Mehlala ea Lipotso le Puisano ea Kamano ea Nako ea Sehlahisoa
Kamano ea Sehlahisoa le Nako, e tsejoang hape e le Pearson Correlation, ke mokhoa oa lipalo-palo o sebelisoang ho lekanya matla le tataiso ea kamano e otlolohileng lipakeng tsa mefuta e 'meli. Mokhoa ona o na le thuso masimong a fapaneng, ho tloha lipatlisisong tsa thuto le tlhahlobo ea khoebo ho isa tlhahlobong ea liteko mahlaleng a tlhaho. Sengoloa sena se tla tšohla mathata a 'maloa a mehlala le litharollo tsa ona bakeng sa ho bala Kamano ea Sehlahisoa le Nako.
Pendahuluan
Pele re kena lipotsong tsa mohlala, ke mohopolo o motle ho utloisisa mohopolo oa motheo oa Kamano ea Motsotso oa Sehlahisoa. Foromo e akaretsang e sebelisoang ho bala coefficient ea kamano ea Pearson (\(r\)) ke:
\[ r = \frac{n(\sum{XY}) – (\sum{X})(\sum{Y})}{\sqrt{[n\sum{X^2} – (\sum{X})^2][n\sum{Y^2} – (\sum{Y})^2]}} \]
Moo:
– \( n \) ke palo ea lipara tsa data.
– \( \sum{XY} \) ke kakaretso ya dihlahiswa tsa \( X \) le \( Y \).
– \( \sum{X} \) ke kakaretso ya diphetoho \( X \).
– \( \sum{Y} \) ke kakaretso ya diphetoho \( Y \).
– \( \sum{X^2} \) ke kakaretso ya dikwere tsa phetoho \( X \).
– \( \sum{Y^2} \) ke kakaretso ya dikwere tsa phetoho \( Y \).
Koefficient ea kamano ea Pearson (\( r \)) kamehla e pakeng tsa -1 le 1. Kamano e ntle e bontša hore li-variable ka bobeli li tsamaea ka lehlakoreng le le leng, ha kamano e mpe e bontša hore ha phetoho e le 'ngoe e ntse e eketseha, e 'ngoe ea fokotseha. Haeba \( r = 0 \), joale ha ho na kamano e otlolohileng pakeng tsa li-variable tse peli.
Mohlala oa Potso ea 1
Lintlha
Lintlha tse latelang ke lintlha tsa tlhahlobo ea lipalo le fisiks bakeng sa baithuti ba 5:
| Seithuti | Lipalo (X) | Fisiks (Y) |
|——-|——————-|———-|
| 1 | 85 | 90 |
| 2 | 78 | 85 |
| 3 | 85 | 80 |
| 4 | 70 | 70 |
| 5 | 80 | 88 |
Mehato ea ho Rarolla
1. Ho Bala Likarolo tsa Bohlokoa:
– \( \sum{X} \) = 85 + 78 + 85 + 70 + 80 = 398
– \( \sum{Y} \) = 90 + 85 + 80 + 70 + 88 = 413
– \( \sum{XY} \) = (85\ 90) + (78\ 85) + (85\ 80) + (70\ 70) + (80\ 88) = 7650 + 6630 + 6800 + 4900 + 7040 = 33020
– \( \sum{X^2} \) = (85^2) + (78^2) + (85^2) + (70^2) + (80^2) = 7225 + 6084 + 7225 + 4900 + 6400 = 31834
– \( \sum{Y^2} \) = (90^2) + (85^2) + (80^2) + (70^2) + (88^2) = 8100 + 7225 + 6400 + 4900 + 7744 = 34369
2. Kenya foromong:
\[ r = \frac{n(\sum{XY}) – (\sum{X})(\sum{Y})}{\sqrt{[n\sum{X^2} – (\sum{X})^2][n\sum{Y^2} – (\sum{Y})^2]}} \]
\[ r = \frac{5(33020) – (398)(413)}{\sqrt{[5(31834) – (398)^2][5(34369) – (413)^2]}} \]
3. Ho Bala Liphetho:
– Palo: \( 5(33020) – (398)(413) = 165100 – 164474 = 626 \)
– Sehlohlolo:
– \( n\sum{X^2} – (\sum{X})^2 = 5(31834) – (398)^2 = 159170 – 158404 = 766 \)
– \( n\sum{Y^2} – (\sum{Y})^2 = 5(34369) – (413)^2 = 171845 – 170569 = 1276 \)
– \( \sqrt{766 \times 1276} \hoo e ka bang \sqrt{976856} \hoo e ka bang 989.36 \)
\[ r = \frac{626}{989.36} \hoo e ka bang 0.633 \]
Ka hona, coefficient ea kamano ea Pearson pakeng tsa lintlha tsa liteko tsa lipalo le fisiks ke 0.633, e leng se bontšang hore ho na le kamano e mahareng pakeng tsa li-variable tse peli.
