{"id":68,"date":"2023-08-05T20:30:12","date_gmt":"2023-08-05T20:30:12","guid":{"rendered":"https:\/\/statorials.org\/tr\/varyans\/"},"modified":"2023-08-05T20:30:12","modified_gmt":"2023-08-05T20:30:12","slug":"varyans","status":"publish","type":"post","link":"https:\/\/statorials.org\/tr\/varyans\/","title":{"rendered":"Varyans"},"content":{"rendered":"

Bu yaz\u0131m\u0131zda varyans olarak da adland\u0131r\u0131lan varyans\u0131n ne oldu\u011funu ve nas\u0131l hesapland\u0131\u011f\u0131n\u0131 a\u00e7\u0131klayaca\u011f\u0131z. Varyans hesaplamas\u0131n\u0131n somut bir \u00f6rne\u011fi olan varyans form\u00fcl\u00fcn\u00fc bulacaks\u0131n\u0131z ve ayr\u0131ca \u00e7evrimi\u00e7i bir hesap makinesiyle herhangi bir veri k\u00fcmesinin varyans\u0131n\u0131 hesaplayabileceksiniz.<\/p>\n

Ayr\u0131ca, farkl\u0131 bir \u015fekilde yap\u0131ld\u0131\u011f\u0131 i\u00e7in grupland\u0131r\u0131lm\u0131\u015f verilerin varyans\u0131n\u0131 nas\u0131l bulaca\u011f\u0131n\u0131z\u0131 da g\u00f6steriyoruz. Son olarak pop\u00fclasyon varyans\u0131 ile \u00f6rneklem varyans\u0131 aras\u0131ndaki fark\u0131, varyans ile standart sapma aras\u0131ndaki fark\u0131 ve bu istatistiksel \u00f6l\u00e7\u00fcm\u00fcn \u00f6zelliklerini \u00f6\u011fretiyoruz.<\/p>\n

<\/span> Varyans nedir?<\/span><\/h2>\n

\u0130statistikte varyans, bir rastgele de\u011fi\u015fkenin de\u011fi\u015fkenli\u011fini g\u00f6steren bir da\u011f\u0131l\u0131m \u00f6l\u00e7\u00fcs\u00fcd\u00fcr.<\/strong> Varyans, art\u0131klar\u0131n karelerinin toplam\u0131n\u0131n toplam g\u00f6zlem say\u0131s\u0131na b\u00f6l\u00fcnmesine e\u015fittir.<\/p>\n

Kal\u0131nt\u0131n\u0131n istatistiksel bir veri noktas\u0131n\u0131n de\u011feri ile veri k\u00fcmesinin ortalamas\u0131 aras\u0131ndaki fark olarak anla\u015f\u0131ld\u0131\u011f\u0131n\u0131 unutmay\u0131n.<\/p>\n

Olas\u0131l\u0131k teorisinde varyans\u0131n sembol\u00fc Yunanca sigma kare harfidir (\u03c3 2<\/sup> ). Genellikle Var(X)<\/em> olarak da temsil edilmesine ra\u011fmen X<\/em> , varyans\u0131n hesapland\u0131\u011f\u0131 rastgele de\u011fi\u015fkendir.<\/p>\n

Genel olarak bir rastgele de\u011fi\u015fkenin varyans de\u011ferinin yorumlanmas\u0131<\/strong> basittir. Varyans de\u011feri ne kadar b\u00fcy\u00fck olursa veri o kadar da\u011f\u0131n\u0131k olur. Tam tersi, varyans de\u011feri ne kadar k\u00fc\u00e7\u00fckse, veri serisindeki da\u011f\u0131l\u0131m da o kadar az olacakt\u0131r. Ancak varyans\u0131 yorumlarken ayk\u0131r\u0131<\/em> de\u011ferlere dikkat edilmelidir \u00e7\u00fcnk\u00fc bunlar varyans de\u011ferini \u00e7arp\u0131tabilir.<\/p>\n

varyans, da\u011f\u0131l\u0131m d\u0131\u015f\u0131nda dikkate al\u0131nan di\u011fer \u00f6l\u00e7\u00fcler ise aral\u0131k, standart sapma, ortalama sapma ve de\u011fi\u015fim katsay\u0131s\u0131d\u0131r.<\/p>\n

<\/span>Bo\u015fluk nas\u0131l hesaplan\u0131r<\/span><\/h2>\n

Fark\u0131 hesaplamak i\u00e7in a\u015fa\u011f\u0131daki ad\u0131mlar\u0131n ger\u00e7ekle\u015ftirilmesi gerekir:<\/p>\n

    \n
  1. Veri k\u00fcmesinin aritmetik ortalamas\u0131n\u0131<\/a> bulun.<\/span><\/li>\n
  2. De\u011ferler ile veri k\u00fcmesinin ortalamas\u0131 aras\u0131ndaki fark olarak tan\u0131mlanan art\u0131klar\u0131 hesaplay\u0131n.<\/span><\/li>\n
  3. Geri kalan her \u015feyin karesini al\u0131n.<\/span><\/li>\n
  4. \u00d6nceki ad\u0131mda hesaplanan t\u00fcm sonu\u00e7lar\u0131 ekleyin.<\/span><\/li>\n
  5. Toplam veri say\u0131s\u0131na b\u00f6l\u00fcn. Elde edilen sonu\u00e7 veri serisinin varyans\u0131d\u0131r.<\/span><\/li>\n<\/ol>\n

