{"id":642,"date":"2025-01-12T02:33:00","date_gmt":"2025-01-11T18:33:00","guid":{"rendered":"https:\/\/smartwms.cn\/?p=642"},"modified":"2025-02-12T12:44:56","modified_gmt":"2025-02-12T04:44:56","slug":"standard-deviation","status":"publish","type":"post","link":"https:\/\/smartwms.cn\/en\/shares\/standard-deviation\/","title":{"rendered":"\u4ec0\u4e48\u662f\u6807\u51c6\u5dee\uff1f\uff08\u5b9a\u4e49\u3001\u516c\u5f0f\u3001\u4f5c\u7528\u53ca\u793a\u4f8b\uff09"},"content":{"rendered":"<h2 class=\"wp-block-heading\" id=\"h_17830577910_0\">&#21069;&#35328;<\/h2>\n\n\n\n<p>\u672c\u7bc7\u65e8\u5728\u4e3a\u64b0\u5199<em><strong>\u300a\u5b89\u5168\u5e93\u5b58\u3001\u6700\u5c0f\u5e93\u5b58\u3001\u6700\u5927\u5e93\u5b58\u7684\u5b9a\u4e49\u3001\u516c\u5f0f\u3001\u4f5c\u7528\u53ca\u793a\u4f8b\u300b<\/strong><\/em>\u4e00\u6587\u8fdb\u884c\u9884\u5907\u3002\u5728\u8ba1\u7b97\u5b89\u5168\u5e93\u5b58\u7684\u8fc7\u7a0b\u4e2d\uff0c\u5fc5\u987b\u5bf9\u6807\u51c6\u5dee\u8fdb\u884c\u8ba1\u7b97\u3002<\/p>\n\n\n<h2 class=\"wp-block-heading\" id=\"h_17830577910_1\">&#19968;&#12289;&#26631;&#20934;&#24046;&#30340;&#23450;&#20041;<\/h2>\n\n\n\n<p><strong>\u6807\u51c6\u5dee\u662f\u4e00\u79cd\u7528\u4e8e\u8861\u91cf\u6570\u636e\u79bb\u6563\u7a0b\u5ea6\u7684\u7edf\u8ba1\u91cf\uff0c\u53cd\u6620\u4e86\u4e00\u7ec4\u6570\u636e\u76f8\u5bf9\u4e8e\u5176\u5e73\u5747\u503c\u7684\u5206\u6563\u60c5\u51b5\u3002<\/strong>\u7b80\u5355\u6765\u8bf4\uff0c\u6807\u51c6\u5dee\u80fd\u8ba9\u6211\u4eec\u76f4\u89c2\u5730\u4e86\u89e3\u6570\u636e\u7684\u5206\u5e03\u8303\u56f4\u548c\u6ce2\u52a8\u7a0b\u5ea6\u3002<\/p>\n\n\n\n<p>\u5728\u7edf\u8ba1\u5b66\u4e2d\uff0c\u6570\u636e\u7684\u5206\u5e03\u7279\u5f81\u81f3\u5173\u91cd\u8981\u3002<strong>\u5747\u503c\u4ee3\u8868\u4e86\u6570\u636e\u7684\u96c6\u4e2d\u8d8b\u52bf\uff0c\u800c\u6807\u51c6\u5dee\u5219\u63cf\u8ff0\u4e86\u6570\u636e\u7684\u79bb\u6563\u8d8b\u52bf\u3002<\/strong>\u5982\u679c\u4e00\u7ec4\u6570\u636e\u7684\u6807\u51c6\u5dee\u8f83\u5c0f\uff0c\u90a3\u5c31\u8868\u660e\u8fd9\u4e9b\u6570\u636e\u7d27\u5bc6\u56f4\u7ed5\u5728\u5e73\u5747\u503c\u5468\u56f4\uff0c\u6570\u636e\u7684\u4e00\u81f4\u6027\u8f83\u9ad8\uff1b\u53cd\u4e4b\uff0c\u82e5\u6807\u51c6\u5dee\u8f83\u5927\uff0c\u5219\u610f\u5473\u7740\u6570\u636e\u8f83\u4e3a\u5206\u6563\uff0c\u5404\u4e2a\u6570\u636e\u70b9\u4e0e\u5e73\u5747\u503c\u4e4b\u95f4\u7684\u5dee\u5f02\u660e\u663e\u3002<\/p>\n\n\n\n<p>\u4f8b\u5982\uff0c\u5728\u4e00\u4e2a\u73ed\u7ea7\u5b66\u751f\u7684\u8eab\u9ad8\u6570\u636e\u4e2d\uff0c\u82e5\u6807\u51c6\u5dee\u8f83\u5c0f\uff0c\u8bf4\u660e\u5b66\u751f\u4eec\u7684\u8eab\u9ad8\u6bd4\u8f83\u63a5\u8fd1\uff0c\u6574\u4f53\u8eab\u9ad8\u5206\u5e03\u8f83\u4e3a\u96c6\u4e2d\uff1b\u82e5\u6807\u51c6\u5dee\u8f83\u5927\uff0c\u5c31\u8bf4\u660e\u5b66\u751f\u8eab\u9ad8\u5dee\u5f02\u8f83\u5927\uff0c\u6709\u9ad8\u6709\u77ee\uff0c\u5206\u5e03\u8f83\u4e3a\u5206\u6563 