Gabatarwa ga Skewness da Kurtosis
A fannin kididdiga, nazarin bayanai yana da mahimmanci don fassara da fahimtar tsarin da ke cikin bayanan. Manyan ra'ayoyi da ke taimakawa wajen bayyana da taƙaita waɗannan bayanan sune karkacewar bayanai da kurtosis. Waɗannan ma'auni guda biyu suna ba da fahimta mai mahimmanci game da siffa da rarraba bayanai, waɗanda suka wuce matsakaicin karkacewa da daidaitaccen tsari.
Skewness: Fahimtar Rashin Daidaito na Bayanai
Skewness yana bayyana rashin daidaito ko rashin daidaito a cikin rarrabawar wuraren bayanai. A cikin rarrabawa mai daidaito, kamar rarrabawa ta al'ada, ɓangarorin hagu da dama na histogram hotunan madubi ne. Duk da haka, bayanai na gaske ba sa cika dacewa da cikakkiyar daidaituwa, wanda ke haifar da karkacewa.
1. Nau'ikan Skewness:
– Kyakkyawan Skewness (Dama-Skewed): A wannan yanayin, wutsiyar da ke gefen dama na rarrabawa ta fi tsayi ko kiba fiye da gefen hagu. Wannan yana nuna cewa yawancin wuraren bayanai suna mai da hankali ne a gefen hagu, tare da ƙarancin masu wucewa zuwa ga ƙima mafi girma. Misali na iya zama kuɗin shiga na gida, inda yawancin mutane ke samun kuɗi ƙasa da wani ƙayyadadden iyaka, amma kaɗan daga cikin mutane suna samun kuɗi mafi girma.
– Rashin Tsammani (Hagu-Skewed): A nan, wutsiyar da ke gefen hagu na rarrabawar ta fi tsayi ko kuma ta fi kiba. Yawancin bayanan da aka tattara suna kan dama, tare da ƙarancin ƙarancin ƙima. Misali zai iya haɗawa da shekarun ritaya, inda yawancinsu ke yin ritaya tsakanin takamaiman shekaru, amma kaɗan daga cikinsu suna yin ritaya sosai da wuri.
– Sifili Mai Tsanani: Wannan yana faruwa ne a cikin rarrabawa mai daidaito, inda wutsiyoyin hagu da dama ke kama da juna. Misalin gargajiya shine rarrabawa ta al'ada.
2. Lissafin Skewness:
Ana iya ƙididdige skewness ta amfani da dabarar da ke ƙasa:
\[
\text{Skewness} = \frac{n}{(n-1)(n-2)} \sum_{i=1}^n \left( \frac{x_i – \bar{x}}{s} \right)^3
\]
- n shine adadin abubuwan da aka lura
- x_i yana wakiltar kowane abin lura na mutum ɗaya
– \(\bar{x}\) shine ma'anar abubuwan da aka lura
– s shine karkacewar da aka saba
Ƙimar da ta dace tana nuna karkacewar dama, yayin da ƙima mara kyau ke nuna karkacewar hagu.
Kurtosis: Fahimtar Nauyin Wutsiyar Bayanai
Kurtosis yana auna "ƙanƙantar wutsiya" na rarraba bayanai. Yayin da karkacewar ke magana game da daidaito, kurtosis yana magana game da tsayi da kaifi na ƙololuwar tsakiya da nauyin wutsiya. Gabaɗaya ana rarraba Kurtosis zuwa nau'i uku:
1. Nau'in Kurtosis:
– Leptokurtic: Rarrabawar da ke da ƙimar kurtosis mai kyau yana nuna ƙarin ƙimar bayanai sun taru a kusa da matsakaicin, wanda ke haifar da ƙololuwa masu kaifi da wutsiya masu ƙiba. Wannan yana nuna yiwuwar samun mafi girman riba. Ribar kasuwar hannayen jari galibi tana nuna halayen leptokurtic, yana nuna canje-canje masu yawa.
– Platykurtic: Rarrabawa tare da ƙimar kurtosis mara kyau suna nuna kololuwa masu faɗi da wutsiya masu siriri, ma'ana ma'aunin bayanai sun fi yaɗuwa daidai gwargwado a matsakaicin. Saitin lambobi bazuwar galibi suna nuna halayen platykurtic.
