Isibonelo Sombuzo Wengxoxo Ngokusatshalaliswa Okuvamile
Ukusatshalaliswa okuvamile, okwaziwa nangokuthi ukusatshalaliswa kwe-Gaussian, ukusatshalaliswa kwamathuba okusetshenziswa kakhulu kuzibalo. Lokhu kusatshalaliswa kunesimo sensimbi esilinganayo, okubonisa ukuthi idatha ihlelwe ngokuzungeza isilinganiso futhi amathuba okudlulela (amanani akude nesilinganiso) aphansi.
Kulesi sihloko, sizoxoxa ngezinkinga ezahlukahlukene zezibonelo ezihilela ukusatshalaliswa okuvamile kanye nendlela yokuzixazulula. Sizoqala ngokwethula imiqondo eyisisekelo bese siqhubekela ezibonelweni eziyinkimbinkimbi kakhulu.
Izisekelo Zokusabalalisa Okuvamile
Ukusatshalaliswa okuvamile kuwukusatshalaliswa okuqhubekayo okunamapharamitha amabili: isilinganiso kanye nokuphambuka okujwayelekile (SD). Isilinganiso sinquma isikhungo sokusabalalisa, kuyilapho ukuphambuka okujwayelekile kunquma ububanzi bokusatshalaliswa.
Izici ezibalulekile zokusatshalaliswa okuvamile:
1. Ukulingana: Ukusatshalaliswa okuvamile kufana nokulingana mayelana nesilinganiso.
2. Umthetho Wokubusa (Umthetho Wokubusa):
– Cishe u-68% wedatha usengaphakathi kokuphambuka okujwayelekile kwesilinganiso.
– Cishe u-95% wedatha ungaphakathi kokuphambuka okujwayelekile okubili kwesilinganiso.
– Cishe u-99.7% wedatha usekuhlukaneni okujwayelekile okuthathu kwesilinganiso.
Imibuzo Eyisibonelo Nengxoxo
Isibonelo Umbuzo 1: Ukubala i-Z-Score
Umbuzo: Isivivinyo sinamaphuzu aphakathi angu-70 kanye nokuphambuka okujwayelekile okungu-10. Umfundi uthola amaphuzu angu-80. Iyini amaphuzu ka-Z omfundi?
Isixazululo:
I-Z-score iyisilinganiso sokuthi inani lingakanani ukuphambuka okujwayelekile kusuka ku-mean.
Ifomula ye-Z-score:
\[ Z = \frac{X – \mu}{\sigma} \]
Kuphi:
– \( X \) inani elibonwayo.
– \( \mu \) isilinganiso.
– \( \sigma \) ukuphambuka okujwayelekile.
Kuyaziwa:
– \( X = 80 \)
– \( \mu = 70 \)
– \( \sigma = 10 \)
Ukusetshenziswa kwefomula:
\[ Z = \frac{80 – 70}{10} = 1 \]
Ngakho-ke, amaphuzu ka-Z omfundi angu-1, okusho ukuthi amaphuzu angu-80 ayindlela eyodwa yokuphambuka ejwayelekile ngaphezu kwesilinganiso.
Isibonelo Umbuzo 2: Amathuba Enani Elithile
Umbuzo: Ekusabalaleni okuvamile okunesilinganiso esingu-100 kanye nokuphambuka okujwayelekile okungu-15, iyini amathuba okuthola inani elingaphansi kuka-85?
Isixazululo:
Izinyathelo:
1. Bala i-Z-score ngenani \( X = 85 \):
\[ Z = \frac{85 – 100}{15} = \frac{-15}{15} = -1 \]
2. Sebenzisa ithebula lika-Z noma isibali sezibalo ukuthola amathuba ahambisana nesikolo sika-Z esingu--1. Kuthebula lika-Z, amathuba esikolo sika-Z esingu--1 cishe angama-0.1587.
Ngakho-ke, amathuba okuthola inani elingaphansi kuka-85 angama-0.1587 noma angu-15.87%.
