Ukuhlaziywa Kokuhlukahluka kanye Nokuphambuka Okujwayelekile Ekusabalalisweni Kwedatha
Kuzibalo, ukuqonda ukusatshalaliswa kwedatha kubaluleke njengokuqonda amanani aphakathi njenge-mean noma i-median. Amasethi amabili edatha angaba nesilinganiso esifanayo, kodwa ukusatshalaliswa kwawo kuhlukile kakhulu: elinye lingase lihlanganiswe ngokuqinile eduze kwesilinganiso, kanti elinye lingase lisakazeke kabanzi. Yilapho ukungafani kanye nokuphambuka okujwayelekile kungena khona—kuyizilinganiso ezibalulekile zokuthi idatha ihluka kangakanani enanini layo eliphakathi. Lesi sihloko sixoxa ngemibono yabo, amafomula, izincazelo, kanye nezibonelo zokusetshenziswa kwabo ekuhlaziyweni kwedatha.
1. Kungani Ukusatshalaliswa Kwedatha Kubalulekile?
Ukusabalala kwedatha kunikeza ulwazi mayelana nokungaguquguquki kanye nengozi. Isibonelo, kumongo wamaphuzu okuhlolwa, isilinganiso samakilasi A no-B singaba ngu-80. Kodwa-ke, uma umehluko wamaphuzu ekilasi A uncane, iningi labafundi lenza ngendlela efanayo. Ngokuphambene nalokho, uma umehluko wamaphuzu ekilasi B umkhulu, kungenzeka ukuthi abanye abafundi banamaphuzu aphezulu kakhulu kanti abanye banamaphuzu aphansi kakhulu. Ebhizinisini, ukusabalala kwedatha yokuthengisa kubonisa ukuzinza kwemali engenayo; kwezezimali, ukusabalala kwembuyiselo yokutshalwa kwezimali kubonisa izinga lengozi.
Ngokuqonda ukuhlukahluka kanye nokuphambuka okujwayelekile, abenzi bezinqumo bangenza okulandelayo:
– Hlola ukuthi inqubo ethile izinzile noma cha (isib. ukukhiqizwa kwefektri).
- Ukuqhathanisa ukuvumelana phakathi kwamaqembu (isib. izindlela ezimbili zokufunda).
- Ukuthola idatha engaphandle okufanele ihlolwe.
- Ukulinganisa ukungaqiniseki ezibikezelweni nasemamodelini.
2. Umqondo Oyisisekelo Wokwehluka
Ukwehluka kulinganisa ukuphambuka okuyisikwele okumaphakathi kwesethi ngayinye yedatha kusuka ku-mean. Ukuphambuka umehluko phakathi kwamanani edatha kanye ne-mean. Uma amanani amaningi ekude ne-mean, ukwehluka kuzoba kukhulu. Uma amanani eseduze ne-mean, ukwehluka kuzoba kuncane.
Ake sithi kukhona idatha: \(x_1, x_2, …, x_n\) enesilinganiso se-\(\bar{x}\). Ukuphambuka kwedatha ngayinye kungu-\(x_i – \bar{x}\). Kodwa-ke, uma ukuphambuka kwengezwa ngqo, umphumela uhlala u-zero ngoba kukhona ukuphambuka okuhle nokubi okukhanselana. Ukuze kunqotshwe lokhu, ukuphambuka kuhlukaniswa ngesikwele ukuze konke kube kuhle. Yilapho ukuphambuka kuzalwa khona.
a) Ukwehluka Kwenani Labantu
Uma idatha ibhekwa njengemele lonke inani labantu, ukuhlukahluka kwenani labantu kubhalwa kanje:
\[
\sigma^2 = \frac{\sum_{i=1}^{N}(x_i – \mu)^2}{N}
\]
Kuphi:
– \(N\) inani ledatha yabantu,
– \(\mu\) isilinganiso sabantu,
– \(\sigma^2\) ukuhlukahluka kwenani labantu.
b) Ukwehluka Kwesampula
Uma idatha iyisampula evela kubantu abaningi, kusetshenziswa ukuhlukahluka kwesampula:
\[
s^2 = \frac{\sum_{i=1}^{n}(x_i – \bar{x})^2}{n-1}
\]
Isihlukanisi \(n-1\) sibizwa ngokuthi ukulungiswa kweBessel, futhi sisetshenziselwa ukuqinisekisa ukuthi isilinganiso sokwehluka kwabantu asikhethi. Empeleni, ngoba isilinganiso sesampula sibalwa kusukela kudatha uqobo, kukhona "ukulahlekelwa amazinga enkululeko," ngakho isihlukanisi silungiswa ngokufanele.
