Misali na tambayar tattaunawa kan ma'aunin ƙuduri
Ma'aunin ƙaddara (R²) muhimmin ma'auni ne a cikin nazarin komawa baya, yana nuna yadda tsarin komawa baya ke bayyana ainihin bambancin bayanan. A cikin wannan labarin, za mu bayyana manufar ma'aunin ƙaddara ta hanyar cikakken misali da tattaunawa.
Asali na Ma'anar Ma'aunin Tabbatarwa
Ana auna ma'aunin tantancewa ko \( R^2 \) akan sikelin 0 zuwa 1, inda:
– \( R^2 = 0 \) yana nuna cewa tsarin komawa baya ba zai iya bayyana bambancin bayanan kwata-kwata ba.
– \( R^2 = 1 \) ya nuna cewa tsarin komawa baya yana iya bayyana duk bambancin bayanan daidai.
Tsarin da ake amfani da shi wajen ƙididdige ma'aunin ƙayyadewa shine:
\[ R^2 = 1 – \frac{SSR}{SST} \]
Ina:
– SSR (Jimillar Sauran Murabba'i) shine jimlar murabba'ai na bambance-bambancen da ke tsakanin ƙimar da samfurin ya annabta da ainihin ƙimar.
– SST (Jimillar Jimlar Murabba'i) shine jimlar jimlar murabba'ai na bambance-bambancen tsakanin ainihin ƙimar da matsakaicin ainihin ƙimar.
Misalin matsalolin
Bari mu tattauna wata matsala ta misali domin mu fahimci lissafin ma'aunin tantancewa sosai.
Misalin matsalar:
A ce muna da bayanai kan adadin lokutan karatu (X) da kuma sakamakon jarrabawa (Y) na ɗalibai 10:
| Dalibai | Lokacin Karatu (X) | Maki na Jarabawa (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 |
Za mu ƙirƙiri samfurin juyi mai sauƙi na layi inda ake hasashen sakamakon jarrabawa (Y) bisa ga lokutan karatu (X).
Tattaunawa
1. Gina Tsarin Layi Mai Sauƙi
Tsarin juyi mai sauƙi na layi yana da siffar:
\[ Y = a + bX \]
Ina:
– \( Y \) shine sakamakon gwajin da aka annabta.
– \( X \) shine adadin lokutan karatu.
– \( a \) shine tsangwama (ma'aunin haɗuwa akan axis na Y lokacin da X = 0).
– \( b \) shine gangara (juyawar layin komawa baya).
Don ƙididdige sigogi \( a \) da \( b \), muna amfani da dabarar da ke ƙasa:
\[ b = \frac{n(\sum{XY}) – (\sum{X})(\sum{Y})}{n(\sum{X^2}) – (\sum{X})^2} \]
\[ a = \frac{\sum{Y} – b(\sum{X})}{n} \]
Inda \( n \) shine adadin bayanai (a wannan yanayin n = 10).
Daga teburin za mu iya lissafawa:
– \(\sum{X} = 36\)
– \(\sum{Y} = 704\)
– \(\sum{X^2} = 140\)
– \(\sum{Y^2} = 50428\)
– \(\sum{XY} = 2576\)
Bari mu fara lissafin b:
\[ b = \frac{10(2576) – (36)(704)}{10(140) – (36)^2} \]
\[ b = \frac{25760 – 25344}{1400 – 1296} \]
\[ b = \frac{416}{104} \]
\[ b = 4 \]
Sannan, muna lissafin:
\[ a = \frac{704 – 4(36)}{10} \]
\[ a = \frac{704 – 144}{10} \]
\[a = \frac{560}{10} \]
\[ a = 56 \]
Don haka, samfurin komawar layi da muka samu shine:
\[ Y = 56 + 4X \]
2. Lissafa Ƙimar da Aka Yi Hasashenta (Y')
Na gaba, muna lissafin ƙimar da aka annabta \( Y' \) ga kowane \( X \):
| Dalibai | Lokacin Karatu (X) | Sakamakon Jarabawa (Y) | Hasashen Sakamakon (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. Lissafin SSR da SST
Na gaba, muna lissafin SSR da SST don samun \( R^2 \).
SSR:
\[ SSR = \sum{(Y – Y')^2} \]
\[ SSR = (58 – 64)^2 + (64 – 68)^2 + (70 – 72)^2 + (85 – 76)^2 + (57 – 64)^2 + (68 – 68)^2 + (72 – 72)^2 + (90 – 76)^2 + (62 – 68)^2 + (78 – 72)^2 \]
\[ SSR = 36 + 16 + 4 + 81 + 49 + 0 + 0 + 196 + 36 + 36 \]
\[ SSR = 454 \]
SST:
\[ SST = \sum{(Y - \bar{Y})^2} \]
Ina:
\[ \bar{Y} = \frac{\sum{Y}}{n} = \frac{704}{10} = 70.4 \]
\[ SST = (58 – 70.4)^2 + (64 – 70.4)^2 + (70 – 70.4)^2 + (85 – 70.4)^2 + (57 – 70.4)^2 + (68 – 70.4)^2 + (72 – 70.4)^2 + (90 – 70.4)^2 + (62 – 70.4)^2 + (78 – 70.4)^2 \]
\[ SST = 153.76 + 40.96 + 0.16 + 213.16 + 178.56 + 5.76 + 2.56 + 384.16 + 70.56 + 57.76 \]
\[ SST = 1107.44 \]
4. Lissafin Ma'aunin Tabbatarwa \( R^2 \):
\[ R^2 = 1 – \frac{SSR}{SST} \]
\[ R^2 = 1 – \frac{454}{1107.44} \]
\[ R^2 = 1 – 0.41 \]
\[ R^2 = 0.59 \]
Kammalawa
Daga lissafin da ke sama, mun sami ma'aunin tantancewa (R2 = 0.59). Wannan yana nuna cewa samfurin komawar layi da muka ƙirƙira zai iya bayyana kusan kashi 59% na bambancin da ke cikin sakamakon jarrabawa bisa ga lokutan karatu. Sauran kashi 41% na bambancin na iya kasancewa saboda wasu abubuwan da ba a haɗa su cikin samfurin ba.
Ta hanyar fahimtar matakai da lissafin da ke sama, za mu iya ganin mahimmancin ma'aunin ƙuduri wajen tantance yadda tsarin komawa baya da muke ginawa yake bayyana ainihin bambancin bayanan. Kayan aiki ne mai matuƙar amfani a nazarin ƙididdiga da ƙirar bayanai.