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\\mu)^2}{N}}\\)<\/p>\n\n\n\n<p><strong>\u7b26\u53f7\u542b\u4e49\uff1a<\/strong><\/p>\n\n\n\n<p>\\(\\sigma\\)\uff1a\u4ee3\u8868\u603b\u4f53\u6807\u51c6\u5dee\uff0c\u5b83\u662f\u8861\u91cf\u603b\u4f53\u6570\u636e\u79bb\u6563\u7a0b\u5ea6\u7684\u5173\u952e\u6307\u6807\u3002<\/p>\n\n\n\n<p>\\(\\sum\\)\uff1a\u662f\u6c42\u548c\u7b26\u53f7\uff0c\u8868\u793a\u5bf9\u4e00\u7cfb\u5217\u6570\u503c\u8fdb\u884c\u7d2f\u52a0\u64cd\u4f5c\u3002<\/p>\n\n\n\n<p>\\(x_i\\)\uff1a\u8868\u793a\u603b\u4f53\u4e2d\u7684\u7b2ci\u4e2a\u6570\u636e\u503c\uff0c\u5176\u4e2di\u662f\u4ece 1 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1\u53ef\u4ee5\u4f7f\u6837\u672c\u6807\u51c6\u5dee\u66f4\u597d\u5730\u65e0\u504f\u4f30\u8ba1\u603b\u4f53\u6807\u51c6\u5dee\uff0c\u63d0\u9ad8\u4f30\u8ba1\u7684\u51c6\u786e\u6027\u3002<\/p>\n\n\n<h2 class=\"wp-block-heading\" 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 class=\"wp-block-heading\" id=\"h_17830577910_4\">&#22235;&#12289;&#26631;&#20934;&#24046;&#30340;&#37325;&#35201;&#24615;<\/h2>\n\n\n\n<p><strong>\uff08\u4e00\uff09\u5728\u7edf\u8ba1\u5b66\u4e2d\u7684\u6838\u5fc3\u5730\u4f4d<\/strong><\/p>\n\n\n\n<p>\u6807\u51c6\u5dee\u662f\u7edf\u8ba1\u5b66\u7684\u6838\u5fc3\u6982\u5ff5\u4e4b\u4e00\uff0c\u5b83\u4e0e\u5747\u503c\u3001\u4e2d\u4f4d\u6570\u7b49\u5171\u540c\u6784\u6210\u4e86\u63cf\u8ff0\u6027\u7edf\u8ba1\u7684\u91cd\u8981\u5185\u5bb9\u3002\u5728\u6574\u4e2a\u7edf\u8ba1\u5b66\u4f53\u7cfb\u4e2d\uff0c\u6807\u51c6\u5dee\u8d2f\u7a7f\u4e8e\u5404\u79cd\u7edf\u8ba1\u65b9\u6cd5\u548c\u7406\u8bba\u4e4b\u4e2d\u3002\u65e0\u8bba\u662f\u53c2\u6570\u4f30\u8ba1\u4e2d\u5bf9\u603b\u4f53\u53c2\u6570\u7684\u63a8\u65ad\uff0c\u8fd8\u662f\u5047\u8bbe\u68c0\u9a8c\u4e2d\u5bf9\u539f\u5047\u8bbe\u7684\u5224\u65ad\uff0c\u4ea6\u6216\u662f\u65b9\u5dee\u5206\u6790\u4e2d\u5bf9\u4e0d\u540c\u7ec4\u6570\u636e\u5dee\u5f02\u7684\u5206\u6790\uff0c\u6807\u51c6\u5dee\u90fd\u626e\u6f14\u7740\u4e0d\u53ef\u6216\u7f3a\u7684\u89d2\u8272\u3002\u5b83\u4e3a\u6211\u4eec\u7406\u89e3\u548c\u5206\u6790\u6570\u636e\u63d0\u4f9b\u4e86\u5173\u952e\u7684\u89c6\u89d2\uff0c\u662f\u8fdb\u884c\u79d1\u5b66\u7edf\u8ba1\u63a8\u65ad\u7684\u57fa\u77f3\u3002<\/p>\n\n\n\n<p><strong>\uff08\u4e8c\uff09\u5728\u51b3\u7b56\u5236\u5b9a\u4e2d\u7684\u5e94\u7528\u4ef7\u503c<\/strong><\/p>\n\n\n\n<p>\u5728\u5546\u4e1a\u3001\u7ecf\u6d4e\u3001\u5de5\u7a0b\u7b49\u4f17\u591a\u9886\u57df\u7684\u51b3\u7b