Imibuzo Eyisibonelo Exoxa Ngokuhlukahluka Nokuphambuka Okujwayelekile Kwedatha Yeqembu
I-Pendahuluan
Kuzibalo, ukuhlukahluka kanye nokuphambuka okujwayelekile kuyizilinganiso ezimbili zezibalo ezibalulekile ekuqondeni ukuhlakazeka, noma ukusabalala kwedatha kusuka ku-mean. Ukuhlukahluka kulinganisa ukuthi idatha isakazeke kude kangakanani kusuka ku-mean, kuyilapho ukuphambuka okujwayelekile kuyimpande yesikwele yokungafani, okunikeza isilinganiso esisezinyunithi ezifanayo nedatha yokuqala.
Incazelo
– Ukwehluka (σ² noma S²): Isilinganiso sezikwele somehluko phakathi kwenani ngalinye ledatha kanye nesilinganiso sedatha.
– Ukuphambuka Okujwayelekile (σ noma S): Kuyimpande yesikwele yokwehluka.
Ifomula Yokwehluka kanye Nokuphambuka Okujwayelekile Kwedatha Yeqembu
Kudatha yeqembu, sisebenzisa imvamisa yedatha ekilasini ngalinye. Nansi ifomula:
I-Varian
\[ S^2 = \frac{ \sum f_i \left( x_i – \bar{x} \right)^2 }{ N-1 } \]
Ukuphambuka Okujwayelekile
\[ S = \sqrt{S^2} \]
Kuphi:
– \( f_i \) = imvamisa yekilasi ngalinye.
– \( x_i \) = iphuzu eliphakathi lekilasi ngalinye.
– \( \bar{x} \) = isilinganiso sedatha yeqembu.
– \( N \) = inani eliphelele ledatha.
Imibuzo Eyisibonelo Nengxoxo
Ake sithi sinedatha yesisindo seqembu labantu abaqoqwe ngezigaba.
| Isikhawu Sesisindo (kg) | Imvamisa (f) |
|————————|————–|
| 50 – 54 | 2 |
| 55 – 59 | 5 |
| 60 – 64 | 8 |
| 65 – 69 | 7 |
| 70 – 74 | 3 |
Isinyathelo sokuqala ukuthola indawo ephakathi yekilasi ngalinye ( \( x_i \) ) bese kubalwa isilinganiso (\( \bar{x} \)).
1. Ukubala i-Midpoint ( \( x_i \) )
\[ \text{Midpoint} = \frac{\text{Lower Limit} + \text{Upper Limit}}{2} \]
| Isikhawu Sesisindo (kg) | Imvamisa (f) | Iphuzu Eliphakathi (\( x_i \) ) |
|——————|————–|———————|
| 50 – 54 | 2 | 52 |
| 55 – 59 | 5 | 57 |
| 60 – 64 | 8 | 62 |
| 65 – 69 | 7 | 67 |
| 70 – 74 | 3 | 72 |
2. Ukubala Isilinganiso ( \( \bar{x} \) )
\[ \bar{x} = \frac{ \sum f_i x_i }{ N } \]
Inani eliphelele ledatha \( N \):
\[ N = 2 + 5 + 8 + 7 + 3 = 25 \]
\[ \sum f_i x_i = (2 \izikhathi ezingu-52) + (5 \izikhathi ezingu-57) + (8 \izikhathi ezingu-62) + (7 \izikhathi ezingu-67) + (3 \izikhathi ezingu-72) \]
\[ = 104 + 285 + 496 + 469 + 216 = 1570 \]
Ngakho-ke, isilinganiso (\( \bar{x} \)):
\[ \bar{x} = \frac{1570}{25} = 62.8 \]
3. Ukubala Ukwehluka ( \( S^2 \) )
Sidinga ukubala \( \sum f_i ( x_i – \bar{x} )^2 \):
\[
\begin{align }
(x_i – \bar{x})^2: & (52 – 62.8)^2 = 118.84 \\
& (57 – 62.8)^2 = 33.64 \\
& (62 – 62.8)^2 = 0.64 \\
& (67 – 62.8)^2 = 17.64 \\
& (72 – 62.8)^2 = 84.64
\end{align }
\]
Ukuphindaphinda ngobuningi:
\[
\begin{align }
f_i (x_i – \bar{x})^2: & 2 \times 118.84 = 237.68 \\
& 5 \izikhathi 33.64 = 168.2 \\
kanye no-8 \izikhathi 0.64 = 5.12 \\
& 7 \izikhathi 17.64 = 123.48 \\
& 3 \izikhathi 84.64 = 253.92
\end{align }
\]
\[
\sum f_i (x_i – \bar{x})^2 = 237.68 + 168.2 + 5.12 + 123.48 + 253.92 = 788.4
\]
Manje sesingakwazi ukubala ukuhlukahluka (\( S^2 \)):
\[ S^2 = \frac{ 788.4 }{ 25 – 1 } = \frac{ 788.4 }{ 24 } \cishe 32.85 \]
4. Ukubala Ukuphambuka Okujwayelekile ( \( S \) )
Ukuphambuka okujwayelekile ( \( S \)):
\[ S = \sqrt{ S^2 } \]
\[ S = \sqrt{ 32.85 } \cishe 5.73 \]
Isiphetho
Kusukela kudatha yesibonelo engenhla, sinalokhu:
- Ukubala isisindo somzimba esimaphakathi: 62.8 kg
– Ukubala umehluko: 32.85 kg²
– Ukubala ukuphambuka okujwayelekile: 5.73 kg
Incazelo yokuphambuka okujwayelekile iwukuthi ukuphambuka okujwayelekile kwedatha yesisindo kusuka ku-average cishe kungama-5.73 kg. Lokhu kubonisa ukusabalala kwedatha maqondana ne-average, okungasiza ekunqumeni ukuthi idatha yethu iguquguquka kangakanani.
Ukuqonda okuphelele kokwehluka kanye nokuphambuka okujwayelekile kubalulekile, ikakhulukazi kulabo abasebenza kwizibalo, ucwaningo, kanye nokuhlola, njengoba bezama ukuqonda idatha ngesimo samaqembu noma ukusatshalaliswa. Ukwazi ukuthi ungabala futhi uhumushe kanjani lezi zilinganiso ezimbili kungasiza ekwenzeni izinqumo ezingcono ngokusekelwe kudatha ekhona.
