{"id":146,"date":"2023-08-04T23:38:31","date_gmt":"2023-08-04T23:38:31","guid":{"rendered":"https:\/\/statorials.org\/tr\/y-basiklik-asimetrisi\/"},"modified":"2023-08-04T23:38:31","modified_gmt":"2023-08-04T23:38:31","slug":"y-basiklik-asimetrisi","status":"publish","type":"post","link":"https:\/\/statorials.org\/tr\/y-basiklik-asimetrisi\/","title":{"rendered":"Asimetri ve d\u00fczle\u015fme"},"content":{"rendered":"

Bu makale istatistikte \u00e7arp\u0131kl\u0131k ve bas\u0131kl\u0131\u011f\u0131n ne oldu\u011funu a\u00e7\u0131klamaktad\u0131r. B\u00f6ylece, bu iki kavram\u0131n tan\u0131m\u0131n\u0131, \u00e7arp\u0131kl\u0131k ve bas\u0131kl\u0131\u011f\u0131n nas\u0131l hesaplanaca\u011f\u0131n\u0131, form\u00fcllerinin neler oldu\u011funu ve ayr\u0131ca herhangi bir veri \u00f6rne\u011finin \u00e7arp\u0131kl\u0131k ve bas\u0131kl\u0131\u011f\u0131n\u0131 hesaplamak i\u00e7in \u00e7evrimi\u00e7i bir hesap makinesi bulacaks\u0131n\u0131z. <\/p>\n

<\/span> \u00c7arp\u0131kl\u0131k ve bas\u0131kl\u0131k nedir?<\/span><\/h2>\n

\u00c7arp\u0131kl\u0131k ve bas\u0131kl\u0131k,<\/strong> bir da\u011f\u0131l\u0131m\u0131n \u015feklini grafik \u00e7izmeye gerek kalmadan tan\u0131mlamak i\u00e7in kullan\u0131lan iki istatistiksel \u00f6l\u00e7\u00fcd\u00fcr. Daha spesifik olarak \u00e7arp\u0131kl\u0131k, bir da\u011f\u0131l\u0131m\u0131n simetri (veya \u00e7arp\u0131kl\u0131k) derecesini belirtirken bas\u0131kl\u0131k, bir da\u011f\u0131l\u0131m\u0131n ortalama etraf\u0131ndaki konsantrasyon derecesini g\u00f6sterir.<\/p>\n

\u0130statistiklerde \u00e7arp\u0131kl\u0131k ve bas\u0131kl\u0131\u011fa \u015fekil \u00f6l\u00e7\u00fcleri<\/a> de denir.<\/p>\n

\ud83d\udc49Herhangi bir veri k\u00fcmesinin \u00e7arp\u0131kl\u0131\u011f\u0131n\u0131 ve bas\u0131kl\u0131\u011f\u0131n\u0131 hesaplamak i\u00e7in a\u015fa\u011f\u0131daki \u00e7evrimi\u00e7i hesap makinesini kullanabilirsiniz.<\/u><\/p>\n

<\/span> Asimetri<\/span><\/h2>\n

\u0130statistiklerde \u00e7arp\u0131kl\u0131k<\/strong> , bir da\u011f\u0131l\u0131m\u0131n ortalamas\u0131na g\u00f6re simetri (veya asimetri) derecesini g\u00f6steren bir \u00f6l\u00e7\u00fcd\u00fcr. Basit\u00e7e s\u00f6ylemek gerekirse \u00e7arp\u0131kl\u0131k, bir da\u011f\u0131l\u0131m\u0131n simetri (veya asimetri) derecesini grafiksel olarak g\u00f6stermeye gerek kalmadan belirlemek i\u00e7in kullan\u0131lan istatistiksel bir parametredir.<\/p>\n

