{"id":69367,"date":"2016-03-11T08:49:45","date_gmt":"2016-03-11T00:49:45","guid":{"rendered":"http:\/\/highscope.ch.ntu.edu.tw\/wordpress\/?p=69367"},"modified":"2021-10-06T16:08:02","modified_gmt":"2021-10-06T08:08:02","slug":"%e6%af%8d%e9%ab%94%e8%ae%8a%e7%95%b0%e6%95%b8v-s-%e6%a8%a3%e6%9c%ac%e8%ae%8a%e7%95%b0%e6%95%b8","status":"publish","type":"post","link":"http:\/\/localhost\/%e6%af%8d%e9%ab%94%e8%ae%8a%e7%95%b0%e6%95%b8v-s-%e6%a8%a3%e6%9c%ac%e8%ae%8a%e7%95%b0%e6%95%b8\/","title":{"rendered":"\u6bcd\u9ad4\u8b8a\u7570\u6578v.s.\u6a23\u672c\u8b8a\u7570\u6578"},"content":{"rendered":"

\u6bcd\u9ad4\u8b8a\u7570\u6578(\\(\\sigma^2\\))v.s.\u6a23\u672c\u8b8a\u7570\u6578(\\(s^2\\))<\/strong><\/span>
\n\u570b\u7acb\u81fa\u7063\u5927\u5b78\u8fb2\u85dd\u5b78\u7cfb \u5433\u535a\u96c5<\/strong><\/span><\/p>\n

\u4e00\u3001\u524d\u8a00<\/strong><\/p>\n

\u6bcf\u7576\u6536\u96c6\u5b8c\u4e00\u7b46\u8cc7\u6599\u5f8c\uff0c\u53ef\u80fd\u6703\u975e\u5e38\u96f6\u4e82\u3001\u8907\u96dc\uff0c\u5f88\u96e3\u770b\u51fa\u8a72\u7b46\u8cc7\u6599\u7684\u7279\u6027\uff0c\u90a3\u6211\u5011\u53c8\u5982\u4f55\u6574\u7406\u9019\u4e9b\u8cc7\u6599\u5462\uff1f\u5e38\u5e38\u6703\u756b\u5716\u8868\u793a\u8cc7\u6599\u7684\u5206\u5e03\u60c5\u5f62\uff0c\u4e5f\u6703\u8a08\u7b97\u5176\u5e73\u5747\u6578 (mean)\u3001\u4e2d\u4f4d\u6578 (median)\u3001\u773e\u6578 (mode)\u2026\u7b49\u4f86\u770b\u8a72\u7b46\u8cc7\u6599\u7684\u4e2d\u5fc3\u4f4d\u7f6e\uff0c\u540c\u6642\uff0c\u9084\u6703\u8a08\u7b97\u5168\u8ddd (range)\u3001\u8b8a\u7570\u6578 (variance)…\u7b49\uff0c\u4f86\u770b\u8a72\u7b46\u8cc7\u6599\u7684\u5206\u6563\u7a0b\u5ea6\uff0c\u5982\u6b64\u4e00\u4f86\uff0c\u8cc7\u6599\u6536\u96c6\u8005\u53ef\u4ee5\u7c21\u55ae\u6558\u8ff0\u8a72\u8cc7\u6599\u7684\u7279\u6027\uff0c\u8b93\u6709\u8208\u8da3\u8005\u53ef\u4ee5\u5feb\u901f\u4e86\u89e3\uff0c\u53d6\u5f97\u6240\u9700\u7684\u8cc7\u8a0a\uff0c\u800c\u9019\u985e\u7684\u6578\u64da\u5206\u6790\u53ef\u7d71\u7a31\u70ba\u6558\u8ff0\u7d71\u8a08\u5b78 (Descriptive Statistics)\u3002<\/p>\n

\u4eca\u5929\u6211\u5011\u8981\u7279\u5225\u8ac7\u8ad6\u8b8a\u7570\u6578\uff0c\u8b8a\u7570\u6578\u5728\u9ad8\u4e2d\u8ab2\u672c\u88e1\u8868\u793a\u6210\uff1a<\/p>\n

\\(\\sigma^2=\\displaystyle \\sum_{i=i}^{N}\\frac{(x_i-\\mu)^2}{N}~~~~~~~~~(1.1)\\)<\/p>\n

