{"id":2535,"date":"2023-07-21T20:39:52","date_gmt":"2023-07-21T20:39:52","guid":{"rendered":"https:\/\/statorials.org\/tr\/rde-merkezi-limit-teoremi\/"},"modified":"2023-07-21T20:39:52","modified_gmt":"2023-07-21T20:39:52","slug":"rde-merkezi-limit-teoremi","status":"publish","type":"post","link":"https:\/\/statorials.org\/tr\/rde-merkezi-limit-teoremi\/","title":{"rendered":"R&#39;de merkezi limit teoremi nas\u0131l uygulan\u0131r (\u00f6rneklerle)"},"content":{"rendered":"<p><\/p>\n<hr>\n<p><span style=\"color: #000000;\"><strong>Merkezi limit teoremi,<\/strong> <a href=\"https:\/\/statorials.org\/tr\/ornekleme-dagilimi-1\/\" target=\"_blank\" rel=\"noopener\">pop\u00fclasyon da\u011f\u0131l\u0131m\u0131<\/a> normal olmasa bile, \u00f6rneklem b\u00fcy\u00fckl\u00fc\u011f\u00fc yeterince b\u00fcy\u00fckse, bir \u00f6rneklem ortalamas\u0131n\u0131n \u00f6rnekleme da\u011f\u0131l\u0131m\u0131n\u0131n yakla\u015f\u0131k olarak normal oldu\u011funu belirtir.<\/span><\/p>\n<p> <span style=\"color: #000000;\">Merkezi limit teoremi ayr\u0131ca \u00f6rnekleme da\u011f\u0131l\u0131m\u0131n\u0131n a\u015fa\u011f\u0131daki \u00f6zelliklere sahip olaca\u011f\u0131n\u0131 belirtir:<\/span><\/p>\n<p> <span style=\"color: #000000;\"><strong>1.<\/strong> \u00d6rnekleme da\u011f\u0131l\u0131m\u0131n\u0131n ortalamas\u0131 n\u00fcfus da\u011f\u0131l\u0131m\u0131n\u0131n ortalamas\u0131na e\u015fit olacakt\u0131r:<\/span><\/p>\n<p style=\"text-align: center;\"> <strong><span style=\"color: #000000;\"><span style=\"text-decoration: overline;\">x<\/span> = \u00b5<\/span><\/strong><\/p>\n<p> <span style=\"color: #000000;\"><strong>2.<\/strong> \u00d6rnekleme da\u011f\u0131l\u0131m\u0131n\u0131n standart sapmas\u0131, n\u00fcfus da\u011f\u0131l\u0131m\u0131n\u0131n standart sapmas\u0131n\u0131n \u00f6rneklem b\u00fcy\u00fckl\u00fc\u011f\u00fcne b\u00f6l\u00fcnmesine e\u015fit olacakt\u0131r:<\/span><\/p>\n<p style=\"text-align: center;\"> <strong><span style=\"color: #000000;\">s = \u03c3<\/span> <span style=\"color: #000000;\">\/n<\/span><\/strong><\/p>\n<p> <span style=\"color: #000000;\">A\u015fa\u011f\u0131daki \u00f6rnek, merkezi limit teoreminin R&#8217;de nas\u0131l uygulanaca\u011f\u0131n\u0131 g\u00f6sterir.<\/span><\/p>\n<h3> <span style=\"color: #000000;\"><strong>\u00d6rnek: Merkezi limit teoreminin R&#8217;de uygulanmas\u0131<\/strong><\/span><\/h3>\n<p> <span style=\"color: #000000;\">Bir kaplumba\u011fan\u0131n kabu\u011funun geni\u015fli\u011finin minimum 2 in\u00e7 ve maksimum 6 in\u00e7 geni\u015flikte <a href=\"https:\/\/statorials.org\/tr\/uniforma-dagitimi\/\" target=\"_blank\" rel=\"noopener\">d\u00fczg\u00fcn bir da\u011f\u0131l\u0131m<\/a> izledi\u011fini varsayal\u0131m.<\/span><\/p>\n<p> <span style=\"color: #000000;\">Yani rastgele bir kaplumba\u011fa se\u00e7ip kabu\u011funun geni\u015fli\u011fini \u00f6l\u00e7ersek, <em>geni\u015fli\u011finin<\/em> de muhtemelen 2 ile 6 in\u00e7 aras\u0131nda olmas\u0131 muhtemeldir.