– Mesokurtic: Rarrabawar da ke da ƙimar kurtosis kusa da sifili, kamar rarrabawar al'ada, tana nuna matsakaicin kololuwa da kauri wutsiya.
2. Lissafi Kurtosis:
Ana ƙididdige Kurtosis ta amfani da dabarar:
\[
\text{Kurtosis} = \frac{n(n+1)}{(n-1)(n-2)(n-3)} \sum_{i=1}^n \left( \frac{x_i – \bar{x}}{s} \right)^4 – 3 \frac{(n-1)^2}{(n-2)(n-3)}
\]
Kamar karkacewar, n shine adadin abubuwan da aka lura, x_i shine kowane lura, \(\bar{x}\) shine matsakaicin, kuma s shine karkacewar da aka saba amfani da ita. Ana amfani da cirewa da uku don sanya kurtosis na rarrabawar al'ada daidai yake da sifili. Ƙimar da ta dace tana nufin rarrabawar leptokurtic, yayin da ƙima mara kyau ke nuna rarrabawar platykurtic.
Abubuwan Taimako da Aikace-aikace
Skewness da kurtosis ba wai kawai gine-ginen falsafa bane, har ma suna da tasiri mai amfani a fannoni daban-daban, tun daga fannin kuɗi zuwa kula da inganci da kuma ilmin halitta.
1. Kuɗi:
– A fannin kula da haɗari, ana amfani da skewness da kurtosis don yin koyi da hasashen ribar kuɗi. Skewness yana sanar da haɗarin canje-canje masu kyau ko marasa kyau, yayin da kurtosis ke taimakawa wajen fahimtar yiwuwar aukuwar mummunan yanayi ko kuma abubuwan da suka faru na baƙi.
– Manajan fayil suna la'akari da rashin daidaito da kurtosis lokacin zabar kadarori, suna ƙoƙarin samun daidaito mai kyau tsakanin haɗari da riba.
2. Kula da Inganci:
– A fannin masana'antu, yin nazari kan karkacewar bayanai daga hanyoyin samarwa na iya nuna rashin inganci ko karkacewar inganci. Kurtosis na iya gano yiwuwar wasu abubuwan da ka iya nuna lahani ko kurakurai masu wuya.
3. Ilimin Halittu da Magani:
– A fannin ilimin cututtuka, karkacewar na iya nuna yaduwar cututtuka, yayin da kurtosis na iya nuna kasancewar masu yaɗawa masu yawa. A fannin kwayoyin halitta, waɗannan matakan suna taimakawa wajen nazarin yadda aka rarraba halaye a cikin al'umma.
4. Kimiyyar Muhalli:
– Masu bincike suna amfani da rashin daidaito wajen nazarin bayanai kan gurɓataccen iska, suna bayyana yanayin da gurɓataccen iska ke faruwa. Kurtosis yana taimakawa wajen fahimtar abubuwan da ke faruwa a muhalli masu wahala amma masu mahimmanci, kamar malalar mai ko ganin nau'ikan halittu masu wuya.
Kammalawa
Fahimtar karkacewa da kurtosis yana ba da kyakkyawan hangen nesa game da bayanai, yana taimakawa wajen yanke shawara mafi kyau a fannoni daban-daban. Yayin da karkacewa ke nuna rashin daidaito, kurtosis yana nuna halayen rarrabawa. Tare, waɗannan matakan suna ba da mahimman bayanai fiye da ƙididdigar taƙaitacciyar al'ada, suna haɓaka ikonmu na fassara da hasashe abubuwan da ke faruwa masu rikitarwa.
Kamar kowace kayan aikin kididdiga, yana da matuƙar muhimmanci a tuna cewa rashin daidaito da kurtosis suna da matuƙar amfani idan aka yi amfani da su tare da sauran ƙididdiga masu bayani. Wannan hanyar ta cikakke tana tabbatar da cikakken fahimtar rarraba bayanai, tana haifar da yanke shawara mai kyau da kuma haɓaka sabbin abubuwan da aka gano.
