{"id":97,"date":"2023-08-05T12:39:47","date_gmt":"2023-08-05T12:39:47","guid":{"rendered":"https:\/\/statorials.org\/tr\/ortalama-medyan-ve-mod\/"},"modified":"2023-08-05T12:39:47","modified_gmt":"2023-08-05T12:39:47","slug":"ortalama-medyan-ve-mod","status":"publish","type":"post","link":"https:\/\/statorials.org\/tr\/ortalama-medyan-ve-mod\/","title":{"rendered":"Ortalama, medyan ve mod"},"content":{"rendered":"

Bu makalede ortalama, medyan ve modun ne oldu\u011fu a\u00e7\u0131klanmaktad\u0131r. Ortalamay\u0131, medyan\u0131 ve modu nas\u0131l elde edece\u011finizi, bunlar\u0131n ne i\u00e7in kullan\u0131ld\u0131\u011f\u0131n\u0131 ve bu \u00fc\u00e7 istatistiksel \u00f6l\u00e7\u00fc aras\u0131ndaki fark\u0131n ne oldu\u011funu \u00f6\u011freneceksiniz. Ek olarak, sonundaki \u00e7evrimi\u00e7i hesap makinesini kullanarak herhangi bir istatistiksel \u00f6rne\u011fin ortalamas\u0131n\u0131, medyan\u0131n\u0131 ve modunu hesaplayabileceksiniz. <\/p>\n

<\/span> Ortalama, medyan ve mod nedir?<\/span><\/h2>\n

Ortalama, medyan ve mod, merkezi konumun istatistiksel \u00f6l\u00e7\u00fcmleridir.<\/strong> Ba\u015fka bir deyi\u015fle ortalama, medyan ve mod, istatistiksel bir \u00f6rne\u011fin tan\u0131mlanmas\u0131na yard\u0131mc\u0131 olan de\u011ferlerdir, \u00f6zellikle merkezi de\u011ferlerinin ne oldu\u011funu g\u00f6sterirler.<\/p>\n

Ortalama, medyan ve mod \u015fu \u015fekilde tan\u0131mlan\u0131r:<\/p>\n

    \n
  • Ortalama<\/strong> : \u00d6rnekteki t\u00fcm verilerin ortalamas\u0131d\u0131r.<\/span><\/li>\n
  • Medyan<\/strong> : Bu, en k\u00fc\u00e7\u00fckten en b\u00fcy\u00fc\u011fe do\u011fru s\u0131ralanan t\u00fcm verilerin ortadaki de\u011feridir.<\/span><\/li>\n
  • Mod<\/strong> : Veri setinde en \u00e7ok tekrarlanan de\u011ferdir.<\/span><\/li>\n<\/ul>\n

    Bu \u00fc\u00e7 istatistiksel \u00f6l\u00e7\u00fcm a\u015fa\u011f\u0131da daha ayr\u0131nt\u0131l\u0131 olarak a\u00e7\u0131klanmaktad\u0131r.<\/p>\n

    <\/span> Yar\u0131m<\/span><\/h3>\n

    Ortalamay\u0131 hesaplamak<\/strong> i\u00e7in t\u00fcm de\u011ferleri toplay\u0131n ve ard\u0131ndan toplam veri say\u0131s\u0131na b\u00f6l\u00fcn. Ortalaman\u0131n form\u00fcl\u00fc bu nedenle a\u015fa\u011f\u0131daki gibidir:<\/p>\n<\/p>\n

    \"\\displaystyle\\overline{x}=\\frac{\\displaystyle\\sum_{i=1}^N<\/p>\n<\/p>\n

    \ud83d\udc49Herhangi bir veri setinin ortalamas\u0131n\u0131, medyan\u0131n\u0131 ve modunu hesaplamak i\u00e7in a\u015fa\u011f\u0131daki hesap makinesini kullanabilirsiniz.<\/u><\/p>\n

    Ortalama sembol, x harfinin \u00fczerinde yatay bir bantt\u0131r<\/p>\n<\/p>\n

    \"(\\overline{x}).\"<\/p>\n

    Ayr\u0131ca ortalama simgesiyle \u00f6rnek ortalamas\u0131n\u0131 pop\u00fclasyon ortalamas\u0131ndan ay\u0131rt edebilirsiniz: bir \u00f6rne\u011fin ortalamas\u0131 simgesiyle ifade edilir.<\/p>\n

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

    Bir n\u00fcfusun ortalamas\u0131 Yunan harfini kullan\u0131rken<\/p>\n

    \"\\mu.\"<\/p>\n<\/p>\n

    Ortalama ayn\u0131 zamanda aritmetik ortalama<\/strong> veya ortalama<\/strong> olarak da bilinir. Ayr\u0131ca istatistiksel bir da\u011f\u0131l\u0131m\u0131n ortalamas\u0131 onun matematiksel beklentisine e\u015fde\u011ferdir.<\/p>\n

