Imibuzo Nengxoxo Yezibonelo: Ukwehluka Nokuphambuka Okujwayelekile Kwedatha Eyodwa
Izibalo ziyigatsha lezibalo eliphathelene nokuqoqwa, ukuhlaziywa, ukuchazwa, ukwethula, kanye nokuhlela idatha. Imiqondo emibili ebalulekile kuzibalo ukuhlukahluka kanye nokuphambuka okujwayelekile. Lesi sihloko sizoxoxa ngokuningiliziwe ukuthi ungabala kanjani ukuhlukahluka kanye nokuphambuka okujwayelekile kwesethi yedatha eyodwa ngezinkinga eziningana zezibonelo.
Ukuqonda Ukwehluka Nokuphambuka Okujwayelekile
Ukwehluka kuyisilinganiso esilinganisa ukuthi izinombolo eziseqoqweni ledatha zisakazeke kude kangakanani kusukela ku-average noma isilinganiso. Ukwehluka kuvezwa ngamayunithi ayisikwele edatha yokuqala, okwenza kube nzima ukukuhumusha.
Ukuphambuka okujwayelekile kuyimpande yesikwele yokwehluka. Kunikeza isilinganiso esiqondakalayo sokuthi idatha ivame ukuphambuka kangakanani ku-mean ngoba amayunithi ayo afana namayunithi okuqala edatha.
Ifomula Ejwayelekile
Kudatha eyodwa, ifomula yokwehluka \( \sigma^2 \) kanye nokuphambuka okujwayelekile \( \sigma \) kwabantu imi kanje:
1. Ukwehluka (σ²):
\( \sigma^2 = \frac{1}{N} \sum_{i=1}^{N} (x_i – \mu)^2 \)
2. Ukuphambuka Okujwayelekile (σ):
\( \sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (x_i – \mu)^2} \)
Di mana:
– \( N \) inani ledatha kubantu.
– \( x_i \) inani ledatha elihambisanayo.
– \( \mu \) isilinganiso noma isilinganiso sedatha.
Uma sibala ukuhlukahluka kanye nokuphambuka okujwayelekile kwesampula, khona-ke ifomula engenhla iguqulwa kancane:
1. Ukwehluka Kwesampula (s²):
\( s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i – \bar{x})^2 \)
2. Isampula Yokuphambuka Okujwayelekile:
\( s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i – \bar{x})^2} \)
Di mana:
– \( n \) inani ledatha kusampula.
– \( \bar{x} \) isilinganiso noma isilinganiso sesampula.
Imibuzo Eyisibonelo Nengxoxo
Isibonelo Sombuzo 1:
Njengoba kunikezwe idatha elandelayo:
8, 10, 10, 10, 12, 14
Bala ukuhlukahluka kanye nokuphambuka okujwayelekile kwedatha!
Izinyathelo Zesixazululo:
1. Ukubala Isilinganiso (Isilinganiso) \( \mu \):
\[
\mu = \frac{8 + 10 + 10 + 10 + 12 + 14}{6} = \frac{64}{6} = 10.67
\]
2. Bala Umehluko Phakathi Kwedatha Nesilinganiso, Bese Kube Yisikwele:
\[
(8 – 10.67)^2 = 7.1289
\]
\[
(10 – 10.67)^2 = 0.4489
\]
\[
(10 – 10.67)^2 = 0.4489
\]
\[
(10 – 10.67)^2 = 0.4489
\]
\[
(12 – 10.67)^2 = 1.7689
\]
\[
(14 – 10.67)^2 = 11.1089
\]
3. Ukwengeza Yonke Imiphumela Eyisikwele:
\[
\ isamba (x_i – \mu)^2 = 7.1289 + 0.4489 + 0.4489 + 0.4489 + 1.7689 + 11.1089 = 21.3534
\]
4. Ukubala Ukwehluka (σ²) Kwedatha Yabantu:
\[
\sigma^2 = \frac{21.3534}{6} = 3.559
\]
Qaphela: Njengoba le datha ibizwa ngokuthi idatha yabantu, sihlukanisa ngo-6.
5. Ukubala Ukuphambuka Okujwayelekile (σ):
\[
\sigma = \sqrt{3.559} \cishe kube ngu-1.886
\]
Ngakho-ke, ukuhluka kwedatha kungu-3.559 kanti ukuphambuka okujwayelekile kungu-1.886.
Isibonelo Sombuzo 2:
Njengoba kunikezwe idatha elandelayo yesampula:
5, 6, 8, 9, 10, 11
Bala umehluko kanye nokuphambuka okujwayelekile kwesampula!
Izinyathelo Zesixazululo:
1. Ukubala Isilinganiso (Isilinganiso) \( \bar{x} \) :
\[
\bar{x} = \frac{5 + 6 + 8 + 9 + 10 + 11}{6} = \frac{49}{6} = 8.167
\]
2. Bala Umehluko Phakathi Kwedatha Nesilinganiso, Bese Kube Yisikwele:
\[
(5 – 8.167)^2 = 10.035
\]
\[
(6 – 8.167)^2 = 4.694
\]
\[
(8 – 8.167)^2 = 0.028
\]
\[
(9 – 8.167)^2 = 0.694
\]
\[
(10 – 8.167)^2 = 3.361
\]
\[
(11 – 8.167)^2 = 7.945
\]
3. Ukwengeza Yonke Imiphumela Eyisikwele:
\[
\isamba (x_i – \ibha{x})^2 = 10.035 + 4.694 + 0.028 + 0.694 + 3.361 + 7.945 = 26.757
\]
4. Ukubala Ukwehluka Kwesampula (s²):
\[
s^2 = \frac{26.757}{5} = 5.351
\]
Qaphela: Njengoba lokhu kuyidatha yesampula, sihlukanisa ngo-5 (n-1).
5. Ukubala Ukuphambuka Okujwayelekile:
\[
s = \sqrt{5.351} \cishe 2.313
\]
Ngakho-ke, ukuhluka kwedatha yesampula kungu-5.351 kanti ukuphambuka okujwayelekile kungu-2.313.
Isiphetho
Ukubala ukuhlukahluka kanye nokuphambuka okujwayelekile kubalulekile ekuqondeni ukuthi idatha esabalele ingaphakathi kwesethi ethile. Ngenkathi ukuhlukahluka kunikeza isilinganiso sethiyori sokuhlakazeka ezikweleni zamayunithi okuqala, ukuphambuka okujwayelekile kuhumusha isilinganiso sokuhlakazeka ngokwemayunithi okuqala edatha, okwenza kube lula ukukuqonda. Ekuhlaziyweni kwedatha, lezi zilinganiso ezimbili zivame ukusetshenziselwa ukuhlola ukuhlukahluka kwedatha nokwenza izinqumo zezibalo.
Ngokuqonda izinyathelo namafomula adingekayo, singabala kalula ukuhlukahluka kanye nokuphambuka okujwayelekile kwezimo ezahlukahlukene esibhekana nazo ekuqoqweni nasekuhlaziyweni kwedatha kwansuku zonke. Ngethemba ukuthi lesi sihloko sizokunikeza ukuqonda okujulile ngemibono yokuhlukahluka kanye nokuphambuka okujwayelekile kumasethi edatha eyodwa.