<\/span><\/p>\n<p> <span style=\"color: #000000;\">A\u015fa\u011f\u0131daki kod, R&#8217;de 2 ila 6 in\u00e7 aras\u0131nda e\u015fit olarak da\u011f\u0131t\u0131lm\u0131\u015f 1.000 kaplumba\u011fan\u0131n kabuk geni\u015fliklerinin \u00f6l\u00e7\u00fcmlerini i\u00e7eren bir veri k\u00fcmesinin nas\u0131l olu\u015fturulaca\u011f\u0131n\u0131 g\u00f6sterir:<\/span> <\/p>\n<pre style=\"background-color: #ececec; font-size: 15px;\"> <strong><span style=\"color: #008080;\">#make this example reproducible\n<span style=\"color: #000000;\">set. <span style=\"color: #3366ff;\">seeds<\/span> (0)\n\n<span style=\"color: #008080;\">#create random variable with sample size of 1000 that is uniformly distributed\n<\/span>data &lt;- runif(n=1000, min=2, max=6)\n\n<span style=\"color: #008080;\">#create histogram to visualize distribution of turtle shell widths\n<\/span>hist(data, col=' <span style=\"color: #ff0000;\">steelblue<\/span> ', main=' <span style=\"color: #ff0000;\">Histogram of Turtle Shell Widths<\/span> ')\n<\/span><\/span><\/strong><\/pre>\n<p><img decoding=\"async\" loading=\"lazy\" class=\" wp-image-22527 aligncenter\" src=\"https:\/\/statorials.org\/wp-content\/uploads\/2023\/08\/central1.jpg\" alt=\"\" width=\"386\" height=\"392\" srcset=\"\" sizes=\"auto, \"><\/p>\n<p> <span style=\"color: #000000;\">Kaplumba\u011fa kabu\u011fu geni\u015fliklerinin da\u011f\u0131l\u0131m\u0131n\u0131n normalde hi\u00e7 da\u011f\u0131lmad\u0131\u011f\u0131n\u0131 unutmay\u0131n.<\/span><\/p>\n<p> <span style=\"color: #000000;\">\u015eimdi bu pop\u00fclasyondan 5 kaplumba\u011fan\u0131n tekrarlanan rastgele \u00f6rneklerini ald\u0131\u011f\u0131m\u0131z\u0131 ve \u00f6rnek ortalamas\u0131n\u0131 tekrar tekrar \u00f6l\u00e7t\u00fc\u011f\u00fcm\u00fcz\u00fc hayal edin.<\/span><\/p>\n<p> <span style=\"color: #000000;\">A\u015fa\u011f\u0131daki kod, bu i\u015flemin R&#8217;de nas\u0131l yap\u0131laca\u011f\u0131n\u0131 ve \u00f6rnek ortalamalar\u0131n da\u011f\u0131l\u0131m\u0131n\u0131 g\u00f6rselle\u015ftirmek i\u00e7in bir histogram\u0131n nas\u0131l olu\u015fturulaca\u011f\u0131n\u0131 g\u00f6sterir:<\/span> <\/p>\n<pre style=\"background-color: #ececec; font-size: 15px;\"> <strong><span style=\"color: #008080;\"><span style=\"color: #000000;\"><span style=\"color: #008080;\">#create empty vector to hold sample means\n<\/span><\/span><span style=\"color: #000000;\">sample5 &lt;- c()\n\n<span style=\"color: #008080;\">#take 1,000 random samples of size n=5\n<\/span>n = 1000\n<span style=\"color: #008000;\">for<\/span> (i <span style=\"color: #008000;\">in<\/span> 1:n){\nsample5[i] = mean(sample(data, 5, replace= <span style=\"color: #008000;\">TRUE<\/span> ))\n}\n\n<span style=\"color: #008080;\">#calculate mean and standard deviation of sample means\n<\/span>mean(sample5)\n\n[1] 4.008103\n\nsd(sample5)\n\n[1] 0.5171083 \n\n<span style=\"color: #008080;\">#create histogram to visualize sampling distribution of sample means\n<\/span>hist(sample5, col = ' <span style=\"color: #ff0000;\">steelblue<\/span> ', xlab=' <span style=\"color: #ff0000;\">Turtle Shell Width<\/span> ', main=' <span style=\"color: #ff0000;\">Sample size = 5<\/span> ')<\/span><\/span><\/strong> <\/pre>\n<p><img decoding=\"async\" loading=\"lazy\" class=\" wp-image-22528 aligncenter\" src=\"https:\/\/statorials.org\/wp-content\/uploads\/2023\/08\/central2.jpg\" alt=\"\" width=\"404\" height=\"410\" srcset=\"\" sizes=\"auto, \"><\/p>\n<p> <span style=\"color: #000000;\">\u00d6rneklerin geldi\u011fi da\u011f\u0131l\u0131m normal da\u011f\u0131lmam\u0131\u015f olmas\u0131na ra\u011fmen, \u00f6rnek ortalamalar\u0131n\u0131n \u00f6rnekleme da\u011f\u0131l\u0131m\u0131n\u0131n normal da\u011f\u0131lm\u0131\u015f g\u00f6r\u00fcnd\u00fc\u011f\u00fcne dikkat edin.<\/span><\/p>\n<p> <span style=\"color: #000000;\">Ayr\u0131ca bu \u00f6rnekleme da\u011f\u0131l\u0131m\u0131 i\u00e7in \u00f6rnek ortalamas\u0131n\u0131 ve \u00f6rnek standart sapmas\u0131n\u0131 da not edin:<\/span><\/p>\n<ul>\n<li> <span style=\"color: #000000;\"><strong>x\u0304<\/strong> : 4,008<\/span><\/li>\n<li> <span style=\"color: #000000;\"><strong>s<\/strong> : 0,517<\/span><\/li>\n<\/ul>\n<p> <span style=\"color: #000000;\">\u015eimdi kulland\u0131\u011f\u0131m\u0131z \u00f6rnek boyutunu n=5&#8217;ten n=30&#8217;a \u00e7\u0131kard\u0131\u011f\u0131m\u0131z\u0131 ve \u00f6rnek ortalamalar\u0131n\u0131n histogram\u0131n\u0131 yeniden olu\u015fturdu\u011fumuzu varsayal\u0131m:<\/span> <\/p>\n<pre style=\"background-color: #ececec; font-size: 15px;\"> <strong><span style=\"color: #008080;\">#create empty vector to hold sample means\n<span style=\"color: #000000;\">sample30 &lt;- c()\n\n<span style=\"color: #008080;\">#take 1,000 random samples of size n=30\n<\/span>n = 1000\n<span style=\"color: #008000;\">for<\/span> (i <span style=\"color: #008000;\">in<\/span> 1:n){\nsample30[i] = mean(sample(data, 30, replace= <span style=\"color: #008000;\">TRUE<\/span> ))\n}\n\n<span style=\"color: #008080;\">#calculate mean and standard deviation of sample means\n<\/span>mean(sample30)\n\n[1] 4.000472\n\nsd(sample30)\n\n[1] 0.2003791\n\n<span style=\"color: #008080;\">#create histogram to visualize sampling distribution of sample means\n<\/span>hist(sample30, col = ' <span style=\"color: #ff0000;\">steelblue<\/span> ', xlab=' <span style=\"color: #ff0000;\">Turtle Shell Width<\/span> ', main=' <span style=\"color: #ff0000;\">Sample size = 30<\/span> ')<\/span><\/span><\/strong> <\/pre>\n<p><img decoding=\"async\" loading=\"lazy\" class=\" wp-image-22529 aligncenter\" src=\"https:\/\/statorials.org\/wp-content\/uploads\/2023\/08\/central3.jpg\" alt=\"\" width=\"427\" height=\"435\" srcset=\"\" sizes=\"auto, \"><\/p>\n<p> <span style=\"color: #000000;\">\u00d6rnekleme da\u011f\u0131l\u0131m\u0131 yine normal olarak