3. Ukuphambuka Okujwayelekile: Umsuka Wokuhlukahluka
Ukwehluka kunesici esisodwa esiwusizo: amayunithi ayo ayizikwele zamayunithi edatha. Uma idatha iku-"rupiah", ukwehluka kuku-"rupiah²", okunzima ukukuhumusha ngokuqondile. Ngakho-ke, sisebenzisa ukuphambuka okujwayelekile, okuyimpande yesikwele yokwehluka.
a) Ukuphambuka Kwezinga Lomphakathi
\[
\sigma = \sqrt{\sigma^2}
\]
b) Ukuphambuka Okujwayelekile Kwesampula
\[
s = \sqrt{s^2}
\]
Ukuphambuka okujwayelekile kunamayunithi afanayo nedatha yokuqala, okwenza kube lula ukuyiqonda. Ukuphambuka okujwayelekile okuphezulu kubonisa idatha esabalele kakhulu; ukuphambuka okujwayelekile okuphansi kubonisa isethi yedatha eningi kakhulu.
4. Isibonelo Esilula Sokubala
Isibonelo, idatha yamaphuzu okuhlolwa: 70, 75, 80, 85, 90.
1) Bala isilinganiso:
\[
\bar{x} = \frac{70+75+80+85+90}{5} = 80
\]
2) Bala ukuphambuka kwenani ngalinye kusuka ku-mean:
– 70: \(70-80=-10\)
– 75: \(75-80=-5\)
– 80: \(80-80=0\)
– 85: \(85-80=5\)
– 90: \(90-80=10\)
3) Yenza isikwele ukuphambuka:
– 100, 25, 0, 25, 100
4) Engeza:
\[
\isamba (x_i-\ibha{x})^2 = 250
\]
5) Ukwehluka kwesampula:
\[
s^2 = \frac{250}{5-1} = 62.5
\]
6) Isampula yokuphambuka okujwayelekile:
\[
s = \sqrt{62.5} \cishe 7.91
\]
Incazelo: isilinganiso samaphuzu singama-80, kanti “ngokuvamile” amaphuzu ahluka ngamaphuzu angaba ngu-7-8 kusilinganiso.
5. Ukuhunyushwa Kokuhlukahluka Nokuphambuka Okujwayelekile
Ukwehluka kanye nokuphambuka okujwayelekile akuzona nje izinombolo; kumele zihunyushwe ngokomongo.
– Ukuphambuka okuncane okujwayelekile: ukuhambisana okuphezulu. Isibonelo, inqubo yokukhiqiza enokuphambuka okuncane kakhulu okujwayelekile ngobukhulu bomkhiqizo ikhombisa ikhwalithi ezinzile.
– Ukuphambuka okukhulu okujwayelekile: ukuhlukahluka okuphezulu. Ekutshalweni kwezimali, ukuphambuka okuphezulu kwenzuzo kusho ukuguquguquka okuphezulu (ingozi ephezulu).
– Ukuqhathanisa phakathi kwamaqembu: uma amaqembu amabili enokwehluka okujwayelekile okufanayo kodwa okuhlukile, iqembu elinokuhluka okuncane liyafana kakhulu.
Noma kunjalo, kubalulekile ukukhumbula ukuthi ukuphambuka okujwayelekile kuyazwela ezintweni ezingaphandle. Inani elilodwa elidlulele lingakhulisa kakhulu ukuphambuka kanye nokuphambuka okujwayelekile. Ngakho-ke, ukuhlaziywa kokusabalalisa kuvame ukuhambisana nokubonakala (ama-histogram, ama-boxplot) noma izindlela eziqinile njenge-IQR (ububanzi be-interquartile).