\u5176\u4e2d \\(x_i\\)\u00a0\u70ba\u5404\u89c0\u6e2c\u503c(\u4e00\u5171 \\(N\\) \u500b\u89c0\u6e2c\u503c\uff0c\u4ea6\u5373\u65cf\u7fa4\u4e2d\u4e00\u5171\u6709 \\(N\\) \u500b\u89c0\u6e2c\u503c)\uff1b\\(\\mu\\)(\u8b80\u4f5cmu)\u70ba\u65cf\u7fa4\u5e73\u5747\u6578\uff0c\u53ef\u8868\u793a\u6210\uff1a<\/p>\n

\\(\\mu=\\displaystyle\\frac{1}{N}(x_1+x_2+\\cdots+x_N)=\\frac{1}{N}\\sum_{i=1}^{N}x_i~~~~~~~~~(1.2)\\)<\/p>\n

\u4e0a\u8ff0\u6240\u63d0\u53ca\u7684\u8b8a\u7570\u6578\u70ba\u6bcd\u9ad4\u8b8a\u7570\u6578\uff0c\u4e8b\u5be6\u4e0a\u9084\u6709\u6a23\u672c\u8b8a\u7570\u6578\uff0c\u516c\u5f0f\u8868\u793a\u6210\uff1a<\/p>\n

\\(s^2=\\displaystyle\\sum_{i=1}^{n}\\frac{(x_i-\\bar{x})^2}{n-1}~~~~~~~~~(1.3)\\)<\/p>\n

\u5176\u4e2d \\(x_i\\)\u00a0\u70ba\u5404\u89c0\u6e2c\u503c(\u5171 \\(n\\) \u500b\u89c0\u6e2c\u503c)\uff1b\\(\\bar{x}\\) \u70ba\u6a23\u672c\u5e73\u5747\u6578\u3002<\/p>\n

\\(\\bar{x}=\\displaystyle\\frac{1}{n}(x_1+x_2+\\cdots+x_n)=\\frac{1}{n}\\sum_{i=1}^{n}x_i~~~~~~~~~(1.4)\\)<\/p>\n

\u4e8c\u3001\u6bcd\u9ad4\u8b8a\u7570\u6578v.s.\u6a23\u672c\u8b8a\u7570\u6578<\/strong><\/p>\n

\u5927\u5bb6\u6216\u8a31\u6703\u5f88\u7591\u60d1\uff0c\u70ba\u4ec0\u9ebc\u6703\u6709\u6bcd\u9ad4\u8b8a\u7570\u6578\u8207\u6a23\u672c\u8b8a\u7570\u6578\u5462\uff1f\u4ed6\u5011\u5f7c\u6b64\u9593\u5b58\u5728\u54ea\u4e9b\u5dee\u7570\u5462\uff1f<\/p>\n

\u5f80\u5f80\u6211\u5011\u6b32\u95dc\u6ce8\u7684\u65cf\u7fa4\u8cc7\u6599\u91cf\u5f88\u5927\u751a\u81f3\u662f\u7121\u9650\u5927\uff0c\u800c\u4e14\u65cf\u7fa4\u7684\u5e73\u5747\u6578(\\(\\mu\\))\u5be6\u969b\u4e0a\u5e38\u5e38\u7121\u6cd5\u77e5\u9053\uff0c\u70ba\u4e86\u6e1b\u5c11\u8abf\u67e5\u6210\u672c\u8207\u589e\u52a0\u6548\u7387\uff0c\u5e38\u5e38\u6703\u85c9\u7531\u62bd\u6a23(sampling)\u53d6\u5f97\u6a23\u672c\u8cc7\u6599\uff0c\u5e0c\u671b\u80fd\u85c9\u7531\u6a23\u672c\u8cc7\u6599\uff0c\u7372\u5f97\u6a23\u672c\u5e73\u5747\u6578\u8207\u6a23\u672c\u8b8a\u7570\u6578\uff0c\u5229\u7528\u6a23\u672c\u5e73\u5747\u6578(\\(\\bar{x}\\))\u4f86\u4f30\u8a08\u65cf\u7fa4\u5e73\u5747\u6578(\\(\\mu\\))\uff0c\u8207\u5229\u7528\u6a23\u672c\u8b8a\u7570\u6578(\\(s^2\\))\u4f86\u4f30\u8a08\u6bcd\u9ad4\u8b8a\u7570\u6578(\\(\\sigma^2\\))\uff0c\u9032\u800c\u4e86\u89e3\u6574\u500b\u65cf\u7fa4\u7684\u72c0\u6cc1(\u5716\u4e00)\u3002\u81f3\u65bc\u600e\u6a23\u624d\u662f\u597d\u7684\u62bd\u6a23\uff0c\u624d\u80fd\u6e96\u78ba\u4f30\u8a08\u65cf\u7fa4\uff0c\u8acb\u8a73\u898b\u5176\u4ed6\u7ae0\u7bc0\uff0c\u5728\u6b64\u4e0d\u52a0\u4ee5\u8457\u58a8\u3002<\/p>\n