da\u011f\u0131t\u0131l\u0131r , ancak \u00f6rnek standart sapmas\u0131 daha da k\u00fc\u00e7\u00fckt\u00fcr:<\/span><\/p>\n<ul>\n<li> <span style=\"color: #000000;\">: <strong>0,200<\/strong><\/span><\/li>\n<\/ul>\n<p> <span style=\"color: #000000;\">Bunun nedeni, \u00f6nceki \u00f6rne\u011fe (n=5) k\u0131yasla daha b\u00fcy\u00fck bir \u00f6rneklem boyutu (n=30) kullanmam\u0131zd\u0131r, dolay\u0131s\u0131yla \u00f6rnek ortalamalar\u0131n\u0131n standart sapmas\u0131 daha da k\u00fc\u00e7\u00fckt\u00fcr.<\/span><\/p>\n<p> <span style=\"color: #000000;\">E\u011fer giderek daha b\u00fcy\u00fck \u00f6rnekler kullanmaya devam edersek, \u00f6rnek standart sapmas\u0131n\u0131n giderek k\u00fc\u00e7\u00fcld\u00fc\u011f\u00fcn\u00fc g\u00f6rece\u011fiz.<\/span><\/p>\n<p> <span style=\"color: #000000;\">Bu, pratikte merkezi limit teoremini g\u00f6stermektedir.<\/span><\/p>\n<h3> <span style=\"color: #000000;\"><strong>Ek kaynaklar<\/strong><\/span><\/h3>\n<p> <span style=\"color: #000000;\">A\u015fa\u011f\u0131daki kaynaklar merkezi limit teoremi hakk\u0131nda ek bilgi sa\u011flar:<\/span><\/p>\n<p> <a href=\"https:\/\/statorials.org\/tr\/merkezi-limit-teoremi\/\" target=\"_blank\" rel=\"noopener\">Merkezi Limit Teoremine Giri\u015f<\/a><br \/> <a href=\"https:\/\/statorials.org\/tr\/merkezi-limit-teoremi-hesaplayicisi\/\" target=\"_blank\" rel=\"noopener\">Merkezi Limit Teoremi Hesaplay\u0131c\u0131<\/a><br \/> <a href=\"https:\/\/statorials.org\/tr\/merkezi-limit-teoremi-gercek-ornekleri\/\" target=\"_blank\" rel=\"noopener\">Merkezi Limit Teoremini Ger\u00e7ek Hayatta Kullanmaya \u0130li\u015fkin 5 \u00d6rnek<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Merkezi limit teoremi, pop\u00fclasyon da\u011f\u0131l\u0131m\u0131 normal olmasa bile, \u00f6rneklem b\u00fcy\u00fckl\u00fc\u011f\u00fc yeterince b\u00fcy\u00fckse, bir \u00f6rneklem ortalamas\u0131n\u0131n \u00f6rnekleme da\u011f\u0131l\u0131m\u0131n\u0131n yakla\u015f\u0131k olarak normal oldu\u011funu belirtir. Merkezi limit teoremi ayr\u0131ca \u00f6rnekleme da\u011f\u0131l\u0131m\u0131n\u0131n a\u015fa\u011f\u0131daki \u00f6zelliklere sahip olaca\u011f\u0131n\u0131 belirtir: 1. \u00d6rnekleme da\u011f\u0131l\u0131m\u0131n\u0131n ortalamas\u0131 n\u00fcfus da\u011f\u0131l\u0131m\u0131n\u0131n ortalamas\u0131na e\u015fit olacakt\u0131r: x = \u00b5 2. \u00d6rnekleme da\u011f\u0131l\u0131m\u0131n\u0131n standart sapmas\u0131, n\u00fcfus da\u011f\u0131l\u0131m\u0131n\u0131n standart sapmas\u0131n\u0131n \u00f6rneklem [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[11],"tags":[],"class_list":["post-2535","post","type-post","status-publish","format-standard","hentry","category-rehber"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.3 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>R&#039;de Merkezi Limit Teoremi Nas\u0131l Uygulan\u0131r (\u00d6rneklerle) - 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