Tusaale su'aal dood ah oo ku saabsan isku-dhafka go'aaminta
Isu-dheellitirka go'aaminta (R²) waa halbeeg muhiim ah oo ku jira falanqaynta dib-u-celinta, taasoo muujinaysa sida qaabka dib-u-celinta uu u sharraxayo kala duwanaanshaha dhabta ah ee xogta. Maqaalkan, waxaan ku sharxi doonnaa fikradda isku-dheellitirka go'aaminta iyada oo loo marayo tusaale iyo dood faahfaahsan.
Fikradda Aasaasiga ah ee Isku-xidhka Go'aaminta
Isugeynta go'aaminta ama \( R^2 \) waxaa lagu cabbiraa miisaan u dhexeeya 0 ilaa 1, halkaasoo:
– \( R^2 = 0 \) waxay muujinaysaa in qaabka dib-u-celinta uusan awood u lahayn inuu sharaxo kala duwanaanshaha xogta gabi ahaanba.
– \( R^2 = 1 \) waxay muujinaysaa in qaabka dib-u-celinta uu awood u leeyahay inuu si fiican u sharaxo dhammaan kala duwanaanshaha xogta.
Qaacidada aasaasiga ah ee lagu xisaabinayo isku-xidhka go'aaminta waa:
\[ R^2 = 1 – \frac{SSR}{SST} \]
Halkee:
– SSR (Wadarta Haraaga Labajibbaaran) waa wadarta labajibbaaran ee kala duwanaanshaha u dhexeeya qiimayaasha uu saadaaliyay qaabka iyo qiimayaasha dhabta ah.
– SST (Wadarta Wadarta Labajibbaaranayaasha) waa wadarta wadarta labajibbaaranayaasha ee kala duwanaanshaha qiimaha dhabta ah iyo celceliska qiimayaasha dhabta ah.
Tusaalaha dhibaatooyinka
Aan ka wada hadalno tusaale dhibaato si aan si qoto dheer u fahanno xisaabinta isku-xidhka go'aaminta.
Tusaale ahaan dhibaatooyinka:
Ka soo qaad inaan hayno xog ku saabsan tirada saacadaha waxbarashada (X) iyo dhibcaha imtixaanka (Y) ee 10 arday:
| Ardayda | Saacadaha Waxbarashada (X) | Dhibcaha Imtixaanka (Y) |
|——-|———————–|———————–|
| 1 | 2 | 58 |
| 2 | 3 | 64 |
| 3 | 4 | 70 |
| 4 | 5 | 85 |
| 5 | 2 | 57 |
| 6 | 3 | 68 |
| 7 | 4 | 72 |
| 8 | 5 | 90 |
| 9 | 3 | 62 |
| 10 | 4 | 78 |
Waxaan abuuri doonnaa qaab fudud oo dib-u-celin toosan ah halkaas oo dhibcaha imtixaanka (Y) lagu saadaaliyo iyadoo lagu saleynayo saacadaha waxbarashada (X).
Dood
1. Dhisidda Qaab Fudud oo Hoos u Dhaca Toosan
Qaabka dib-u-celinta toosan ee fudud wuxuu leeyahay foomka:
\[ Y = a + bX \]
Halkee:
– \( Y \) waa dhibcaha imtixaanka la saadaaliyay.
– \( X \) waa tirada saacadaha waxbarashada.
– \( a \) waa barta isgoyska ee dhidibka Y marka X = 0).
– \( b \) waa janjeera (u janjeera xariiqda dib u noqoshada).
Si loo xisaabiyo xuduudaha \( a \) iyo \( b \), waxaan isticmaalnaa qaacidada soo socota:
\[ b = \frac{n(\sum{XY}) – (\sum{X})(\sum{Y})}{n(\sum{X^2}) – (\sum{X})^2} \]
\[ a = \frac{\sum{Y} – b(\sum{X})}{n} \]
Halkee \( n \) yahay tirada xogta (xaaladdan n = 10).