56\u8fc7\u7a0b\u4e2d\uff0c\u6807\u51c6\u5dee\u90fd\u5177\u6709\u6781\u9ad8\u7684\u5e94\u7528\u4ef7\u503c\u3002\u4f01\u4e1a\u5728\u5236\u5b9a\u751f\u4ea7\u8ba1\u5212\u65f6\uff0c\u9700\u8981\u8003\u8651\u4ea7\u54c1\u8d28\u91cf\u6570\u636e\u7684\u6807\u51c6\u5dee\uff0c\u4ee5\u786e\u4fdd\u751f\u4ea7\u7684\u7a33\u5b9a\u6027\u548c\u4ea7\u54c1\u8d28\u91cf\u7684\u4e00\u81f4\u6027\uff1b\u6295\u8d44\u8005\u5728\u8fdb\u884c\u6295\u8d44\u51b3\u7b56\u65f6\uff0c\u4f1a\u53c2\u8003\u5404\u7c7b\u8d44\u4ea7\u7684\u6536\u76ca\u7387\u6807\u51c6\u5dee\u6765\u8bc4\u4f30\u98ce\u9669\uff0c\u8fdb\u800c\u5236\u5b9a\u5408\u7406\u7684\u6295\u8d44\u7ec4\u5408\u7b56\u7565\uff1b\u653f\u5e9c\u90e8\u95e8\u5728\u5236\u5b9a\u5b8f\u89c2\u7ecf\u6d4e\u653f\u7b56\u65f6\uff0c\u4e5f\u4f1a\u4f9d\u636e\u76f8\u5173\u7ecf\u6d4e\u6570\u636e\u7684\u6807\u51c6\u5dee\u6765\u5206\u6790\u7ecf\u6d4e\u8fd0\u884c\u7684\u7a33\u5b9a\u6027\u548c\u6ce2\u52a8\u6027\uff0c\u4e3a\u653f\u7b56\u7684\u5236\u5b9a\u63d0\u4f9b\u79d1\u5b66\u4f9d\u636e\u3002<\/p>\n\n\n\n<p><strong>\uff08\u4e09\uff09\u5bf9\u7814\u7a76\u548c\u5b9e\u9a8c\u7684\u91cd\u8981\u610f\u4e49<\/strong><\/p>\n\n\n\n<p>\u5728\u79d1\u5b66\u7814\u7a76\u548c\u5b9e\u9a8c\u4e2d\uff0c\u6807\u51c6\u5dee\u5bf9\u4e8e\u8bc4\u4f30\u5b9e\u9a8c\u7ed3\u679c\u7684\u51c6\u786e\u6027\u548c\u53ef\u9760\u6027\u8d77\u7740\u5173\u952e\u4f5c\u7528\u3002\u79d1\u7814\u4eba\u5458\u901a\u8fc7\u5206\u6790\u5b9e\u9a8c\u6570\u636e\u7684\u6807\u51c6\u5dee\uff0c\u53ef\u4ee5\u5224\u65ad\u5b9e\u9a8c\u8fc7\u7a0b\u662f\u5426\u5b58\u5728\u8f83\u5927\u7684\u8bef\u5dee\uff0c\u5b9e\u9a8c\u7ed3\u679c\u662f\u5426\u5177\u6709\u53ef\u91cd\u590d\u6027\u3002\u5982\u679c\u5b9e\u9a8c\u6570\u636e\u7684\u6807\u51c6\u5dee\u8f83\u5c0f\uff0c\u8bf4\u660e\u5b9e\u9a8c\u7ed3\u679c\u8f83\u4e3a\u7a33\u5b9a\u3001\u53ef\u9760\uff0c\u5177\u6709\u8f83\u9ad8\u7684\u53ef\u4fe1\u5ea6\uff1b\u53cd\u4e4b\uff0c\u5982\u679c\u6807\u51c6\u5dee\u8f83\u5927\uff0c\u5219\u9700\u8981\u5bf9\u5b9e\u9a8c\u65b9\u6cd5\u3001\u4eea\u5668\u8bbe\u5907\u3001\u64cd\u4f5c\u6d41\u7a0b\u7b49\u65b9\u9762\u8fdb\u884c\u5168\u9762\u68c0\u67e5\uff0c\u627e\u51fa\u53ef\u80fd\u5bfc\u81f4\u6570\u636e\u6ce2\u52a8\u7684\u539f\u56e0\uff0c\u4ee5\u63d0\u9ad8\u5b9e\u9a8c\u7684\u51c6\u786e\u6027\u548c\u79d1\u5b66\u6027\u3002<\/p>\n\n\n\n<p><strong>\uff08\u56db\uff09\u5728\u98ce\u9669\u7ba1\u7406\u4e2d\u7684\u5173\u952e\u4f5c\u7528<\/strong><\/p>\n\n\n\n<p>\u5728\u98ce\u9669\u7ba1\u7406\u9886\u57df\uff0c\u6807\u51c6\u5dee\u662f\u8861\u91cf\u98ce\u9669\u7684\u91cd\u8981\u6307\u6807\u4e4b\u4e00\u3002\u5b83\u80fd\u591f\u5e2e\u52a9\u6211\u4eec\u91cf\u5316\u98ce\u9669\u7684\u5927\u5c0f\uff0c\u4ece\u800c\u91c7\