\"69367_p1\"<\/a>

\u5716\u4e00 \u65cf\u7fa4\u8207\u6a23\u672c\u7684\u95dc\u4fc2(\u672c\u6587\u4f5c\u8005\u5433\u535a\u96c5\u88fd\uff09<\/p><\/div>\n

\u800c\u4ee5\u6a23\u672c\u8cc7\u6599\u6c42\u5176\u8b8a\u7570\u6578\uff0c\u7a31\u4e4b\u6a23\u672c\u8b8a\u7570\u6578\uff0c\u53c8\u53ef\u7a31\u70ba\u5747\u65b9(mean square)\uff0c\u5982\u5f0f\u5b501.3\u3002<\/p>\n

\u5747\u65b9\u516c\u5f0f\u4e2d\u5728\u5206\u5b50\u90e8\u5206\uff0c\u6211\u5011\u7a31\u4e4b\u70ba\u5e73\u65b9\u548c(sum of squares)\uff0c\u5c07\u6bcf\u4e00\u500b\u89c0\u6e2c\u503c\u8207\u6a23\u672c\u5e73\u5747\u6578\u4e4b\u5dee\u4e88\u4ee5\u5e73\u65b9\u518d\u52a0\u7e3d\u8d77\u4f86\uff1b\u5747\u65b9\u5728\u5206\u6bcd\u90e8\u5206\u662f \\(n-1\\) \u800c\u4e0d\u662f \\(n\\)\uff0c\u5176\u539f\u56e0\u70ba\u5982\u679c\u4ee5 \\(n\\) \u53d6\u4ee3 \\(n-1\\) \u6703\u9020\u6210\u7576\u4ee5\u6a23\u672c\u8b8a\u7570\u6578\u4f86\u4f30\u8a08\u6bcd\u7fa4\u9ad4\u8b8a\u7570\u6578\u6642\uff0c\u6703\u767c\u751f\u4f4e\u4f30(underestimate)\u7684\u73fe\u8c61\u8a3b\u4e00<\/sup>\uff0c\u800c\u9019\u88e1\u7684 \\(n-1\\) \u5728\u7d71\u8a08\u5b78\u4e0a\u7a31\u4e4b\u81ea\u7531\u5ea6(degree of freedom) \u8a3b\u4e8c<\/sup>\u3002\u5728\u6578\u7406\u7d71\u8a08\u4e0a\u53ef\u4ee5\u8b49\u660e\u4ee5\u81ea\u7531\u5ea6\u4f5c\u70ba\u9664\u6578\u6240\u8a08\u7b97\u51fa\u4f86\u7684\u5747\u65b9\uff0c\u624d\u662f\u65cf\u7fa4\u7684\u7121\u504f\u4f30\u503c\u8a3b\u4e09<\/sup>\uff0c\u4ea6\u5373 \\(s^2\\)\u00a0\u624d\u662f \\(\\sigma^2\\)\u00a0\u7684\u826f\u597d\u4f30\u503c\u3002<\/p>\n