    <\/span> Ortalama \u00f6rnek<\/span><\/h4>\n
      \n
    • Bir \u00f6\u011frenci bir okul y\u0131l\u0131 boyunca \u015fu notlar\u0131 ald\u0131: matematikte 9, dilde 7, tarihte 6, ekonomide 8 ve fen bilimlerinde 7,5. T\u00fcm notlar\u0131n\u0131z\u0131n ortalamas\u0131 nedir?<\/li>\n<\/ul>\n

      Aritmetik ortalamay\u0131 bulmak i\u00e7in t\u00fcm notlar\u0131 toplay\u0131p dersteki toplam ders say\u0131s\u0131na (5) b\u00f6lmemiz gerekir. Bu nedenle aritmetik ortalama form\u00fcl\u00fcn\u00fc uygular\u0131z:<\/p>\n<\/p>\n

      \"\\displaystyle\\overline{x}=\\frac{\\displaystyle\\sum_{i=1}^N<\/p>\n<\/p>\n

      Verileri form\u00fclde yerine koyar\u0131z ve aritmetik ortalamay\u0131 hesaplar\u0131z:<\/p>\n<\/p>\n

      \"\\overline{x}=\\cfrac{9+7+5+8+7,5}{5}=7,3\"<\/p>\n<\/p>\n

      G\u00f6rd\u00fc\u011f\u00fcn\u00fcz gibi aritmetik ortalamada her de\u011fere ayn\u0131 a\u011f\u0131rl\u0131k atan\u0131r, yani her veri par\u00e7as\u0131 b\u00fct\u00fcn i\u00e7inde ayn\u0131 a\u011f\u0131rl\u0131\u011fa sahiptir.<\/p>\n

      <\/span> Medyan<\/span><\/h3>\n

      Medyan<\/strong> , en k\u00fc\u00e7\u00fckten en b\u00fcy\u00fc\u011fe do\u011fru s\u0131ralanan t\u00fcm verilerin ortadaki de\u011feridir. Ba\u015fka bir deyi\u015fle medyan s\u0131ral\u0131 veri setini iki e\u015fit par\u00e7aya b\u00f6ler.<\/p>\n

      Medyan\u0131n hesaplanmas\u0131, toplam veri say\u0131s\u0131n\u0131n \u00e7ift veya tek olmas\u0131na ba\u011fl\u0131d\u0131r:<\/p>\n

        \n
      • Toplam veri say\u0131s\u0131 tek<\/strong> ise medyan verinin tam ortas\u0131nda kalan de\u011fer olacakt\u0131r. Yani s\u0131ralanan verinin (n+1)\/2 konumundaki de\u011feri.<\/span><\/li>\n

        \"Me=x_{\\frac{n+1}{2}\"<\/p>\n<\/p>\n

      • Toplam veri noktas\u0131 say\u0131s\u0131 \u00e7ift ise<\/strong> medyan, merkezde bulunan iki veri noktas\u0131n\u0131n ortalamas\u0131 olacakt\u0131r. Yani s\u0131ral\u0131 verinin n\/2 ve n\/2+1 konumlar\u0131nda bulunan de\u011ferlerin aritmetik ortalamas\u0131d\u0131r.<\/span><\/li>\n

        \"Me=\\cfrac{x_{\\frac{n}{2}}+x_{\\frac{n}{2}+1}}{2}\"<\/p>\n<\/p>\n<\/ul>\n

        Alt\u0131n<\/p>\n<\/p>\n

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

        \u00f6rnekteki veri \u00f6\u011felerinin toplam say\u0131s\u0131d\u0131r.<\/p>\n

        Me<\/em> terimi genellikle bir de\u011ferin t\u00fcm g\u00f6zlemlerin medyan\u0131 oldu\u011funu belirtmek i\u00e7in bir sembol olarak kullan\u0131l\u0131r.<\/p>\n

        \ud83d\udc49Herhangi bir veri setinin ortalamas\u0131n\u0131, medyan\u0131n\u0131 ve modunu hesaplamak i\u00e7in a\u015fa\u011f\u0131daki hesap makinesini kullanabilirsiniz.<\/u><\/p>\n

        <\/span>Medyan \u00f6rne\u011fi<\/span><\/h4>\n
          \n
        • A\u015fa\u011f\u0131daki verilerin ortancas\u0131n\u0131 bulun: 3, 4, 1, 6, 7, 4, 8, 2, 8, 4, 5<\/li>\n<\/ul>\n