u53d6\u76f8\u5e94\u7684\u98ce\u9669\u63a7\u5236\u63aa\u65bd\u3002\u5728\u91d1\u878d\u98ce\u9669\u7ba1\u7406\u4e2d\uff0c\u6807\u51c6\u5dee\u88ab\u5e7f\u6cdb\u5e94\u7528\u4e8e\u8861\u91cf\u6295\u8d44\u7ec4\u5408\u7684\u98ce\u9669\u6c34\u5e73\u3002\u901a\u8fc7\u8ba1\u7b97\u6295\u8d44\u7ec4\u5408\u4e2d\u5404\u79cd\u8d44\u4ea7\u7684\u6807\u51c6\u5dee\u4ee5\u53ca\u5b83\u4eec\u4e4b\u95f4\u7684\u76f8\u5173\u6027\uff0c\u6295\u8d44\u8005\u53ef\u4ee5\u51c6\u786e\u8bc4\u4f30\u6574\u4e2a\u6295\u8d44\u7ec4\u5408\u7684\u98ce\u9669\u72b6\u51b5\uff0c\u5e76\u6839\u636e\u81ea\u5df1\u7684\u98ce\u9669\u627f\u53d7\u80fd\u529b\u8fdb\u884c\u5408\u7406\u7684\u8d44\u4ea7\u914d\u7f6e\u3002\u5728\u5de5\u7a0b\u9879\u76ee\u7ba1\u7406\u4e2d\uff0c\u6807\u51c6\u5dee\u53ef\u4ee5\u7528\u4e8e\u8bc4\u4f30\u9879\u76ee\u8fdb\u5ea6\u3001\u6210\u672c\u7b49\u65b9\u9762\u7684\u98ce\u9669\uff0c\u5e2e\u52a9\u9879\u76ee\u7ba1\u7406\u8005\u63d0\u524d\u8bc6\u522b\u6f5c\u5728\u7684\u98ce\u9669\u56e0\u7d20\uff0c\u5236\u5b9a\u6709\u6548\u7684\u98ce\u9669\u5e94\u5bf9\u7b56\u7565\uff0c\u964d\u4f4e\u98ce\u9669\u53d1\u751f\u7684\u53ef\u80fd\u6027\u548c\u5f71\u54cd\u7a0b\u5ea6\u3002<\/p>\n\n\n<h2 class=\"wp-block-heading\" id=\"h_17830577910_5\">&#20116;&#12289;&#22914;&#20309;&#35745;&#31639;&#26631;&#20934;&#24046;<\/h2>\n\n\n\n<p><strong>\uff08\u4e00\uff09\u603b\u4f53\u6807\u51c6\u5dee\u7684\u8ba1\u7b97\u6b65\u9aa4<\/strong><\/p>\n\n\n\n<p><strong>1\uff09\u8ba1\u7b97\u603b\u4f53\u5747\u503c\uff1a<\/strong><\/p>\n\n\n\n<p>\u5c06\u603b\u4f53\u4e2d\u6240\u6709\u6570\u636e\u503c\u76f8\u52a0\uff0c\u7136\u540e\u9664\u4ee5\u6570\u636e\u7684\u603b\u4e2a\u6570\uff0c\u5373\u5f97\u5230\u603b\u4f53\u5747\u503c\u03bc\u3002<\/p>\n\n\n\n<p>\u4f8b\u5982\uff0c\u5bf9\u4e8e\u603b\u4f53\u6570\u636e3, 5, 7, 9, 11\uff0c\u603b\u4f53\u5747\u503c\\(\\mu = \\frac{3 + 5 + 7 + 9 + 11}{5} = 7\\)\u3002<\/p>\n\n\n\n<p><strong>2\uff09\u8ba1\u7b97\u6bcf\u4e2a\u6570\u636e\u503c\u4e0e\u5747\u503c\u7684\u5dee\u503c\uff1a<\/strong><\/p>\n\n\n\n<p>\u5bf9\u4e8e\u603b\u4f53\u4e2d\u7684\u6bcf\u4e2a\u6570\u636e\u503c\\(x_i\\)\uff0c\u8ba1\u7b97\\(x_i &#8211; \\mu\\)\u3002<\/p>\n\n\n\n<p>\u4ee5\u4e0a\u8ff0\u6570\u636e\u4e3a\u4f8b\uff0c\u5dee\u503c\u5206\u522b\u4e3a3 &#8211; 7 = -4\uff0c 5 &#8211; 7 = -2\uff0c 7 &#8211; 7 = 0\uff0c 9 &#8211; 7 = 2\uff0c 11 &#8211; 7 = 4\u3002<\/p>\n\n\n\n<p><strong>3\uff09\u8ba1\u7b97\u5dee\u503c\u7684\u5e73\u65b9\uff1a<\/strong><\/p>\n\n\n\n<p>\u5c06\u6bcf\u4e2a\u5dee\u503c\u8fdb\u884c\u5e73\u65b9\uff0c\u5f97\u5230\\((x_i &#8211; \\mu)^2\\)\u3002<\/p>\n\n\n\n<p>\u4e0a\u8ff0\u5dee\u503c\u7684\u5e73\u65b9\u5206\u522b\u4e3a \\((-4)^2=16\\)\uff0c\\((-2)^2=4\\)\uff0c\\(0^2=0\\) \uff0c\\(2^2=4\\) \uff0c \\((-4)^2=16\\)\u3002<\/p>\n\n\n\n<p><strong>4\uff09\u8ba1\u7b97\u5e73\u65b9\u548c\uff1a<\/strong><\/p>\n\n\n\n<p>\u5c06\u6240\u6709\u5dee\u503c\u7684\u5e73\u65b9\u76f8\u52a0\uff0c\u5373<\/p>\n\n\n\n<p class=\"has-text-align-center\">\\(\\sum_{i=1}^{N} (x_i &#8211; \\mu)^2\\)<\/p>\n\n\n\n<p>\u5728\u8fd9\u4e2a\u4f8b\u5b50\u4e2d\uff0c\u5e73\u65b9\u548c\u4e3a16 + 4 + 0 + 4 + 16 = 40\u3002<\/p>\n\n\n\n<p><strong>5\uff09\u8ba1\u7b97\u603b\u4f53\u6807\u51c6\u5dee\uff1a<\/strong><\/p>\n\n\n\n<p>\u5c06\u5e73\u65b9\u548c\u9664\u4ee5\u603b\u4f53\u6570\u636e\u7684\u4e2a\u6570N\uff0c\u7136\u540e\u5bf9\u7ed3\u679c\u53d6\u5e73\u65b9\u6839\uff0c\u5373<\/p>\n\n\n\n<p class=\"has-text-align-center\">\\(\\sigma = \\sqrt{\\frac{\\sum_{i=1}^{N} (x_i &#8211; \\mu)^2}{N}}\\)<\/p>\n\n\n\n<p>\u6240\u4ee5\uff0c\u8be5\u603b\u4f53\u7684\u6807\u51c6\u5dee\\(\\sigma = \\sqrt{\\frac{40}{5}} = \\sqrt{8} \\approx 2.83\\)\u3002<\/p>\n\n\n\n<p><strong>\uff08\u4e8c\uff09\u6837\u672c\u6807\u51c6\u5dee\u7684\u8ba1\u7b97\u6b65\u9aa4<\/strong><\/p>\n\n\n\n<p>\u6837\u672c\u6807\u51c6\u5dee\u7684\u8ba1\u7b97\u6b65\u9aa4\u4e0e\u603b\u4f53\u6807\u51c6\u5dee\u7c7b\u4f3c\uff0c\u4f46\u5728\u6700\u540e\u4e00\u6b65\u6709\u6240\u4e0d\u540c\u3002<\/p>\n\n\n\n<p><strong>1\uff09\u8ba1\u7b97\u6837\u672c\u5747\u503c\uff1a<\/strong><\/p>\n\n\n\n<p>\u5c06\u6837\u672c\u4e2d\u6240\u6709\u6570\u636e\u503c\u76f8\u52a0\uff0c\u518d\u9664\u4ee5\u6837\u672c\u6570\u636e\u7684\u4e2a\u6570\uff0c\u5f97\u5230\u6837\u672c\u5747\u503c\\(\\overline{x}\\)\u3002\u4f8b\u5982\uff0c\u5bf9\u4e8e\u6837\u672c\u6570\u636e 2, 4, 6, 8 \uff0c\u6837\u672c\u5747\u503c \\(\\overline{x}= \\frac{2 + 4 + 6 + 8 }{4} = 5\\)\u3002<\/p>\n\n\n\n<p><strong>2\uff09\u8ba1\u7b97\u6bcf\u4e2a\u6570\u636e\u503c\u4e0e\u5747\u503c\u7684\u5dee\u503c\uff1a<\/strong><\/p>\n\n\n\n<p>\u8ba1\u7b97\\(x_i &#8211; \\overline{x}\\)\uff0c\u5dee\u503c\u5206\u522b\u4e3a 2 &#8211; 5 = -3 \uff0c 4 &#8211; 5 = -1 \uff0c 6 &#8211; 5 = 1 \uff0c 8 &#8211; 5 = 3 \u3002<\/p>\n\n\n\n<p><strong>3\uff09\u8ba1\u7b97\u5dee\u503c\u7684\u5e73\u65b9\uff1a<\/strong><\/p>\n\n\n\n<p>\u5f97\u5230\\((x_i &#8211; \\overline{x})^2\\)\uff0c\u5206\u522b\u4e3a\\((-3)^2=9\\)\uff0c\\((-1)^2=1\\) \uff0c \\(1^2=1\\)\uff0c \\(3^2=9\\) \u3002<\/p>\n\n\n\n<p><strong>4\uff09\u8ba1\u7b97\u5e73\u65b9\u548c\uff1a<\/strong><\/p>\n\n\n\n<p>\u5c06\u6240\u6709\u5dee\u503c\u7684\u5e73\u65b9\u76f8\u52a0\uff0c\u5373<\/p>\n\n\n\n<p class=\"has-text-align-center\">\\(\\sum_{i=1}^{N} (x_i &#8211; \\mu)^2\\)<\/p>\n\n\n\n<p>\u6b64\u4f8b\u4e2d\u5e73\u65b9\u548c\u4e3a9 + 1 + 1 + 9 = 20 \u3002<\/p>\n\n\n\n<p><strong>5\uff09\u8ba1\u7b97\u6837\u672c\u6807\u51c6\u5dee\uff1a<\/strong><\/p>\n\n\n\n<p>\u5c06\u5e73\u65b9\u548c\u9664\u4ee5\u6837\u672c\u6570\u636e\u4e2a\u6570\u51cf 1\uff08\u5373n &#8211; 1\uff09\uff0c\u7136\u540e\u53d6\u5e73\u65b9\u6839\uff0c\u5373<\/p>\n\n\n\n<p class=\"has-text-align-center\">\\(s = \\sqrt{\\frac{\\sum_{i=1}^{n} (x_i &#8211; \\overline{x})^2}{n-1}}\\)<\/p>\n\n\n\n<p>\u6240\u4ee5\uff0c\u8be5\u6837\u672c\u7684\u6807\u51c6\u5dee \\(s = \\sqrt{\\frac{20}{(4-1)}} = \\sqrt{\\frac{20}{3}} \\approx 2.58\\)\u3002<\/p>\n\n\n\n<p>\u5728\u5b9e\u9645\u8ba1\u7b97\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u501f\u52a9\u7edf\u8ba1\u8f6f\u4ef6\uff08\u5982SPSS\u3001R\u3001Python\u4e2d\u7684\u76f8\u5173\u5e93\uff09\u6216\u7535\u5b50\u8868\u683c\u8f6f\u4ef6\uff08\u5982Excel\uff09\u6765\u5feb\u901f\u51c6\u786e\u5730\u8ba1\u7b97\u6807\u51c6\u5dee\uff0c\u63d0\u9ad8\u8ba1\u7b97\u6548\u7387\u548c\u51c6\u786e\u6027\u3002<\/p>\n\n\n<h2 class=\"wp-block-heading\" id=\"h_17830577910_6\">&#20845;&#12289;&#26631;&#20934;&#24046;&#31034;&#20363;<\/h2>\n\n\n\n<p><strong>\u5b66\u751f\u6210\u7ee9\u5206\u6790\uff1a<\/strong><\/p>\n\n\n\n<p>\u5047\u8bbe\u6709\u4e24\u4e2a\u73ed\u7ea7\u7684\u82f1\u8bed\u8003\u8bd5\u6210\u7ee9\u5982\u4e0b\uff1a<\/p>\n\n\n\n<p>\u4e00\u73ed\uff1a75\uff0c80\uff0c85\uff0c90\uff0c95<\/p>\n\n\n\n<p>\u4e8c\u73ed\uff1a60\uff0c70\uff0c80\uff0c90\uff0c100<\/p>\n\n\n\n<p><strong>1\uff09\u8ba1\u7b97\u4e00\u73ed\u6210\u7ee9\u7684\u76f8\u5173\u6570\u636e\uff1a<\/strong><\/p>\n\n\n\n<p><strong>\u5747\u503c\uff1a<\/strong><\/p>\n\n\n\n<p class=\"has-text-align-center\">\\(\\overline{x_1} = \\frac{75 + 80 + 85 + 90 + 95}{5} = 85\\)<\/p>\n\n\n\n<p><strong>\u6807\u51c6\u5dee<\/strong>\uff1a<\/p>\n\n\n\n<p>\\(s_1 = \\sqrt{\\frac{(75-85)^2 + (80-85)^2 + (85-85)^2 + (90-85)^2 + (95-85)^2}{5-1}}\\)\\(\\\\ = \\sqrt{\\frac{(-10)^2 + (-5)^2 + 0^2 + 5^2 + 10^2}{4}}\\\\ = \\sqrt{\\frac{100+ 25 + 0 + 25 + 100}{4}}\\\\ = \\sqrt{\\frac{250}{4}}\\\\ = \\sqrt{62.5} \\\\ \\approx 7.91\\)<\/p>\n\n\n\n<p><strong>2\uff09\u8ba1\u7b97\u4e8c\u73ed\u6210\u7ee9\u7684\u76f8\u5173\u6570\u636e\uff1a<\/strong><\/p>\n\n\n\n<p><strong>\u5747\u503c<\/strong>\uff1a<\/p>\n\n\n\n<p class=\"has-text-align-center\">\\(\\overline{x_2} = \\frac{60 + 70 + 80 + 90 + 100}{5} = 80\\)<\/p>\n\n\n\n<p><strong>\u6807\u51c6\u5dee<\/strong>\uff1a<\/p>\n\n\n\n<p>\\(s_2 = \\sqrt{\\frac{(60-80)^2 + (70-80)^2 + (80-80)^2 + (90-80)^2 + (100-80)^2}{5-1}}\\)\\(\\\\ = \\sqrt{\\frac{(-20)^2 + (-10)^2 + 0^2 + 10^2 + 20^2}{4}}\\\\ = \\sqrt{\\frac{400+ 100 + 0 + 100 + 400}{4}}\\\\ = \\sqrt{\\frac{1000}{4}}\\\\ \\approx 