a\u96f6\u3002<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img loading=\"lazy\" decoding=\"async\" class=\"ql-img-inline-formula quicklatex-auto-format\" style=\"vertical-align: -5px;\" title=\"Rendered by QuickLaTeX.com\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-60234622e7a82de8d87a8e5e5af3a686_l3.png\" alt=\"Var(a)=0\\qquad a\\in \\mathbb{R}\" width=\"167\" height=\"19\" \/><\/p>\n<ul>\n<li>\u6807\u91cf\u4e0e\u53d8\u91cf\u7684\u4e58\u79ef\u7684\u65b9\u5dee\u7b49\u4e8e\u8be5\u6807\u91cf\u7684\u5e73\u65b9\u4e58\u4ee5\u53d8\u91cf\u7684\u65b9\u5dee\u3002<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img loading=\"lazy\" decoding=\"async\" class=\"ql-img-inline-formula quicklatex-auto-format\" style=\"vertical-align: -5px;\" title=\"Rendered by QuickLaTeX.com\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-cfa5d6d67c4af8203682ecbd0b76d525_l3.png\" alt=\"Var(aX)=a^2\\cdot Var(X)\\qquad a\\in \\mathbb{R}\" width=\"266\" height=\"20\" \/><\/p>\n<ul>\n<li>\u4e24\u4e2a\u56e0\u53d8\u91cf\u4e4b\u548c\u7684\u65b9\u5dee\u7b49\u4e8e\u6bcf\u4e2a\u53d8\u91cf\u5404\u81ea\u7684\u65b9\u5dee\u4e4b\u548c\u52a0\u4e0a\u4e24\u4e2a\u53d8\u91cf\u4e4b\u95f4\u534f\u65b9\u5dee\u7684\u4e24\u500d\u3002<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img loading=\"lazy\" decoding=\"async\" class=\"ql-img-inline-formula quicklatex-auto-format\" style=\"vertical-align: -5px;\" title=\"Rendered by QuickLaTeX.com\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-ad23ae4c3b77e0d08f974f1c6f858ec1_l3.png\" alt=\"Var(X+Y)=Var(X)+Var(Y)+2Cov(X,Y)\" width=\"378\" height=\"19\" \/><\/p>\n<ul>\n<li>\u56e0\u6b64\uff0c\u5982\u679c\u4e24\u4e2a\u53d8\u91cf\u662f\u72ec\u7acb\u7684\uff0c\u4e3a\u4e86\u786e\u5b9a\u5b83\u4eec\u603b\u548c\u7684\u65b9\u5dee\uff0c\u5c06\u5b83\u4eec\u7684\u65b9\u5dee\u76f8\u52a0\u5c31\u8db3\u591f\u4e86\uff1a<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img loading=\"lazy\" decoding=\"async\" class=\"ql-img-inline-formula quicklatex-auto-format\" style=\"vertical-align: -5px;\" title=\"Rendered by QuickLaTeX.com\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-e1aa58a3af75d4e27a430273183e26b6_l3.png\" alt=\"Var(X+Y)=Var(X)+Var(Y)\" width=\"264\" height=\"19\" \/><\/p>\n<ul>\n<li>\u504f\u5dee\u4e5f\u53ef\u4ee5\u4f7f\u7528\u4ee5\u4e0b\u516c\u5f0f\u901a\u8fc7\u6570\u5b66\u671f\u671b\u6765\u5b9a\u4e49\uff1a<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img loading=\"lazy\" decoding=\"async\" class=\"ql-img-inline-formula quicklatex-auto-format\" style=\"vertical-align: -7px;\" title=\"Rendered by QuickLaTeX.com\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-3adf3028629c39719280e2611df6daf5_l3.png\" alt=\"Var(X)=E\\bigl[(X-\\overline{X})^2\\bigr]\" width=\"186\" height=\"22\" 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