Mafunso ndi Zitsanzo za Kukambirana za Kugwirizana kwa Nthawi Yogulitsa
Kugwirizana kwa Zinthu ndi Nthawi, komwe kumadziwikanso kuti Pearson Correlation, ndi njira yowerengera yomwe imagwiritsidwa ntchito poyesa mphamvu ndi njira ya ubale wolunjika pakati pa zinthu ziwiri. Njirayi ndi yothandiza m'magawo osiyanasiyana, kuyambira kafukufuku wamaphunziro ndi kusanthula bizinesi mpaka kuwunika kwa zoyeserera mu sayansi yachilengedwe. Nkhaniyi ikambirana zitsanzo zingapo za mavuto ndi mayankho awo pakuwerengera Kugwirizana kwa Zinthu ndi Nthawi.
Pendauluan
Tisanalowe mu mafunso a chitsanzo, ndi bwino kumvetsetsa lingaliro loyambira la Product Moment Correlation. Fomula yonse yomwe imagwiritsidwa ntchito powerengera coefficient ya Pearson correlation (\(r\)) ndi iyi:
\[ r = \frac{n(\sum{XY}) – (\sum{X})(\sum{Y})}{\sqrt{[n\sum{X^2} – (\sum{X})^2][n\sum{Y^2} – (\sum{Y})^2]}} \]
Kumene:
– \( n \) ndi chiwerengero cha ma data awiriawiri.
– \( \sum{XY} \) ndi chiwerengero cha zinthu zomwe \( X \) ndi \( Y \).
– \( \sum{X} \) ndi chiwonkhetso cha zosintha \( X \).
– \( \sum{Y} \) ndi chiwonkhetso cha zosintha \( Y \).
– \( \sum{X^2} \) ndi chiwerengero cha mabwalo a variable \( X \).
– \( \sum{Y^2} \) ndi chiwerengero cha mabwalo a variable \( Y \).
Chiŵerengero cha mgwirizano wa Pearson (\( r \)) nthawi zonse chimakhala pakati pa -1 ndi 1. Chiŵerengero chabwino chimasonyeza kuti ma variable onse awiri amayenda mbali imodzi, pomwe chiŵerengero choipa chimasonyeza kuti pamene variable imodzi ikuwonjezeka, inayo ikuchepa. Ngati \( r = 0 \), ndiye kuti palibe mgwirizano wolunjika pakati pa ma variable awiriwa.
Chitsanzo cha Funso 1
Deta
Zotsatirazi ndi ziwerengero za mayeso a masamu ndi fizikisi kwa ophunzira 5:
| Wophunzira | Masamu (X) | Fiziki (Y) |
|——-|——————-|———-|
| 1 | 85 | 90 |
| 2 | 78 | 85 |
| 3 | 85 | 80 |
| 4 | 70 | 70 |
| 5 | 80 | 88 |
Njira Zothetsera Mavuto
1. Kuwerengera Zigawo Zofunika:
– \( \sum{X} \) = 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. Lowetsani mu fomula iyi:
\[ r = \frac{n(\sum{XY}) – (\sum{X})(\sum{Y})}{\sqrt{[n\sum{X^2} – (\sum{X})^2][n\sum{Y^2} – (\sum{Y})^2]}} \]
\[ r = \frac{5(33020) – (398)(413)}{\sqrt{[5(31834) – (398)^2][5(34369) – (413)^2]}} \]
3. Kuwerengera Zotsatira:
– Nambala: \( 5(33020) – (398)(413) = 165100 – 164474 = 626 \)
- Chipembedzo:
– \( n\sum{X^2} – (\sum{X})^2 = 5(31834) – (398)^2 = 159170 – 158404 = 766 \)
– \( n\sum{Y^2} – (\sum{Y})^2 = 5(34369) – (413)^2 = 171845 – 170569 = 1276 \)
– \( \sqrt{766 \times 1276} \pafupifupi \sqrt{976856} \pafupifupi 989.36 \)
\[r = \frac{626}{989.36} \pafupifupi 0.633 \]
Motero, coefficient ya Pearson yolumikizana pakati pa zigoli za mayeso a masamu ndi fizikisi ndi 0.633, zomwe zikusonyeza kuti pali mgwirizano wabwino pakati pa zinthu ziwirizi.
Chitsanzo cha Funso 2
Deta
Izi ndi deta yokhudza mtengo wogulitsa ndi ndalama zogulira kuchokera ku miyezi 6 ku kampani:
| Mwezi | Kutsatsa (X) | Malonda (Y) |
|——-|————–|———————|
| 1 | 2000 | 2500 |
| 2 | 1800 | 2100 |
| 3 | 2200 | 2700 |
| 4 | 2400 | 2900 |
| 5 | 2300 | 3000 |
| 6 | 2500 | 3200 |
Njira Zothetsera Mavuto
1. Kuwerengera Zigawo Zofunika:
– \( \sum{X} \) = 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. Lowetsani mu fomula iyi:
\[ r = \frac{n(\sum{XY}) – (\sum{X})(\sum{Y})}{\sqrt{[n\sum{X^2} – (\sum{X})^2][n\sum{Y^2} – (\sum{Y})^2]}} \]
\[ r = \frac{6(36580000) – (13200)(16400)}{\sqrt{[6(29380000) – (13200)^2][6(45590000) – (16400)^2]}} \]
3. Kuwerengera Zotsatira:
– Nambala: \( 6(36580000) – (13200)(16400) = 219480000 – 216480000 = 3000000 \)
- Chipembedzo:
– \( n\sum{X^2} – (\sum{X})^2 = 6(29380000) – (13200)^2 = 176280000 – 174240000 = 2040000 \)
– \( n\sum{Y^2} – (\sum{Y})^2 = 6(45590000) – (16400)^2 = 273540000 – 268960000 = 4580000 \)
– \( \sqrt{2040000 \times 4580000} \pafupifupi \sqrt{9343200000000} \pafupifupi 3056246.20 \)
\[r = \frac{3000000}{3056246.20} \pafupifupi 0.981 \]
Motero, coefficient ya Pearson pakati pa ndalama zotsatsa ndi mtengo wogulitsa ndi 0.981, zomwe zikusonyeza kuti pali mgwirizano wamphamvu kwambiri pakati pa zinthu ziwirizi.
Mapeto
Pearson correlation coefficient (\(r\)) ndi chida chothandiza kwambiri pomvetsetsa ubale wa mzere pakati pa ma variable awiri. Mu zitsanzo zomwe zaperekedwa, tikuwona momwe tingawerengere mtengo wa \(r\) ndikuwutanthauzira. Ubale wapamwamba (pafupi ndi 1 kapena -1) umasonyeza ubale wolimba, pomwe ubale wotsika (pafupi ndi 0) umasonyeza ubale wofooka. Ndikofunikira kudziwa kuti ubwenzi sikutanthauza chifukwa; zimangosonyeza kuti pali ubale pakati pa ma variable awiriwa.