Shaxda waxaan ka xisaabin karnaa:
– \(\sum{X} = 36\)
– \(\sum{Y} = 704\)
– \(\sum{X^2} = 140\)
– \(\sum{Y^2} = 50428\)
– \(\sum{XY} = 2576\)
Aan marka hore xisaabino b:
\[ b = \frac{10(2576) – (36)(704)}{10(140) – (36)^2} \]
\[ b = \frac{25760 – 25344}{1400 – 1296} \]
\[ b = \frac{416}{104} \]
\[ b = 4 \]
Kadib, waxaan xisaabineynaa:
\[ a = \frac{704 – 4(36)}{10} \]
\[ a = \frac{704 – 144}{10} \]
\[ a = \frac{560}{10} \]
\[ a = 56 \]
Markaa, qaabka dib-u-celinta toosan ee aan helno waa:
\[ Y = 56 + 4X \]
2. Xisaabi Qiimaha La Saadaaliyay (Y')
Marka xigta, waxaan xisaabinaynaa qiimaha la saadaaliyay \( Y' \) mid kasta \( X \):
| Ardayda | Saacadaha Waxbarashada (X) | Dhibcaha Imtixaanka (Y) | Dhibcaha Saadaasha (Y') |
|——-|———————–|—————————–|——————————-|
| 1 | 2 | 58 | \( 56 + 4(2) = 64 \) |
| 2 | 3 | 64 | \( 56 + 4(3) = 68 \) |
| 3 | 4 | 70 | \( 56 + 4(4) = 72 \) |
| 4 | 5 | 85 | \( 56 + 4(5) = 76 \) |
| 5 | 2 | 57 | \( 56 + 4(2) = 64 \) |
| 6 | 3 | 68 | \( 56 + 4(3) = 68 \) |
| 7 | 4 | 72 | \( 56 + 4(4) = 72 \) |
| 8 | 5 | 90 | \( 56 + 4(5) = 76 \) |
| 9 | 3 | 62 | \( 56 + 4(3) = 68 \) |
| 10 | 4 | 78 | \( 56 + 4(4) = 72 \) |
3. Xisaabinta SSR iyo SST
Marka xigta, waxaan xisaabinaynaa SSR iyo SST si aan u helno \( R^2 \).
SSR:
\[ SSR = \wadarta{(Y – Y')^2} \]
\[ SSR = (58 – 64)^2 + (64 – 68)^2 + (70 – 72)^2 + (85 – 76)^2 + (57 – 64)^2 + (68 – 68)^2 + (72 – 72)^2 + (90 – 76)^2 + (62 – 68)^2 + (78 – 72)^2 \]
\[ SSR = 36 + 16 + 4 + 81 + 49 + 0 + 0 + 196 + 36 + 36 \]
\[ SSR = 454 \]
SST:
\[ SST = \sum{(Y - \bar {Y})^2} \]
Halkee:
\[ \bar{Y} = \frac{\sum{Y}}{n} = \frac{704}{10} = 70.4 \]
\[ SST = (58 – 70.4)^2 + (64 – 70.4)^2 + (70 – 70.4)^2 + (85 – 70.4)^2 + (57 – 70.4)^2 + (68 – 70.4)^2 + (72 – 70.4)^2 + (90 – 70.4)^2 + (62 – 70.4)^2 + (78 – 70.4)^2 \]
\[ SST = 153.76 + 40.96 + 0.16 + 213.16 + 178.56 + 5.76 + 2.56 + 384.16 + 70.56 + 57.76 \]
\[ SST = 1107.44 \]
4. Xisaabinta Isku-xidhka Go'aaminta \( R^2 \):
\[ R^2 = 1 – \frac{SSR}{SST} \]
\[ R^2 = 1 – \frac{454}{1107.44} \]
\[ R^2 = 1 – 0.41 \]
\[ R^2 = 0.59 \]
Gabagabo
Xisaabinta kor ku xusan, waxaan ka helnay isku-xidhka go'aaminta (R2 = 0.59). Tani waxay muujinaysaa in qaabka dib-u-celinta toosan ee aan abuurnay uu sharxi karo qiyaastii 59% kala duwanaanshaha dhibcaha imtixaanka iyadoo lagu saleynayo saacadaha waxbarashada. 41% ee soo hartay ee kala duwanaanshaha waxaa sababi kara arrimo kale oo aan ku jirin qaabka.
Markaan fahanno tallaabooyinka iyo xisaabinta kor ku xusan, waxaan arki karnaa muhiimadda isku-dhafka go'aaminta marka la qiimeynayo sida ugu wanaagsan ee qaabka dib-u-celinta ee aan u dhisno uu u sharraxayo kala duwanaanshaha dhabta ah ee xogta. Waa qalab aad waxtar u leh falanqaynta tirakoobka iyo qaabaynta xogta.