Dolay\u0131s\u0131yla asimetrik bir da\u011f\u0131l\u0131m, ortalaman\u0131n solunda, sa\u011f\u0131ndakiyle kar\u015f\u0131la\u015ft\u0131r\u0131ld\u0131\u011f\u0131nda farkl\u0131 say\u0131da de\u011fere sahip olan da\u011f\u0131l\u0131md\u0131r. \u00d6te yandan simetrik bir da\u011f\u0131l\u0131mda ortalaman\u0131n solunda ve sa\u011f\u0131nda ayn\u0131 say\u0131da de\u011fer bulunur.<\/p>\n

B\u00f6ylece \u00fc\u00e7 t\u00fcr asimetriyi<\/strong> ay\u0131rt ediyoruz:<\/p>\n

    \n
  • Pozitif asimetri<\/strong> : Da\u011f\u0131l\u0131m\u0131n ortalaman\u0131n sa\u011f\u0131nda soluna g\u00f6re daha farkl\u0131 de\u011ferleri vard\u0131r.<\/span><\/li>\n
  • Simetri<\/strong> : Da\u011f\u0131l\u0131m, ortalaman\u0131n solunda ve ortalaman\u0131n sa\u011f\u0131nda ayn\u0131 say\u0131da de\u011fere sahiptir.<\/span><\/li>\n
  • Negatif \u00e7arp\u0131kl\u0131k<\/strong> : Da\u011f\u0131l\u0131m\u0131n ortalaman\u0131n solunda sa\u011f\u0131na g\u00f6re daha farkl\u0131 de\u011ferleri vard\u0131r.<\/span> <\/li>\n<\/ul>\n
    \"asimetri<\/figure>\n

    <\/span> asimetri katsay\u0131s\u0131<\/span><\/h3>\n

    \u00c7arp\u0131kl\u0131k katsay\u0131s\u0131<\/strong> veya asimetri indeksi<\/strong> , bir da\u011f\u0131l\u0131m\u0131n asimetrisini belirlemeye yard\u0131mc\u0131 olan istatistiksel bir katsay\u0131d\u0131r. B\u00f6ylece asimetri katsay\u0131s\u0131 hesaplanarak da\u011f\u0131l\u0131m\u0131n grafiksel olarak g\u00f6sterilmesine gerek kalmadan da\u011f\u0131l\u0131m\u0131n ne t\u00fcr bir asimetri sundu\u011funu bilmek m\u00fcmk\u00fcnd\u00fcr.<\/p>\n

    Asimetri katsay\u0131s\u0131n\u0131 hesaplamak i\u00e7in farkl\u0131 form\u00fcller olsa da ve a\u015fa\u011f\u0131da hepsini g\u00f6rece\u011fiz, kullan\u0131lan form\u00fcl ne olursa olsun asimetri katsay\u0131s\u0131n\u0131n yorumlanmas\u0131 her zaman \u015fu \u015fekilde yap\u0131l\u0131r: <\/p>\n

    \n
      \n
    • \u00c7arp\u0131kl\u0131k katsay\u0131s\u0131 pozitif ise da\u011f\u0131l\u0131m pozitif \u00e7arp\u0131kt\u0131r<\/strong> .<\/span><\/li>\n
    • Asimetri katsay\u0131s\u0131 s\u0131f\u0131ra e\u015fitse da\u011f\u0131l\u0131m simetriktir<\/strong> .<\/span><\/li>\n
    • \u00c7arp\u0131kl\u0131k katsay\u0131s\u0131 negatif ise da\u011f\u0131l\u0131m negatif \u00e7arp\u0131kt\u0131r<\/strong> .<\/span><\/li>\n<\/ul>\n<\/div>\n

      Fisher’in asimetri katsay\u0131s\u0131<\/h4>\n

      Fisher’in \u00e7arp\u0131kl\u0131k katsay\u0131s\u0131, ortalamaya g\u00f6re \u00fc\u00e7\u00fcnc\u00fc momentin numune standart sapmas\u0131na b\u00f6l\u00fcnmesine e\u015fittir. Bu nedenle Fisher’in asimetri katsay\u0131s\u0131n\u0131n form\u00fcl\u00fc<\/strong> \u015f\u00f6yledir:<\/p>\n<\/p>\n

