Mibvunzo yemuenzaniso uye Kukurukurirana kweChigadzirwa Chenguva Yekubatana
Kubatana Kwechigadzirwa Nenguva, inozivikanwawo sePearson Correlation, inzira yekuverenga nhamba inoshandiswa kuyera simba negwara rehukama hwakatsetseka pakati pezvinhu zviviri. Nzira iyi inobatsira muzvikamu zvakasiyana-siyana, kubva pakutsvagisa kwedzidzo nekuongorora mabhizinesi kusvika pakuongororwa kwekuyedza mune zvesainzi yezvisikwa. Chinyorwa chino chichakurukura mienzaniso yakati wandei yezvinetso nemhinduro dzazvo pakuverenga Kubatana Kwechigadzirwa Nenguva.
Pendauluan
Tisati tapinda mumibvunzo yemuenzaniso, zvakanaka kuti tinzwisise pfungwa huru yeProduct Moment Correlation. Fomura inoshandiswa kuverenga Pearson correlation coefficient (\(r\)) ndeiyi:
\[ r = \frac{n(\sum{XY}) – (\sum{X})(\sum{Y})}{\sqrt{[n\sum{X^2} – (\sum{X})^2][n\sum{Y^2} – (\sum{Y})^2]}} \]
Dimana:
– \( n \) ndiyo nhamba yemapeya edata.
– \( \sum{XY} \) ihuwandu hwezvibereko zve \( X \) uye \( Y \).
– \( \sum{X} \) ihuwandu hwezvinhu zvinoshanduka \( X \).
– \( \sum{Y} \) ihuwandu hwezvinhu zvinoshanduka \( Y \).
– \( \sum{X^2} \) ihuwandu hwezvikwere zvechinoshanduka \( X \).
– \( \sum{Y^2} \) ihuwandu hwezvikwere zvechinoshanduka \( Y \).
Pearson's correlation coefficient (\( r \)) inogara iri pakati pe -1 ne 1. Positive correlation inoratidza kuti ma variable ese ari maviri anofamba akananga kumwe chete, nepo negative correlation ichiratidza kuti sezvo imwe variable ichiwedzera, imwe inoderera. Kana \( r = 0 \), saka hapana linear correlation pakati pema variable maviri.
Muenzaniso Mubvunzo 1
Data
Inotevera ndiyo data remapoinzi ebvunzo yemasvomhu nefizikisi yevadzidzi vashanu:
| Mudzidzi | Masvomhu (X) | Fizikisi (Y) |
|——-|——————-|———-|
| 1 | 85 | 90 |
| 2 | 78 | 85 |
| 3 | 85 | 80 |
| 4 | 70 | 70 |
| 5 | 80 | 88 |
Matanho Ekugadzirisa
1. Kuverenga Zvikamu Zvakakosha:
– \( \sum{X} \) = 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. Pinda mufomura:
\[ r = \frac{n(\sum{XY}) – (\sum{X})(\sum{Y})}{\sqrt{[n\sum{X^2} – (\sum{X})^2][n\sum{Y^2} – (\sum{Y})^2]}} \]
\[ r = \frac{5(33020) – (398)(413)}{\sqrt{[5(31834) – (398)^2][5(34369) – (413)^2]}} \]
3. Kuverenga Mhedzisiro:
– Nhamba: \( 5(33020) – (398)(413) = 165100 – 164474 = 626 \)
- Dhinomineta:
– \( n\sum{X^2} – (\sum{X})^2 = 5(31834) – (398)^2 = 159170 – 158404 = 766 \)
– \( n\sum{Y^2} – (\sum{Y})^2 = 5(34369) – (413)^2 = 171845 – 170569 = 1276 \)
– \( \sqrt{766 \times 1276} \approx \sqrt{976856} \approx 989.36 \)
\[r = \frac{626}{989.36} \inenge 0.633 \]
Saka, coefficient yePearson correlation pakati pezvibodzwa zvebvunzo yemasvomhu nefizikisi i0.633, zvichiratidza kuti pane correlation iri pakati pezviviri izvi.
Muenzaniso Mubvunzo 2
Data
Inotevera idata pamusoro pemutengo wekutengesa uye mari inoshandiswa pakushambadzira kubva pamwedzi mitanhatu kukambani:
| Mwedzi | Kushambadzira (X) | Kutengesa (Y) |
|——-|————–|———————|
| 1 | 2000 | 2500 |
| 2 | 1800 | 2100 |
| 3 | 2200 | 2700 |
| 4 | 2400 | 2900 |
| 5 | 2300 | 3000 |
| 6 | 2500 | 3200 |
Matanho Ekugadzirisa
1. Kuverenga Zvikamu Zvakakosha:
– \( \sum{X} \) = 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. Pinda mufomura:
\[ r = \frac{n(\sum{XY}) – (\sum{X})(\sum{Y})}{\sqrt{[n\sum{X^2} – (\sum{X})^2][n\sum{Y^2} – (\sum{Y})^2]}} \]
\[ r = \frac{6(36580000) – (13200)(16400)}{\sqrt{[6(29380000) – (13200)^2][6(45590000) – (16400)^2]}} \]
3. Kuverenga Mhedzisiro:
– Nhamba: \( 6(36580000) – (13200)(16400) = 219480000 – 216480000 = 3000000 \)
- Dhinomineta:
– \( n\sum{X^2} – (\sum{X})^2 = 6(29380000) – (13200)^2 = 176280000 – 174240000 = 2040000 \)
– \( n\sum{Y^2} – (\sum{Y})^2 = 6(45590000) – (16400)^2 = 273540000 – 268960000 = 4580000 \)
– \( \sqrt{2040000 \times 4580000} \approx \sqrt{9343200000000} \approx 3056246.20 \)
\[r = \frac{3000000}{3056246.20} \inenge 0.981 \]
Saka, chiyero chePearson correlation coefficient pakati pemari inoshandiswa pakushambadzira uye kukosha kwekutengesa i0.981, izvo zvinoratidza kuti pane hukama hwakasimba pakati pezvinhu zviviri izvi.
Mhedziso
Pearson correlation coefficient (\(r\)) chishandiso chinobatsira zvikuru pakunzwisisa hukama huri pakati pezvinhu zviviri. Mumienzaniso yakapihwa, tinoona maverengero e \(r\) kukosha uye kududzira. Kuwirirana kwepamusoro (pedyo ne1 kana -1) kunoratidza hukama hwakasimba, nepo kuwiriranisa kwakaderera (pedyo ne0) kunoratidza hukama husina kusimba. Zvakakosha kuziva kuti kuwiriranisa hakurevi causation; zvinongoratidza kuti pane hukama pakati pezvinhu zviviri izvi.