    \n
  • \u8a3b\u4e00\uff1a\u7531\u65bc
    \n\\(\\begin{array}{ll}\\displaystyle \\sum_{i=1}^{n}(x_i-\\bar{x})^2&\\displaystyle=\\sum_{i=1}^{n}(x_i^2-2x_i\\bar{x}+\\bar{x}^2)=\\sum_{i=1}^{n}x_i^2-n\\bar{x}^2\\\\&=\\displaystyle \\sum_{i=1}^n(x_i-\\mu)^2-n(\\bar{x}-\\mu)^2\\end{array}\\)
    \n\\(\\rightarrow\\displaystyle \\sum_{i=1}^{n}(x_i-\\bar{x})^2\\le \\sum_{i=1}^{n}(x_i-\\mu)^2\\)
    \n\u82e5\u4ee5 \\(\\sum_{i=1}^{n}(x_i-\\bar{x})^2\/n\\) \u4f5c\u70ba\u6a23\u672c\u7684\u8b8a\u7570\u6578\uff0c\u7531\u4e0a\u5f0f\u53ef\u77e5\u6703\u767c\u751f\u4f4e\u4f30\u7684\u73fe\u8c61\u3002<\/li>\n
  • \u8a3b\u4e8c\uff1a\u81ea\u7531\u5ea6\u662f\u6307\u6a23\u672c\u5167\u7368\u7acb\uff0c\u4e14\u80fd\u5920\u81ea\u7531\u8b8a\u52d5\u7684\u96e2\u5747\u5dee \\((x_i-\\bar{x})\\) \u4e4b\u500b\u6578\u3002\u4f8b\u5982\uff1a\u6a23\u672c\u4e2d\u6709\u56db\u500b\u89c0\u6e2c\u503c\uff0c\u6a23\u672c\u5e73\u5747\u70ba6\uff0c\u5176\u4e2d\u4e09\u500b\u89c0\u6e2c\u503c\u70ba4\u30018\u820710\uff0c\u6700\u5f8c\u4e00\u500b\u89c0\u6e2c\u503c\u4e00\u5b9a\u662f6*4-(4+8+10)=2\u3002\u56e0\u6b64\uff0c\u7576\u6a23\u672c\u5927\u5c0f\u70ba4(=n)\u6642\uff0c\u53ea\u67093(=n-1)\u500b\u96e2\u5747\u5dee\u53ef\u4ee5\u81ea\u7531\u8b8a\u52d5\uff0c\u6b64\u6642\u81ea\u7531\u5ea6\u7b49\u65bc3\u3002<\/li>\n
  • \u8a3b\u4e09\uff1a\u7121\u504f\u4f30\u503c\u7684\u4ecb\u7d39\uff0c\u8acb\u8a73\u898b\u53e6\u7bc7\u6587\u7ae0\u3002<\/li>\n<\/ul>\n

    \u800c\u6709\u6642\u70ba\u4e86\u8a08\u7b97\u65b9\u4fbf\uff0c\u6211\u5011\u4e5f\u53ef\u4ee5\u5c07\u6a23\u672c\u8b8a\u7570\u6578\u7684\u516c\u5f0f\u8868\u793a\u6210\uff1a<\/p>\n

    \\(\\begin{array}{ll} s^2 &=\\displaystyle \\sum_{i=1}^{n}\\frac{(x_i-\\bar{x})^2}{n-1}\\\\&=\\displaystyle \\frac{1}{n-1}\\times(\\sum_{i=1}^n x_{i}^2-2\\sum_{i=1}^nx_i\\bar{x}+n\\bar{x}^2)\\\\&=\\displaystyle \\frac{1}{n-1}\\times(\\sum_{i=1}^nx_i^2-n\\bar{x}^2)\\end{array}\\)<\/p>\n

    \u53e6\u5916\uff0c\u6bcd\u9ad4\u8b8a\u7570\u6578\u7684\u6b63\u5e73\u65b9\u6839\uff0c\u7a31\u4e4b\u70ba\u6bcd\u9ad4\u6a19\u6e96\u5dee(\\(\\sigma\\))\uff1b\u6a23\u672c\u8b8a\u7570\u6578\u7684\u6b63\u5e73\u65b9\u6839\uff0c\u7a31\u4e4b\u70ba\u6a23\u672c\u6a19\u6e96\u5dee(\\(s\\))\u3002<\/p>\n

