Fa'ata'ita'iga o Fesili ma Talanoaga o le Product Moment Correlation
O le Product-Moment Correlation, e ta'ua fo'i o le Pearson Correlation, o se metotia fa'amaufa'ailoga e fa'aaogaina e fua ai le malosi ma le itu o se mafutaga fa'asolosolo i le va o fesuia'iga e lua. E aoga lenei metotia i matā'upu eseese, mai su'esu'ega fa'alea'oa'oga ma au'ili'iliga fa'apisinisi i le iloiloga o fa'ata'ita'iga i saienisi fa'alenatura. O lenei tusiga o le a talanoaina ai ni fa'ata'ita'iga o fa'afitauli ma a latou fofo mo le fuafuaina o le Product-Moment Correlation.
Pendahuluan
A o le’i oo atu i fesili fa’ata’ita’i, o se manatu lelei le malamalama i le manatu autū o le Product Moment Correlation. O le fua fa’atatau lautele e fa’aaogaina e fuafua ai le Pearson correlation coefficient (\(r\)) o le:
\[ r = \frac{n(\sum{XY}) – (\sum{X})(\sum{Y})}{\sqrt{[n\sum{X^2} – (\sum{X})^2][n\sum{Y^2} – (\sum{Y})^2]}} \]
O fea:
– \( n \) o le aofaʻi o paga o faʻamatalaga.
– \( \sum{XY} \) o le aofaʻi lea o oloa o \( X \) ma le \( Y \).
– \( \sum{X} \) o le aofaʻi lea o fesuiaʻiga \( X \).
– \( \sum{Y} \) o le aofaʻi lea o fesuiaʻiga \( Y \).
– \( \sum{X^2} \) o le aofaʻi lea o sikuea o le fesuiaʻiga \( X \).
– \( \sum{Y^2} \) o le aofaʻi lea o sikuea o le fesuiaʻiga \( Y \).
O le fua fa'atatau o le feso'ota'iga a Pearson (\( r \)) e masani lava i le va o le -1 ma le 1. O se feso'ota'iga lelei e fa'ailoa mai ai o fesuia'iga uma e lua e fealua'i i le itu e tasi, ae o se feso'ota'iga leaga e fa'ailoa mai ai a fa'ateleina le tasi fesuia'iga, e fa'aitiitia le isi. Afai \( r = 0 \), ona leai lea o se feso'ota'iga laina i le va o fesuia'iga e lua.
Fa'ata'ita'iga Fesili 1
Faʻamatalaga
O faʻamaumauga nei o togi o le suega matematika ma le fisiki mo tamaiti aʻoga e toʻa 5:
| Tamaititi Aʻoga | Matematika (X) | Fisiki (Y) |
|——-|——————-|—————-|
| 1 | 85 | 90 |
| 2 | 78 | 85 |
| 3 | 85 | 80 |
| 4 | 70 | 70 |
| 5 | 80 | 88 |
Laasaga e Foia ai
1. Fuafuaina o Vaega Taua:
– \( \sum{X} \) = 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. Ulufale i le fua fa'atatau:
\[ r = \frac{n(\sum{XY}) – (\sum{X})(\sum{Y})}{\sqrt{[n\sum{X^2} – (\sum{X})^2][n\sum{Y^2} – (\sum{Y})^2]}} \]
\[ r = \frac{5(33020) – (398)(413)}{\sqrt{[5(31834) – (398)^2][5(34369) – (413)^2]}} \]
3. Fuafuaina o Taunuuga:
– Numera: \( 5(33020) – (398)(413) = 165100 – 164474 = 626 \)
– Fa'ailoga:
– \( n\sum{X^2} – (\sum{X})^2 = 5(31834) – (398)^2 = 159170 – 158404 = 766 \)
– \( n\sum{Y^2} – (\sum{Y})^2 = 5(34369) – (413)^2 = 171845 – 170569 = 1276 \)
– \( \sqrt{766 \times 1276} \approx \sqrt{976856} \approx 989.36 \)
\[ r = \frac{626}{989.36} \approx 0.633 \]
O lea la, o le fua fa'atatau o le feso'ota'iga a Pearson i le va o togi o le suega matematika ma le fisiki e 0.633, lea e fa'ailoa mai ai o lo'o i ai se feso'ota'iga lelei feololo i le va o fesuia'iga e lua.
Fa'ata'ita'iga Fesili 2
Faʻamatalaga
O faʻamatalaga nei i le tau o faʻatauga ma tupe faʻaalu i faʻasalalauga mai le 6 masina i se kamupani:
| Masina | Faasalalauga (X) | Faatauga (Y) |
|——-|—————–|——————|
| 1 | 2000 | 2500 |
| 2 | 1800 | 2100 |
| 3 | 2200 | 2700 |
| 4 | 2400 | 2900 |
| 5 | 2300 | 3000 |
| 6 | 2500 | 3200 |
Laasaga e Foia ai
1. Fuafuaina o Vaega Taua:
– \( \sum{X} \) = 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. Ulufale i le fua fa'atatau:
\[ r = \frac{n(\sum{XY}) – (\sum{X})(\sum{Y})}{\sqrt{[n\sum{X^2} – (\sum{X})^2][n\sum{Y^2} – (\sum{Y})^2]}} \]
\[ r = \frac{6(36580000) – (13200)(16400)}{\sqrt{[6(29380000) – (13200)^2][6(45590000) – (16400)^2]}} \]
3. Fuafuaina o Taunuuga:
– Numera: \( 6(36580000) – (13200)(16400) = 219480000 – 216480000 = 3000000 \)
– Fa'ailoga:
– \( n\sum{X^2} – (\sum{X})^2 = 6(29380000) – (13200)^2 = 176280000 – 174240000 = 2040000 \)
– \( n\sum{Y^2} – (\sum{Y})^2 = 6(45590000) – (16400)^2 = 273540000 – 268960000 = 4580000 \)
– \( \sqrt{2040000 \times 4580000} \approx \sqrt{9343200000000} \approx 3056246.20 \)
\[ r = \frac{3000000}{3056246.20} \approx 0.981 \]
O le mea lea, o le fua fa'atatau o le feso'ota'iga a Pearson i le va o tupe fa'aalu i fa'asalalauga ma le tau o fa'atauga e 0.981, lea e fa'ailoa mai ai o lo'o i ai se feso'ota'iga lelei malosi i le va o fesuia'iga e lua.
I'uga
O le Pearson correlation coefficient (\(r\)) o se meafaigaluega aoga tele mo le malamalama i le sootaga tuusao i le va o fesuiaiga e lua. I faʻataʻitaʻiga ua tuʻuina atu, tatou te vaʻaia ai le auala e fuafua ai le tau o le \(r\) ma faʻamatalaina. O le maualuga o le fesoʻotaʻiga (e latalata i le 1 poʻo le -1) e faʻaalia ai se sootaga malosi, ae o le maualalo o le fesoʻotaʻiga (e latalata i le 0) e faʻaalia ai se sootaga vaivai. E taua le maitauina o le fesoʻotaʻiga e le faʻaalia ai le mafuaʻaga; e naʻo le faʻaalia ai o loʻo i ai se sootaga i le va o fesuiaiga e lua.