          Hesaplama yapmadan \u00f6nce ilk yapmam\u0131z gereken verileri s\u0131n\u0131fland\u0131rmak yani say\u0131lar\u0131 k\u00fc\u00e7\u00fckten b\u00fcy\u00fc\u011fe do\u011fru s\u0131ral\u0131yoruz.<\/p>\n<\/p>\n

          \"1<\/p>\n<\/p>\n

          Bu durumda 11 g\u00f6zlemimiz var, yani toplam veri say\u0131s\u0131 tektir. Bu nedenle medyan\u0131n konumunu hesaplamak i\u00e7in a\u015fa\u011f\u0131daki form\u00fcl\u00fc uygular\u0131z:<\/p>\n<\/p>\n

          \"\\cfrac{n+1}{2}=\\cfrac{11+1}{2}=6\"<\/p>\n<\/p>\n

          Bu nedenle medyan alt\u0131nc\u0131 konumda yer alan veri olacakt\u0131r ve bu durumda bu de\u011fer 4’e kar\u015f\u0131l\u0131k gelir. <\/p>\n<\/p>\n

          \"Me=x_6=4\"<\/p>\n<\/p>\n

          <\/span> Moda<\/span><\/h3>\n

          \u0130statistikte mod<\/strong> , veri k\u00fcmesindeki mutlak frekans\u0131 en y\u00fcksek olan de\u011ferdir, yani mod, bir veri k\u00fcmesinde en \u00e7ok tekrarlanan de\u011ferdir.<\/p>\n

          Bu nedenle, istatistiksel bir veri k\u00fcmesinin modunu hesaplamak i\u00e7in, her veri \u00f6\u011fesinin \u00f6rnekte ka\u00e7 kez g\u00f6r\u00fcnd\u00fc\u011f\u00fcn\u00fc sayman\u0131z yeterlidir; en \u00e7ok tekrarlanan veri, mod olacakt\u0131r.<\/p>\n

          Modun istatistiksel mod<\/strong> veya modal de\u011fer<\/strong> oldu\u011fu da s\u00f6ylenebilir. Benzer \u015fekilde, veriler aral\u0131klar halinde grupland\u0131r\u0131ld\u0131\u011f\u0131nda en \u00e7ok tekrarlanan aral\u0131k, modal aral\u0131k<\/strong> veya modal s\u0131n\u0131ft\u0131r<\/strong> .<\/p>\n

          Genel olarak Mo<\/em> terimi istatistiksel modun sembol\u00fc olarak kullan\u0131l\u0131r; \u00f6rne\u011fin X da\u011f\u0131t\u0131m modu Mo(X)’t\u00fcr.<\/p>\n

          En \u00e7ok tekrarlanan de\u011ferlerin say\u0131s\u0131na g\u00f6re \u00fc\u00e7 t\u00fcr mod ay\u0131rt edilebilir:<\/p>\n

            \n
          • Tek modlu mod<\/strong> : Maksimum tekrar say\u0131s\u0131na sahip yaln\u0131zca bir de\u011fer vard\u0131r. \u00d6rne\u011fin, [1, 4, 2, 4, 5, 3].<\/span><\/li>\n
          • Bimodal mod<\/strong> : Maksimum tekrar say\u0131s\u0131 iki farkl\u0131 de\u011ferde ger\u00e7ekle\u015fir ve her iki de\u011fer de ayn\u0131 say\u0131da tekrarlan\u0131r. \u00d6rne\u011fin, [2, 6, 7, 2, 3, 6, 9].<\/span><\/li>\n
          • Multimodal mod<\/strong> : \u00dc\u00e7 veya daha fazla de\u011fer ayn\u0131 maksimum tekrar say\u0131s\u0131na sahiptir. \u00d6rne\u011fin, [3, 3, 4, 1, 3, 4, 2, 1, 4, 5, 2, 1].<\/span><\/li>\n<\/ul>\n

            \ud83d\udc49Herhangi bir veri setinin ortalamas\u0131n\u0131, medyan\u0131n\u0131 ve modunu hesaplamak i\u00e7in a\u015fa\u011f\u0131daki hesap makinesini kullanabilirsiniz.<\/u><\/p>\n

            <\/span> moda \u00f6rne\u011fi<\/span><\/h4>\n
              \n
            • A\u015fa\u011f\u0131daki veri k\u00fcmesinin modu nedir?<\/li>\n<\/ul>\n

              \"5<\/p>\n<\/p>\n

              Say\u0131lar s\u0131ral\u0131 de\u011fil, bu y\u00fczden yapaca\u011f\u0131m\u0131z ilk \u015fey onlar\u0131 s\u0131ralamak olacak. Bu ad\u0131m zorunlu de\u011fildir ancak moday\u0131 daha kolay bulman\u0131za yard\u0131mc\u0131 olacakt\u0131r.<\/p>\n<\/p>\n

