Zitsanzo za Mafunso Okhudza Kusiyana ndi Kupatuka kwa Deta ya Gulu
Pendauluan
Mu ziwerengero, kusiyana ndi kupotoka kwa muyezo ndi miyeso iwiri ya ziwerengero yomwe ndi yofunika kwambiri kuti mumvetsetse kufalikira, kapena kufalikira, kwa deta kuchokera ku avareji. Kusiyana kumayesa kutalika kwa deta kuchokera ku avareji, pomwe kupotoka kwa muyezo ndi muzu wa kusiyana, zomwe zimapereka muyeso womwe uli mu mayunitsi omwewo ndi deta yoyambirira.
Tanthauzo
– Kusiyana (σ² kapena S²): Ndi avareji ya makwerero a kusiyana pakati pa mtengo uliwonse wa deta ndi avereji ya deta.
– Kupotoka Kokhazikika (σ kapena S): Ndi muzu wa sikweya wa kusiyana.
Fomula Yosinthira ndi Kupatuka Koyenera kwa Deta ya Gulu
Pa deta yamagulu, timagwiritsa ntchito kuchuluka kwa deta mu kalasi iliyonse. Nayi njira:
Zosintha
\[ S^2 = \frac{ \sum f_i \left( x_i – \bar{x} \right)^2 }{ N-1 } \]
Kupatuka Koyenera
\[ S = \sqrt{S^2} \]
Kumene:
– \( f_i \) = kuchuluka kwa kalasi iliyonse.
– \( x_i \) = pakati pa kalasi iliyonse.
– \( \bar{x} \) = avareji ya deta ya gulu.
– \( N \) = chiwerengero chonse cha deta.
Mafunso ndi Kukambirana Zitsanzo
Tiyerekeze kuti tili ndi deta yowerengera kulemera kwa gulu la anthu omwe agawidwa m'magulu.
| Kulemera (kg) | Kuchuluka kwa nthawi (f) |
|————————|————–|
| 50 – 54 | 2 |
| 55 – 59 | 5 |
| 60 – 64 | 8 |
| 65 – 69 | 7 |
| 70 – 74 | 3 |
Gawo loyamba ndikupeza pakati pa kalasi iliyonse ( \( x_i \) ) kenako kuwerengera avareji (\( \bar{x} \)).
1. Kuwerengera Pakati ( \( x_i \) )
\[ \text{Midpoint} = \frac{\text{Lower Limit} + \text{Upper Limit}}{2} \]
| Kulemera (kg) | Kuchuluka kwa nthawi (f) | Pakati (\( x_i \) ) |
|——————|————–|———————|
| 50 – 54 | 2 | 52 |
| 55 – 59 | 5 | 57 |
| 60 – 64 | 8 | 62 |
| 65 – 69 | 7 | 67 |
| 70 – 74 | 3 | 72 |
2. Kuwerengera Avereji ( \( \bar{x} \) )
\[ \bar{x} = \frac{ \sum f_i x_i }{ N } \]
Chiwerengero chonse cha deta \( N \):
\[ N = 2 + 5 + 8 + 7 + 3 = 25 \]
\[ \sum f_i x_i = (2 \nthawi 52) + (5 \nthawi 57) + (8 \nthawi 62) + (7 \nthawi 67) + (3 \nthawi 72) \]
\[ = 104 + 285 + 496 + 469 + 216 = 1570 \]
Kotero, avareji (\( \bar{x} \)):
\[ \bar{x} = \frac{ 1570 }{ 25 } = 62.8 \]
3. Kuwerengera Kusiyana ( \( S^2 \) )
Tifunika kuwerengera \( \sum f_i ( x_i – \bar{x} )^2 \):
\[
\begin{align}
(x_i – \bar{x})^2: & (52 – 62.8)^2 = 118.84 \\
& (57 – 62.8)^2 = 33.64 \\
& (62 – 62.8)^2 = 0.64 \\
& (67 – 62.8)^2 = 17.64 \\
& (72 – 62.8)^2 = 84.64
\end{align}
\]
Kuchulukitsa ndi mafupipafupi:
\[
\begin{align}
f_i (x_i – \bar{x})^2: & 2 \times 118.84 = 237.68 \\
& 5 \nthawi 33.64 = 168.2 \\
& 8 \nthawi 0.64 = 5.12 \\
& 7 \nthawi 17.64 = 123.48 \\
& 3 \nthawi 84.64 = 253.92
\end{align}
\]
\[
\sum f_i (x_i – \bar{x})^2 = 237.68 + 168.2 + 5.12 + 123.48 + 253.92 = 788.4
\]
Tsopano tikhoza kuwerengera kusiyana (\( S^2 \)):
\[ S^2 = \frac{ 788.4 }{ 25 – 1 } = \frac{ 788.4 }{ 24 } \pafupifupi 32.85 \]
4. Kuwerengera Kupotoka Koyenera ( \( S \) )
Kupatuka kwachizolowezi ( \( S \)):
\[ S = \sqrt{ S^2 } \]
\[ S = \sqrt{ 32.85 } \pafupifupi 5.73 \]
Mapeto
Kuchokera ku zitsanzo zomwe zili pamwambapa, tili ndi:
- Kuwerengera kulemera kwa thupi: 62.8 kg
– Kuwerengera kusiyana: 32.85 kg²
- Kuwerengera kupotoka kokhazikika: 5.73 kg
Kutanthauzira kwa kupotoka kokhazikika ndikuti kupotoka kwapakati pa deta yolemera kuchokera ku avareji ndi pafupifupi 5.73 kg. Izi zikusonyeza kufalikira kwa deta poyerekeza ndi avareji, zomwe zingathandize kudziwa momwe deta yathu imasinthira.
Kumvetsetsa bwino kusiyana kwa zinthu ndi kusiyana kwa zinthu n'kofunika kwambiri, makamaka kwa iwo omwe amagwira ntchito mu ziwerengero, kafukufuku, ndi mayeso, pamene akuyesera kumvetsetsa deta m'magulu kapena magawidwe. Kudziwa momwe mungawerengere ndikutanthauzira miyeso iwiriyi kungathandize kupanga zisankho zabwino kutengera deta yomwe ilipo.