    \u4f8b\u984c\uff1a<\/p>\n

    A\u7814\u7a76\u54e1\u60f3\u8981\u4e86\u89e3\u67d0\u4e00\u5730\u534020-30\u6b72\u7684\u5973\u6027\u4e4b\u9ad4\u91cd\uff0c\u4f46\u4ed6\u7684\u6642\u9593\u3001\u7d93\u8cbb\u6709\u9650\uff0c\u6240\u4ee5\u4ed6\u6c7a\u5b9a\u5728\u8a72\u5730\u53bb\u96a8\u6a5f\u62bd\u53d612\u4f4d20-30\u6b72\u7684\u5973\u6027\uff0c\u5f97\u77e5\u5979\u5011\u7684\u9ad4\u91cd55, 45, 60, 48, 43, 52, 48, 43, 50, 50, 48, 58(\u55ae\u4f4d\uff1akg)\uff0c\u8acb\u554f\u901912\u4f4d\u5b78\u751f\u9ad4\u91cd\u7684\u6a23\u672c\u8b8a\u7570\u6578\u70ba\u591a\u5c11?<\/p>\n

    \\(\\begin{array}{ll}\\bar{x} &=\\displaystyle\\frac{1}{n}\\times\\sum_{i=1}^{n}x_i\\\\&=\\displaystyle\\frac{1}{12}(55+45+60+48+43+52+48+43+50+50+48+58)\\\\&=50 \\end{array}\\)<\/p>\n

    \\(\\begin{array}{ll}s^2 &=\\displaystyle\\sum_{i=1}^{n}\\frac{(x_i-\\bar{x})^2}{n-1}=\\frac{1}{12-1}(\\sum_{i=1}^{12}x_i^2-12\\bar{x}^2)\\\\&=\\displaystyle\\frac{1}{11}(30328-12\\times 50^2)\\\\&=29.82 \\end{array}\\)<\/p>\n

    \n

    \u53c3\u8003\u6587\u737b<\/strong><\/p>\n

      \n
    1. \u6c88\u660e\u4f86 (2014) \u751f\u7269\u7d71\u8a08\u5b78\u5165\u9580\u7b2c\u516d\u7248\u3002\u7b2c\u4e09\u7ae0-\u6558\u8ff0\u7d71\u8a08\u5b78\u3002<\/span><\/li>\n
    2. \u90ed\u5bf6\u931a\u3001\u9673\u7389\u654f(2011) \u751f\u7269\u7d71\u8a08\u5b78\u3002\u7b2c4\u7ae0-\u8cc7\u6599\u96c6\u4e2d\u8da8\u52e2\u53ca\u8b8a\u7570\u6027\u7684\u6e2c\u5ea6\u3002<\/span><\/li>\n
    3. \u6c5f\u632f\u6771\u3001\u653f\u6cbb\u5927\u5b78\u7d71\u8a08\u7cfb\uff5c\u6dfa\u8ac7\u81ea\u7531\u5ea6\uff08\u6a23\u672c\u6a19\u6e96\u5dee\u516c\u5f0f\u4e2d\u7684\u5206\u6bcd\u70ba\u4ec0\u9ebc\u8981\u63a1\u7528n-1\uff09\u3002http:\/\/mathcenter.ck.tp.edu.tw\/Resources\/Ctrl\/ePaper\/ePaperOpenFileX.ashx?autoKey=16<\/span><\/li>\n<\/ol>\n<\/div>","protected":false},"excerpt":{"rendered":"

      \u6bcd\u9ad4\u8b8a\u7570\u6578(\\(\\sigma^2\\))v.s.\u6a23\u672c\u8b8a\u7570\u6578(\\(s^2\\)) \u570b\u7acb\u81fa\u7063\u5927\u5b78\u8fb2\u85dd\u5b78\u7cfb \u5433\u535a\u96c5 \u4e00\u3001…<\/p>\n","protected":false},"author":50,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[111,230,229],"tags":[9776,9775,9801,9773,9774,9772,9771,2203],"class_list":["post-69367","post","type-post","status-publish","format-standard","hentry","category-mathematics00","category-math06-01","category-math06","tag-degree-of-freedom","tag-mean-square","tag-population-variance","tag-sample-variance","tag-9774","tag-9772","tag-9771","tag-2203","loop-entry","cat-111","cat-230","cat-229","no-thumbnail"],"views":160884,"_links":{"self":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/69367","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/users\/50"}],"replies":[{"embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/comments?post=69367"}],"version-history":[{"count":1,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/69367\/revisions"}],"predecessor-version":[{"id":86126,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/69367\/revisions\/86126"}],"wp:attachment":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/media?parent=69367"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/categories?post=69367"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/tags?post=69367"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}