              \"2<\/p>\n<\/p>\n

              2 ve 9 say\u0131lar\u0131 iki kez g\u00f6r\u00fcn\u00fcyor, ancak 5 say\u0131s\u0131 \u00fc\u00e7 kez tekrarlan\u0131yor. Bu nedenle veri serisinin modu 5 numarad\u0131r. <\/p>\n<\/p>\n

              \"Mo=5\"<\/p>\n<\/p>\n

              <\/span> Ortalama, medyan ve modun \u00e7\u00f6z\u00fclm\u00fc\u015f al\u0131\u015ft\u0131rmas\u0131<\/span><\/h2>\n

              Art\u0131k ortalaman\u0131n, medyan\u0131n ve modun ne oldu\u011funu bildi\u011finize g\u00f6re, a\u015fa\u011f\u0131da bu istatistiksel \u00f6l\u00e7\u00fcmlerle ilgili ayr\u0131nt\u0131l\u0131 bir al\u0131\u015ft\u0131rma bulacaks\u0131n\u0131z; b\u00f6ylece bunlar\u0131n tam olarak nas\u0131l hesapland\u0131\u011f\u0131n\u0131 g\u00f6rebilirsiniz.<\/p>\n

                \n
              • A\u015fa\u011f\u0131daki istatistiksel veri setinin ortalamas\u0131n\u0131, medyan\u0131n\u0131 ve modunu bulun:<\/li>\n<\/ul>\n

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

                \"0<\/p>\n<\/p>\n

                Verilerin ortalamas\u0131n\u0131 bulmak i\u00e7in hepsini toplay\u0131p toplam veri say\u0131s\u0131na yani 30’a b\u00f6lmemiz gerekiyor:<\/p>\n<\/p>\n

                \"\\displaystyle\\overline{x}=\\frac{\\displaystyle\\sum_{i=1}^N<\/p>\n<\/p>\n

                \u0130kinci olarak \u00f6rnek medyan\u0131 bulal\u0131m. B\u00f6ylece t\u00fcm say\u0131lar\u0131 artan s\u0131raya koyuyoruz:<\/p>\n<\/p>\n

                \"0<\/p>\n<\/p>\n

                \"7<\/p>\n<\/p>\n

                Bu durumda toplam veri say\u0131s\u0131 \u00e7ift oldu\u011fundan, medyan\u0131n bulunaca\u011f\u0131 iki merkezi konumu hesaplamak gerekir. Bunun i\u00e7in a\u015fa\u011f\u0131daki iki form\u00fcl\u00fc kullan\u0131yoruz:<\/p>\n<\/p>\n

                \"\\cfrac{n}{2}=\\cfrac{30}{2}=15\"<\/p>\n<\/p>\n

                \"\\cfrac{n}{2}+1=\\cfrac{30}{2}+1=16\"<\/p>\n<\/p>\n

                Bu nedenle medyan, s\u0131ras\u0131yla 6 ve 7 de\u011ferlerine kar\u015f\u0131l\u0131k gelen on be\u015finci ve on alt\u0131nc\u0131 konumlar aras\u0131nda olacakt\u0131r. Daha do\u011frusu, medyan bu de\u011ferlerin ortalamas\u0131na e\u015fde\u011ferdir:<\/p>\n<\/p>\n

                \"Me=\\cfrac{x_{15}+x_{16}}{2}=\\cfrac{6+7}{2}=6,5\"<\/p>\n<\/p>\n

                Son olarak, modu bulmak i\u00e7in her say\u0131n\u0131n g\u00f6r\u00fcnd\u00fc\u011f\u00fc t\u00fcm zamanlar\u0131 sayman\u0131z yeterlidir. G\u00f6rd\u00fc\u011f\u00fcn\u00fcz gibi 6 ve 8 rakam\u0131 toplamda d\u00f6rt defa kar\u015f\u0131m\u0131za \u00e7\u0131k\u0131yor, bu da maksimum tekrar say\u0131s\u0131d\u0131r. Bu nedenle, bu durumda bu iki modlu bir moddur ve iki say\u0131, veri k\u00fcmesinin modudur: <\/p>\n<\/p>\n

                \"Mo=\\{<\/p>\n<\/p>\n

                <\/span> Ortalama, medyan ve mod hesaplay\u0131c\u0131<\/span><\/h2>\n

                Ortalamas\u0131n\u0131, medyan\u0131n\u0131 ve modunu hesaplamak i\u00e7in herhangi bir istatistiksel \u00f6rnekten verileri a\u015fa\u011f\u0131daki \u00e7evrimi\u00e7i hesap makinesine girin. Veriler bir bo\u015flukla ayr\u0131lmal\u0131 ve ondal\u0131k ay\u0131r\u0131c\u0131 olarak nokta kullan\u0131larak girilmelidir. <\/p>\n