      \"\\displaystyle\\gamma_1=\\frac{\\mu_3}{\\sigma^3}\"<\/p>\n<\/p>\n

      E\u015fde\u011fer olarak, Fisher katsay\u0131s\u0131n\u0131 hesaplamak i\u00e7in a\u015fa\u011f\u0131daki iki form\u00fclden herhangi biri kullan\u0131labilir:<\/p>\n<\/p>\n

      \"\\displaystyle\\gamma_1=\\frac{\\displaystyle<\/p>\n<\/p>\n

      \"\\displaystyle\\gamma_1=\\frac{\\operatorname{E}[X^3]<\/p>\n<\/p>\n

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

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

      matematiksel beklenti,<\/p>\n

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

      aritmetik ortalama,<\/p>\n

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

      standart sapma ve<\/p>\n

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

      toplam veri say\u0131s\u0131.<\/p>\n

      \u00d6te yandan veriler grupland\u0131r\u0131lm\u0131\u015fsa a\u015fa\u011f\u0131daki form\u00fcl\u00fc kullanabilirsiniz:<\/p>\n<\/p>\n

      \"\\displaystyle\\gamma_1=\\frac{\\displaystyle<\/p>\n<\/p>\n

      Bu durumda nerede<\/p>\n<\/p>\n

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

      Bu s\u0131n\u0131f i\u015faretidir ve<\/p>\n

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

      kursun mutlak s\u0131kl\u0131\u011f\u0131.<\/p>\n

      Pearson’un asimetri katsay\u0131s\u0131<\/h4>\n

      Pearson \u00e7arp\u0131kl\u0131k katsay\u0131s\u0131, \u00f6rnek ortalama ile mod aras\u0131ndaki fark\u0131n standart sapmaya (veya standart sapmaya) b\u00f6l\u00fcnmesine e\u015fittir. Pearson asimetri katsay\u0131s\u0131n\u0131n form\u00fcl\u00fc<\/strong> bu nedenle a\u015fa\u011f\u0131daki gibidir:<\/p>\n<\/p>\n

      \"A_p=\\cfrac{\\mu-Mo}{\\sigma}\"<\/p>\n<\/p>\n

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

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

      Pearson katsay\u0131s\u0131d\u0131r,<\/p>\n

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

      aritmetik ortalama,<\/p>\n

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

      moda ve<\/p>\n

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

      Standart sapma.<\/p>\n

      Pearson \u00e7arp\u0131kl\u0131k katsay\u0131s\u0131n\u0131n yaln\u0131zca tek modlu bir da\u011f\u0131l\u0131m olmas\u0131, yani verilerde yaln\u0131zca bir mod olmas\u0131 durumunda hesaplanabilece\u011fini unutmay\u0131n.<\/p>\n

      Bowley’in asimetri katsay\u0131s\u0131<\/h4>\n

      Bowley’in \u00e7arp\u0131kl\u0131k katsay\u0131s\u0131,<\/strong> \u00fc\u00e7\u00fcnc\u00fc \u00e7eyrek art\u0131 birinci \u00e7eyrek eksi medyan\u0131n iki kat\u0131n\u0131n toplam\u0131n\u0131n \u00fc\u00e7\u00fcnc\u00fc ve birinci \u00e7eyrek aras\u0131ndaki farka b\u00f6l\u00fcnmesine e\u015fittir. Dolay\u0131s\u0131yla bu asimetri katsay\u0131s\u0131n\u0131n form\u00fcl\u00fc a\u015fa\u011f\u0131daki gibidir:<\/p>\n<\/p>\n

      \"A_B=\\cfrac{Q_3+Q_1-2\\cdot<\/p>\n<\/p>\n

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

      \"Q_1\"<\/p>\n

      Ve<\/p>\n

      \"Q_3\"<\/p>\n

      Bunlar s\u0131ras\u0131yla birinci ve \u00fc\u00e7\u00fcnc\u00fc \u00e7eyreklerdir ve<\/p>\n