    Sonu\u00e7 olarak, bir veri k\u00fcmesinin varyans\u0131n\u0131 hesaplama form\u00fcl\u00fc<\/strong> \u015f\u00f6yledir: <\/p>\n

    \"varyans\"<\/figure>\n

    Alt\u0131n:<\/p>\n

      \n
    • \n

      \"X\"<\/p>\n

      varyans\u0131n\u0131 hesaplamak istedi\u011finiz rastgele de\u011fi\u015fkendir.<\/li>\n

    • \n

      \"x_i\"<\/p>\n

      veri de\u011feri<\/p>\n

      \"i\"<\/p>\n

      .<\/li>\n

    • \n

      \"n\"<\/p>\n

      toplam g\u00f6zlem say\u0131s\u0131d\u0131r.<\/li>\n

    • \n

      \"\\overline{X}\"<\/p>\n

      rastgele de\u011fi\u015fkenin ortalamas\u0131d\u0131r<\/p>\n

      \"X\"<\/p>\n

      .<\/li>\n<\/ul>\n

      \ud83d\udc49Herhangi bir veri setinin varyans\u0131n\u0131 hesaplamak i\u00e7in a\u015fa\u011f\u0131daki hesaplay\u0131c\u0131y\u0131 kullanabilirsiniz.<\/u><\/p>\n

      Bu nedenle, bir veri serisinden varyans\u0131 \u00e7\u0131karmak i\u00e7in aritmetik ortalaman\u0131n nas\u0131l hesapland\u0131\u011f\u0131n\u0131 bilmeniz \u00f6nemlidir. Bunu nas\u0131l yapaca\u011f\u0131n\u0131z\u0131 hat\u0131rlam\u0131yorsan\u0131z, yukar\u0131da ba\u011flant\u0131s\u0131 verilen makaleye g\u00f6z atabilirsiniz.<\/p>\n

      <\/span> Sapma \u00f6rne\u011fi<\/span><\/h2>\n

      Art\u0131k varyans\u0131n tan\u0131m\u0131n\u0131 bildi\u011fimize g\u00f6re, bir veri serisinin varyans\u0131n\u0131n nas\u0131l elde edildi\u011fini g\u00f6rebilmeniz i\u00e7in ad\u0131m ad\u0131m bir al\u0131\u015ft\u0131rma \u00e7\u00f6zece\u011fiz.<\/p>\n

        \n
      • \u00c7ok uluslu bir \u015firketin son be\u015f y\u0131lda elde etti\u011fi ekonomik sonu\u00e7 malum, \u00e7o\u011funlu\u011fu k\u00e2r etti ama bir y\u0131l ciddi zararlar verdi: 11,5, 2, -9, 7 milyon euro. Bu veri setinin varyans\u0131n\u0131 hesaplay\u0131n.<\/li>\n<\/ul>\n

        Yukar\u0131daki a\u00e7\u0131klamada g\u00f6rd\u00fc\u011f\u00fcm\u00fcz gibi bir veri serisinin varyans\u0131n\u0131 bulmak i\u00e7in yapmam\u0131z gereken ilk \u015fey aritmetik ortalamas\u0131n\u0131 hesaplamakt\u0131r:<\/p>\n

        <\/p>\n<\/p>\n

        \"\\overline{X}=\\cfrac{11+5+2+(-9)+7}{5}=3,2\"<\/p>\n<\/p>\n

        Verilerin ortalama de\u011ferini bildi\u011fimizde varyans form\u00fcl\u00fcn\u00fc kullanabiliriz:<\/p>\n<\/p>\n

        \"Var(X)=\\cfrac{\\displaystyle\\sum_{i=1}^n\\left(x_i-\\overline{X}\\right)^2}{n}\"<\/p>\n<\/p>\n

        Al\u0131\u015ft\u0131rma beyan\u0131n\u0131n sa\u011flad\u0131\u011f\u0131 verileri form\u00fclde de\u011fi\u015ftiririz:<\/p>\n<\/p>\n

        \"Var(X)=\\cfrac{\\displaystyle<\/p>\n<\/p>\n

        Son olarak geriye kalan tek \u015fey varyans\u0131 hesaplamak i\u00e7in gerekli i\u015flemleri \u00e7\u00f6zmektir:<\/p>\n<\/p>\n

        \"\\begin{aligned}Var(X)&=\\cfrac{7,8^2+1,8^2+(-1,2)^2+(-12,2)^2+3,8^2}{5}\\\\[2ex]&=\\cfrac{60,84+3,24+1,44+148,84+14,44}{5}\\\\[2ex]&=<\/p>\n<\/p>\n

        Varyans birimlerinin istatistiksel verilerle ayn\u0131 birimler oldu\u011funu ancak kareleri oldu\u011funu unutmay\u0131n; bu nedenle bu veri grubunun varyans\u0131 45,76 milyon Euro 2’dir<\/sup> . <\/p>\n

        <\/span> Bo\u015fluk Hesaplay\u0131c\u0131<\/span><\/h2>\n

        Varyans\u0131n\u0131 hesaplamak i\u00e7in a\u015fa\u011f\u0131daki hesap makinesine bir istatistiksel veri seti girin. Veriler bir bo\u015flukla ayr\u0131lmal\u0131 ve ondal\u0131k ay\u0131r\u0131c\u0131 olarak nokta kullan\u0131larak girilmelidir. <\/p>\n