15.81\\)<\/p>\n\n\n\n<p>\u4ece\u8ba1\u7b97\u7ed3\u679c\u53ef\u4ee5\u770b\u51fa\uff0c\u4e00\u73ed\u6210\u7ee9\u7684\u6807\u51c6\u5dee\u7ea6\u4e3a7.91\uff0c\u4e8c\u73ed\u6210\u7ee9\u7684\u6807\u51c6\u5dee\u4e3a15.81\u3002\u8fd9\u8868\u660e\u4e00\u73ed\u5b66\u751f\u7684\u6210\u7ee9\u76f8\u5bf9\u66f4\u4e3a\u96c6\u4e2d\uff0c\u56f4\u7ed5\u5e73\u5747\u503c\u7684\u6ce2\u52a8\u8f83\u5c0f\uff1b\u800c\u4e8c\u73ed\u5b66\u751f\u7684\u6210\u7ee9\u5206\u5e03\u66f4\u4e3a\u5206\u6563\uff0c\u5b66\u751f\u4e4b\u95f4\u7684\u6210\u7ee9\u5dee\u5f02\u8f83\u5927\u3002\u6559\u5e08\u53ef\u4ee5\u6839\u636e\u8fd9\u4e9b\u6570\u636e\u8c03\u6574\u6559\u5b66\u7b56\u7565\uff0c\u5bf9\u4e8e\u4e00\u73ed\u53ef\u4ee5\u6ce8\u91cd\u57f9\u4f18\uff0c\u5bf9\u4e8e\u4e8c\u73ed\u5219\u9700\u8981\u52a0\u5f3a\u5bf9\u57fa\u7840\u8584\u5f31\u5b66\u751f\u7684\u8f85\u5bfc\u3002<\/p>\n\n\n<h2 class=\"wp-block-heading\" id=\"h_17830577910_7\">&#19971;&#12289;&#24120;&#35265;&#38382;&#39064;<\/h2>\n\n\n\n<p><strong>1\u3001\u6807\u51c6\u5dee\u4e0e\u65b9\u5dee\u6709\u4ec0\u4e48\u5173\u7cfb\uff1f<\/strong><\/p>\n\n\n\n<p><strong>\u65b9\u5dee\u662f\u6807\u51c6\u5dee\u7684\u5e73\u65b9\uff0c\u5373\u6807\u51c6\u5dee\u662f\u65b9\u5dee\u7684\u5e73\u65b9\u6839\u3002<\/strong>\u65b9\u5dee\u548c\u6807\u51c6\u5dee\u90fd\u7528\u4e8e\u8861\u91cf\u6570\u636e\u7684\u79bb\u6563\u7a0b\u5ea6\uff0c\u5b83\u4eec\u7684\u4f5c\u7528\u76f8\u4f3c\uff0c\u4f46\u6807\u51c6\u5dee\u7684\u5355\u4f4d\u4e0e\u539f\u59cb\u6570\u636e\u7684\u5355\u4f4d\u76f8\u540c\uff0c\u8fd9\u4f7f\u5f97\u6807\u51c6\u5dee\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\u66f4\u5177\u76f4\u89c2\u6027\uff0c\u66f4\u4fbf\u4e8e\u7406\u89e3\u6570\u636e\u7684\u79bb\u6563\u7a0b\u5ea6\u76f8\u5bf9\u4e8e\u539f\u59cb\u6570\u636e\u7684\u610f\u4e49\u3002\u800c\u65b9\u5dee\u7531\u4e8e\u5355\u4f4d\u662f\u539f\u59cb\u6570\u636e\u5355\u4f4d\u7684\u5e73\u65b9\uff0c\u5728\u89e3\u91ca\u4e0a\u76f8\u5bf9\u4e0d\u90a3\u4e48\u76f4\u89c2\u3002<\/p>\n\n\n\n<p><strong>2\u3001\u6807\u51c6\u5dee\u4e3a\u201c0\u201d\u610f\u5473\u7740\u4ec0\u4e48\uff1f<\/strong><\/p>\n\n\n\n<p><strong>\u6807\u51c6\u5dee\u4e3a0\u8868\u793a\u6570\u636e\u96c6\u4e2d\u6240\u6709\u7684\u6570\u636e\u503c\u90fd\u5b8c\u5168\u76f8\u540c\u3002<\/strong>\u56e0\u4e3a\u6807\u51c6\u5dee\u8861\u91cf\u7684\u662f\u6570\u636e\u76f8\u5bf9\u4e8e\u5e73\u5747\u503c\u7684\u79bb\u6563\u7a0b\u5ea6\uff0c\u5982\u679c\u6240\u6709\u6570\u636e\u90fd\u4e00\u6837\uff0c\u90a3\u4e48\u5b83\u4eec\u4e0e\u5e73\u5747\u503c\u7684\u5dee\u503c\u90fd\u4e3a0\uff0c\u7ecf\u8fc7\u8ba1\u7b97\u5f97\u5230\u7684\u6807\u51c6\u5dee\u4e5f\u5c31\u4e3a0\u3002\u4f8b\u5982\u6570\u636e\u96c6 5, 5, 5, 5 \uff0c\u5176\u6807\u51c6\u5dee\u5c31\u662f0\u3002<\/p>\n\n\n\n<p><strong>3\u3001\u6807\u51c6\u5dee\u8d8a\u5927\u8d8a\u597d\u8fd8\u662f\u8d8a\u5c0f\u8d8a\u597d\uff1f<\/strong><\/p>\n\n\n\n<p><strong>\u6807\u51c6\u5dee\u7684\u5927\u5c0f\u4f18\u52a3\u4e0d\u80fd\u4e00\u6982\u800c\u8bba\uff0c\u8fd9\u53d6\u51b3\u4e8e\u5177\u4f53\u7684\u5e94\u7528\u573a\u666f\u548c\u5206\u6790\u76ee\u7684<\/strong>\u3002\u5728\u4e00\u4e9b\u9700\u8981\u6570\u636e\u7a33\u5b9a\u3001\u4e00\u81f4\u6027\u9ad8\u7684\u573a\u666f\u4e2d\uff0c\u5982\u4ea7\u54c1\u8d28\u91cf\u63a7\u5236\u3001\u7cbe\u5bc6\u4eea\u5668\u5236\u9020\u7b49\uff0c\u8f83\u5c0f\u7684\u6807\u51c6\u5dee\u610f\u5473\u7740\u4ea7\u54c1\u8d28\u91cf\u7a33\u5b9a\u3001\u6027\u80fd\u53ef\u9760\uff0c\u662f\u66f4\u7406\u60f3\u7684\u72b6\u6001\uff1b\u800c\u5728\u67d0\u4e9b\u60c5\u51b5\u4e0b\uff0c\u5982\u98ce\u9669\u6295\u8d44\u9886\u57df\uff0c\u8f83\u5927\u7684\u6807\u51c6\u5dee\u53ef\u80fd\u610f\u5473\u7740\u6709\u66f4\u5927\u7684\u6f5c\u5728\u6536\u76ca\u7a7a\u95f4\uff0c\u5bf9\u4e8e\u98ce\u9669\u504f\u597d\u8f83\u9ad8\u7684\u6295\u8d44\u8005\u6765\u8bf4\uff0c\u53ef\u80fd\u4f1a\u66f4\u5173\u6ce8\u6807\u51c6\u5dee\u8f83\u5927\u7684\u6295\u8d44\u6807\u7684\u3002\u6240\u4ee5\uff0c\u9700\u8981\u6839\u636e\u5177\u4f53\u60c5\u51b5\u6765\u8bc4\u4f30\u6807\u51c6\u5dee\u7684\u5927\u5c0f\u662f\u5426\u5408\u9002\u3002<\/p>\n\n\n\n<p><strong>4\u3001\u6837\u672c\u6807\u51c6\u5dee\u4e0e\u603b\u4f53\u6807\u51c6\u5dee\u7684\u533a\u522b\u53ea\u662f\u516c\u5f0f\u4e0a\u5206\u6bcd\u4e0d\u540c\u5417\uff1f<\/strong><\/p>\n\n\n\n<p>\u6837\u672c\u6807\u51c6\u5dee\u4e0e\u603b\u4f53\u6807\u51c6\u5dee\u5728\u516c\u5f0f\u4e0a\u5206\u6bcd\u786e\u5b9e\u5b58\u5728\u5dee\u5f02\uff0c\u603b\u4f53\u6807\u51c6\u5dee\u516c\u5f0f\u5206\u6bcd\u662f\u603b\u4f53\u6570\u91cf N \uff0c\u6837\u672c\u6807\u51c6\u5dee\u516c\u5f0f\u5206\u6bcd\u662f\u6837\u672c\u6570\u91cf\u51cf1\uff0c\u5373 n \u2013 1 \u3002\u4f46\u5b83\u4eec\u7684\u533a\u522b\u4e0d\u6b62\u4e8e\u6b64\u3002<strong>\u603b\u4f53\u6807\u51c6\u5dee\u662f\u5bf9\u6574\u4e2a\u603b\u4f53\u6570\u636e\u79bb\u6563\u7a0b\u5ea6\u7684\u7cbe\u786e\u5ea6\u91cf\uff0c\u53cd\u6620\u7684\u662f\u603b\u4f53\u7684\u771f\u5b9e\u60c5\u51b5\uff1b\u800c\u6837\u672c\u6807\u51c6\u5dee\u662f\u57fa\u4e8e\u4ece\u603b\u4f53\u4e2d\u62bd\u53d6\u7684\u6837\u672c\u6570\u636e\u6765\u8ba1\u7b97\u7684\uff0c\u76ee\u7684\u662f\u901a\u8fc7\u6837\u672c\u53bb\u4f30\u8ba1\u603b\u4f53\u7684\u79bb\u6563\u7a0b\u5ea6\u3002<\/strong>\u7531\u4e8e\u6837\u672c\u53ea\u662f\u603b\u4f53\u7684\u4e00\u90e8\u5206\uff0c\u5b58\u5728\u4e00\u5b9a\u7684\u968f\u673a\u6027\u548c\u4e0d\u786e\u5b9a\u6027\uff0c\u4f7f\u7528 n \u2013 1 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