Mohlala oa Potso ea 2
Lintlha
Lintlha tse latelang ke tsa boleng ba thekiso le litšenyehelo tsa papatso ho tloha likhoeling tse 6 k'hamphaning:
| Khoeli | Papatso (X) | Thekiso (Y) |
|——-|————–|———————|
| 1 | 2000 | 2500 |
| 2 | 1800 | 2100 |
| 3 | 2200 | 2700 |
| 4 | 2400 | 2900 |
| 5 | 2300 | 3000 |
| 6 | 2500 | 3200 |
Mehato ea ho Rarolla
1. Ho Bala Likarolo tsa Bohlokoa:
– \( \sum{X} \) = 2000 + 1800 + 2200 + 2400 + 2300 + 2500 = 13200
– \( \sum{Y} \) = 2500 + 2100 + 2700 + 2900 + 3000 + 3200 = 16400
– \( \sum{XY} \) = (2000\ 2500) + (1800\ 2100) + (2200\ 2700) + (2400\ 2900) + (2300\ 3000) + (2500\ 3200) = 5000000 + 3780000 + 5940000 + 6960000 + 6900000 + 8000000 = 36580000
– \( \sum{X^2} \) = (2000^2) + (1800^2) + (2200^2) + (2400^2) + (2300^2) + (2500^2) = 4000000 + 3240000 + 4840000 + 5760000 + 5290000 + 6250000 = 29380000
– \( \sum{Y^2} \) = (2500^2) + (2100^2) + (2700^2) + (2900^2) + (3000^2) + (3200^2) = 6250000 + 4410000 + 7290000 + 8410000 + 9000000 + 10240000 = 45590000
2. Kenya foromong:
\[ r = \frac{n(\sum{XY}) – (\sum{X})(\sum{Y})}{\sqrt{[n\sum{X^2} – (\sum{X})^2][n\sum{Y^2} – (\sum{Y})^2]}} \]
\[ r = \frac{6(36580000) – (13200)(16400)}{\sqrt{[6(29380000) – (13200)^2][6(45590000) – (16400)^2]}} \]
3. Ho Bala Liphetho:
– Palo: \( 6(36580000) – (13200)(16400) = 219480000 – 216480000 = 3000000 \)
– Sehlohlolo:
– \( n\sum{X^2} – (\sum{X})^2 = 6(29380000) – (13200)^2 = 176280000 – 174240000 = 2040000 \)
– \( n\sum{Y^2} – (\sum{Y})^2 = 6(45590000) – (16400)^2 = 273540000 – 268960000 = 4580000 \)
– \( \sqrt{2040000 \times 4580000} \hoo e ka bang \sqrt{9343200000000} \hoo e ka bang 3056246.20 \)
\[ r = \frac{3000000}{3056246.20} \hoo e ka bang 0.981 \]
Ka hona, coefficient ea kamano ea Pearson pakeng tsa litšenyehelo tsa papatso le boleng ba thekiso ke 0.981, e leng se bontšang hore ho na le kamano e matla haholo pakeng tsa mefuta e 'meli.
Qetello
Palo e lekanang ea kamano ea Pearson (\(r\)) ke sesebelisoa se thusang haholo bakeng sa ho utloisisa kamano e otlolohileng lipakeng tsa li-variable tse peli. Mehlaleng e fanoeng, re bona mokhoa oa ho bala boleng ba \(r\) le ho bo hlalosa. Palo e phahameng (e haufi le 1 kapa -1) e bontša kamano e matla, ha kamano e tlase (e haufi le 0) e bontša kamano e fokolang. Ho bohlokoa ho hlokomela hore kamano ha e bolele sesosa; e bontša feela hore ho na le kamano pakeng tsa li-variable tse peli.