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

      da\u011f\u0131l\u0131m\u0131n medyan\u0131d\u0131r.<\/p>\n

      <\/span> D\u00fczle\u015ftirme<\/span><\/h2>\n

      \u00c7arp\u0131kl\u0131k<\/strong> olarak da adland\u0131r\u0131lan bas\u0131kl\u0131k<\/strong> , bir da\u011f\u0131l\u0131m\u0131n ortalama etraf\u0131nda ne kadar yo\u011funla\u015ft\u0131\u011f\u0131n\u0131 g\u00f6sterir. Ba\u015fka bir deyi\u015fle bas\u0131kl\u0131k, bir da\u011f\u0131l\u0131m\u0131n dik mi yoksa d\u00fcz m\u00fc oldu\u011funu g\u00f6sterir. Spesifik olarak, bir da\u011f\u0131l\u0131m\u0131n bas\u0131kl\u0131\u011f\u0131 ne kadar b\u00fcy\u00fck olursa, o kadar dik (veya daha keskin) olur. <\/p>\n

      \"gurur<\/figure>\n

      \u00dc\u00e7 t\u00fcr dalkavukluk<\/strong> vard\u0131r:<\/p>\n

        \n
      • Leptokurtik<\/strong> : da\u011f\u0131l\u0131m \u00e7ok sivridir, yani veriler ortalaman\u0131n etraf\u0131nda g\u00fc\u00e7l\u00fc bir \u015fekilde yo\u011funla\u015fm\u0131\u015ft\u0131r. Daha do\u011frusu leptokurtik da\u011f\u0131l\u0131mlar normal da\u011f\u0131l\u0131ma g\u00f6re daha keskin da\u011f\u0131l\u0131mlar olarak tan\u0131mlanmaktad\u0131r.<\/span><\/li>\n
      • Mezokurtik<\/strong> : Da\u011f\u0131l\u0131m\u0131n bas\u0131kl\u0131\u011f\u0131 normal da\u011f\u0131l\u0131m\u0131n bas\u0131kl\u0131\u011f\u0131na e\u015fde\u011ferdir. Bu nedenle ne keskin ne de d\u00fcz say\u0131l\u0131r.<\/span><\/li>\n
      • Platykurtic<\/strong> : da\u011f\u0131l\u0131m \u00e7ok d\u00fczd\u00fcr, yani ortalaman\u0131n etraf\u0131ndaki konsantrasyon d\u00fc\u015f\u00fckt\u00fcr. Resmi olarak platikurtik da\u011f\u0131l\u0131mlar normal da\u011f\u0131l\u0131mdan daha d\u00fcz olan da\u011f\u0131l\u0131mlar olarak tan\u0131mlan\u0131r.<\/span><\/li>\n<\/ul>\n

        Farkl\u0131 bas\u0131kl\u0131k t\u00fcrlerinin normal da\u011f\u0131l\u0131m\u0131n bas\u0131kl\u0131\u011f\u0131 referans al\u0131narak tan\u0131mland\u0131\u011f\u0131na dikkat edin. <\/p>\n

        \"iltifat<\/figure>\n

        <\/span> bas\u0131kl\u0131k katsay\u0131s\u0131<\/span><\/h3>\n

        Bas\u0131kl\u0131k katsay\u0131s\u0131n\u0131n form\u00fcl\u00fc<\/strong> a\u015fa\u011f\u0131daki gibidir:<\/p>\n<\/p>\n

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

        Frekans tablolar\u0131nda gruplanan veriler<\/strong> i\u00e7in bas\u0131kl\u0131k katsay\u0131s\u0131 form\u00fcl\u00fc:<\/p>\n<\/p>\n