Isibonelo Umbuzo 3: Ukusebenzisa Imithetho Eyisisekelo
Umbuzo: Kuyaziwa ukuthi ukusatshalaliswa kwamaphuzu okuhlolwa kwezibalo ezikoleni kulandela ukusatshalaliswa okuvamile okunesilinganiso esingu-75 kanye nokuphambuka okujwayelekile okungu-8. Yiliphi inani labafundi abathole amaphuzu aphakathi kuka-67 no-83?
Isixazululo:
I-Langkah-langkah:
1. Bala amaphuzu ka-Z ngamanani angu-67 no-83:
\[ Z_{67} = \frac{67 – 75}{8} = \frac{-8}{8} = -1 \]
\[ Z_{83} = \frac{83 – 75}{8} = \frac{8}{8} = 1 \]
2. Ngokwemithetho ye-empirical, amanani aphakathi kwe--1 SD kanye ne-+1 SD avela embozweni ophakathi cishe ama-68% wabantu.
Ngakho-ke, isilinganiso sabafundi abathole amaphuzu aphakathi kuka-67 no-83 sasicishe sibe ngu-68%.
Isibonelo Umbuzo 4: Ukubala Amanani Kusuka Kuma-Percentile
Umbuzo: Uma ubude obujwayelekile babesilisa abadala ezweni bungu-175 cm kanye nokuphambuka okujwayelekile okungu-7 cm, yikuphi ukuphakama okuku-90th percentile?
Isixazululo:
I-Langkah-langkah:
1. Thola i-Z-score ehambisana nephesenti lama-90. Ngokusekelwe kuthebula le-Z, i-Z-score eseduze no-0.9000 cishe ingu-1.28.
2. Sebenzisa ifomula ukubala inani le-\( X \):
\[ X = \mu + Z \izikhathi \sigma \]
\[ X = 175 + 1.28 \izikhathi ezingu-7 \]
\[ X = 175 + 8.96 \]
\[ X = 183.96 \]
Ngakho-ke, ukuphakama ku-percentile engama-90 kungama-183.96 cm.
Isibonelo Umbuzo 5: Amathuba Esikhathi Esithile
Umbuzo: Njengoba ukusatshalaliswa kwesisindo esisanda kuzalwa kulandela ukusatshalaliswa okuvamile okunesilinganiso esingu-3.5 kg kanye nokuphambuka okujwayelekile okungu-0.5 kg, yimaphi amathuba okuthi ingane inesisindo esiphakathi kuka-3 kg no-4 kg?
Isixazululo:
I-Langkah-langkah:
1. Bala amaphuzu ka-Z ngamanani angu-3 kg kanye no-4 kg:
\[ Z_{3} = \frac{3 – 3.5}{0.5} = \frac{-0.5}{0.5} = -1 \]
\[ Z_{4} = \frac{4 – 3.5}{0.5} = \frac{0.5}{0.5} = 1 \]
2. Amathuba e-Z-score phakathi kuka--1 no-1 ngokusekelwe kuthebula lika-Z cishe angu-0.6826 noma angu-68.26%.
Ngakho-ke, amathuba okuba ingane ibe nesisindo esiphakathi kuka-3 kg no-4 kg angaba ngu-68.26%.
Isiphetho
Ukusatshalaliswa okuvamile kuwumqondo oyisisekelo kwizibalo obalulekile futhi onezinhlelo zokusebenza eziningi zangempela. Kulesi sihloko, sichaze imiqondo eyisisekelo yokusatshalaliswa okuvamile futhi saxazulula izibonelo eziningana ukuze sijulise ukuqonda kwethu.
Ukuqonda ukusatshalaliswa okuvamile akubalulekile nje kuphela ezibalweni kodwa futhi nasezindaweni ezahlukahlukene ezisebenzayo njengezengqondo, ezomnotho, kanye nezinye isayensi yezenhlalo. Ngokuzijwayeza okwanele, ukuxazulula izinkinga zokusatshalaliswa okuvamile kungaba lula futhi kusize ekwenzeni izinqumo eziqhutshwa idatha.