6. Ubudlelwano Nokusatshalaliswa Okuvamile kanye Nemithetho Eyisisekelo
Ekusabalalisweni okuvamile (ijika lensimbi), ukuphambuka okujwayelekile kunencazelo enamandla kakhulu. Kunomthetho we-empirical ovame ukusetshenziswa:
– Cishe u-68% wedatha usebangeni \(\bar{x} \pm 1s\)
– Cishe u-95% wedatha usebangeni \(\bar{x} \pm 2s\)
– Cishe u-99,7% wedatha usebangeni \(\bar{x} \pm 3s\)
Lo mthetho usiza ekwenzeni izincazelo ezisheshayo, isibonelo ukuhlola ukuthi inani "alilona elemvelo" noma lisesebangeni elijwayelekile.
7. Izicelo Ezinkambeni Ezihlukahlukene
1) Imfundo: Ukuqapha ukusatshalaliswa kwamamaki abafundi. Ukuphambuka okuncane kubonisa imiphumela yokufunda elinganayo, kanti ukuphambuka okukhulu kungabonisa izikhala ekuqondeni.
2) Imboni: ukulawulwa kwekhwalithi. Ukwehluka kusetshenziswa ukuhlola ukuhambisana kokukhiqiza.
3) Ezezimali: zilinganisa ukuguquguquka kwentengo yesitoko, imbuyiselo yephothifoliyo, kanye nengozi yokutshalwa kwezimali.
4) Impilo: ukubona ukwehluka komfutho wegazi, amazinga kashukela, noma ezinye izinkomba zemitholampilo kubantu abagulayo.
5) Ucwaningo lwezenhlalo: ukuhlola ukungafani kwezimpendulo zocwaningo kanye nokuhlukahluka kwezici zabaphenduli.
8. Amaphutha Avamile Namathiphu Awusizo
Amanye amaphutha avamile:
– Ukusebenzisa ukuhlukahluka kwesampula (isihlukanisi \(n-1\)) noma idatha iyinani eliphelele, noma okuphambene nalokho.
– Chaza ukuhlukahluka ngaphandle kokucabangela amayunithi aso esikwele; kuphephile ukusebenzisa ukuphambuka okujwayelekile ekuchazeni.
- Unganaki izinto ezingaphandle; kungcono ukuhlola idatha kuqala.
– Qhathanisa ukuphambuka okujwayelekile phakathi kwedatha enezilinganiso ezahlukene ngaphandle kokujwayelekile; kwezinye izimo, sebenzisa i-coefficient of variation (CV) okungukuthi \(CV = \frac{s}{\bar{x}}\times 100\%\) ukuze uthole ukuqhathanisa okufanelekile.
I-Penutup
Ukwehluka kanye nokuphambuka okujwayelekile kuyizinto eziyisisekelo zokuqonda ukusatshalaliswa kwedatha. Ukwehluka kunikeza isisekelo esiqinile sezibalo, kuyilapho ukuphambuka okujwayelekile kunikeza isilinganiso okulula ukusichaza ngoba sifana nedatha yokuqala. Ngokusebenzisa lezi zilinganiso ezimbili, singahlola ngokucacile ukuvumelana, ingozi, kanye nomehluko ezicini zokusabalalisa phakathi kwamasethi edatha. Ekuzijwayezeni kokuhlaziya idatha, ukwehluka kanye nokuphambuka okujwayelekile kusetshenziswa kangcono kanye nezilinganiso zokuthambekela okuphakathi kanye nokubona ukuze kunikezwe isithombe esiphelele sedatha futhi kwenziwe izinqumo ezinolwazi oluthe xaxa.
Uma uthanda, ngingangeza izibonelo zokubala eziyinkimbinkimbi kakhulu (isb. idatha eqoqwe ngamaqoqo), noma ngichaze ubudlelwano bokuphambuka okujwayelekile ne-z-score kanye nokutholwa okungaphandle.