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

        Son olarak aral\u0131klara g\u00f6re grupland\u0131r\u0131lm\u0131\u015f veriler<\/strong> i\u00e7in bas\u0131kl\u0131k katsay\u0131s\u0131 form\u00fcl\u00fc:<\/p>\n<\/p>\n

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

        Alt\u0131n: <\/p>\n

          \n
        • \n

          \"g_2\"<\/p>\n

          bas\u0131kl\u0131k katsay\u0131s\u0131d\u0131r.<\/li>\n

        • \n

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

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

        • \n

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

          serideki i’inci veri noktas\u0131d\u0131r.<\/li>\n

        • \n

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

          da\u011f\u0131l\u0131m\u0131n aritmetik ortalamas\u0131d\u0131r.<\/li>\n

        • \n

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

          da\u011f\u0131l\u0131m\u0131n standart sapmas\u0131 (veya tipik sapmas\u0131).<\/li>\n

        • \n

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

          BT veri setinin mutlak frekans\u0131d\u0131r.<\/li>\n

        • \n

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

          i’inci grubun s\u0131n\u0131f i\u015faretidir.<\/li>\n<\/ul>\n

          T\u00fcm bas\u0131kl\u0131k katsay\u0131s\u0131 form\u00fcllerinde normal da\u011f\u0131l\u0131m\u0131n bas\u0131kl\u0131k de\u011feri oldu\u011fundan 3’\u00fcn \u00e7\u0131kar\u0131ld\u0131\u011f\u0131n\u0131 unutmay\u0131n. B\u00f6ylece bas\u0131kl\u0131k katsay\u0131s\u0131n\u0131n hesaplanmas\u0131 normal da\u011f\u0131l\u0131m\u0131n bas\u0131kl\u0131\u011f\u0131 referans al\u0131narak yap\u0131l\u0131r. Bu nedenle bazen istatistiklerde a\u015f\u0131r\u0131 bas\u0131kl\u0131\u011f\u0131n<\/strong> hesapland\u0131\u011f\u0131 s\u00f6ylenir.<\/p>\n

          Bas\u0131kl\u0131k katsay\u0131s\u0131 hesapland\u0131ktan sonra bunun ne t\u00fcr bas\u0131kl\u0131k oldu\u011funun belirlenmesi i\u00e7in a\u015fa\u011f\u0131daki \u015fekilde yorumlanmas\u0131 gerekir: <\/p>\n

          \n
            \n
          • Bas\u0131kl\u0131k katsay\u0131s\u0131n\u0131n pozitif olmas\u0131 da\u011f\u0131l\u0131m\u0131n leptokurtik<\/strong> oldu\u011fu anlam\u0131na gelir.<\/span><\/li>\n
          • Bas\u0131kl\u0131k katsay\u0131s\u0131n\u0131n s\u0131f\u0131r olmas\u0131 da\u011f\u0131l\u0131m\u0131n mezokurtik<\/strong> oldu\u011fu anlam\u0131na gelir.<\/span><\/li>\n
          • Bas\u0131kl\u0131k katsay\u0131s\u0131n\u0131n negatif olmas\u0131 da\u011f\u0131l\u0131m\u0131n platikurtik<\/strong> oldu\u011fu anlam\u0131na gelir.<\/span> <\/li>\n<\/ul>\n<\/div>\n

            <\/span> \u00c7arp\u0131kl\u0131k ve Bas\u0131kl\u0131k Hesaplay\u0131c\u0131s\u0131<\/span><\/h2>\n

            \u00c7arp\u0131kl\u0131k ve bas\u0131kl\u0131k katsay\u0131s\u0131n\u0131 hesaplamak ve ayr\u0131ca ne t\u00fcr bir da\u011f\u0131l\u0131m oldu\u011funu belirlemek i\u00e7in a\u015fa\u011f\u0131daki hesap makinesine bir veri seti girin. Veriler bir bo\u015flukla ayr\u0131lmal\u0131 ve ondal\u0131k ay\u0131r\u0131c\u0131 olarak nokta kullan\u0131larak girilmelidir. <\/p>\n