He tauira pātai kōrero mō te tauwehenga whakatau
He tawhā nui te tauwehenga whakatau (R²) i roto i te tātari whakatau, e whakaatu ana i te pai o te whakamārama a te tauira whakatau i te rerekētanga tuturu o ngā raraunga. I roto i tēnei tuhinga, ka whakamāramahia e mātou te ariā o te tauwehenga whakatau mā te tauira me te matapaki taipitopito.
Te Ariā Taketake o te Tauwehenga Whakatau
Ka inehia te tauwehenga whakatau, arā, \( R^2 \) rānei, i runga i te tauine mai i te 0 ki te 1, arā:
– Ko te \( R^2 = 0 \) e tohu ana kāore e taea e te tauira whakatauira te whakamārama i te rerekētanga o ngā raraunga.
– E whakaatu ana te \( R^2 = 1 \) ka taea e te tauira whakatauira te whakamārama i ngā rerekētanga katoa o ngā raraunga me te tino tika.
Ko te tātai taketake mō te tatau i te tauwehenga whakatau ko:
\[ R^2 = 1 – \frac{SSR}{SST} \]
Kei hea:
– Ko te SSR (Tapeke o ngā Toenga Tapawhā) ko te tapeke o ngā tapawhā o ngā rerekētanga i waenga i ngā uara i matapaetia e te tauira me ngā uara tuturu.
– Ko te SST (Tapeke o ngā Tapawhā) ko te tapeke o te tapeke o ngā tapawhā o ngā rerekētanga i waenga i te uara tūturu me te toharite o ngā uara tūturu.
Tauira raruraru
Me matapakihia tētahi tauira raruraru kia mārama ake ai tātou ki te tatau i te tauwehenga whakatau.
Tauira raruraru:
Me kī kei a tātou ngā raraunga mō te maha o ngā hāora ako (X) me ngā kaute whakamātautau (Y) o te 10 ākonga:
| Ngā Ākonga | Ngā Hāora Ako (X) | Kaute Whakamātautau (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 |
Ka waihangahia e mātou he tauira whakatauira rārangi māmā, ka matapaetia ngā kaute whakamātautau (Y) i runga i ngā hāora ako (X).
Kōrero
1. Te Hanga i tētahi Tauira Whakamuri Raina Māmā
Ko te āhua o tētahi tauira whakatauira rārangi māmā:
\[ Y = a + bX \]
Kei hea:
– Ko te kaute whakamātautau i matapaetia ko \( Y \).
– Ko te \( X \) te maha o ngā hāora ako.
– Ko te \( a \) te pūwāhi whakawhiti (te pūwāhi whakawhiti i te tuaka Y ina X = 0).
– Ko te \( b \) te pikinga (te pikinga o te rārangi whakahekenga).
Hei tatau i ngā tawhā \( a \) me \( b \), ka whakamahia e mātou te tātai e whai ake nei:
\[ b = \frac{n(\tapeke{XY}) – (\tapeke{X})(\tapeke{Y})}{n(\tapeke{X^2}) – (\tapeke{X})^2} \]
\[ a = \frac{\tapeke{Y} – b(\tapeke{X})}{n} \]
Ko \( n \) te maha o ngā raraunga (i tēnei wā ko n = 10).
Mai i te ripanga ka taea e tātou te tatau:
– \(\tapeke{X} = 36\)
– \(\tapeke{Y} = 704\)
– \(\tapeke{X^2} = 140\)
– \(\tapeke{Y^2} = 50428\)
– \(\tapeke{XY} = 2576\)
Me tatau tātou i te b i te tuatahi:
\[ b = \frac{10(2576) – (36)(704)}{10(140) – (36)^2} \]
\[ b = \frac{25760 – 25344}{1400 – 1296} \]
\[ b = \frac{416}{104} \]
\[ b = 4 \]
Kātahi ka tatauhia e mātou he:
\[ a = \frac{704 – 4(36)}{10} \]
\[ a = \frac{704 – 144}{10} \]
\[ a = \frac{560}{10} \]
\[ a = 56 \]
Nā, ko te tauira whakatauira raina ka whiwhi tātou ko:
\[ Ā = 56 + 4X \]
2. Tātaihia te Uara Matapae (Y')
Muri iho, ka tatauhia e mātou te uara i matapaetia \( Y' \) mō ia \( X \):
| Ngā Ākonga | Ngā Hāora Ako (X) | Kaute Whakamātautau (Y) | Kaute Matapae (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. Te tatau i te SSR me te SST
Muri iho, ka tatauhia e mātou te SSR me te SST kia whiwhi ai i te \( R^2 \).
SSR:
\[ SSR = \tapeke{(Y – Y')^2} \]
\[ SSR = (58 – 64)^2 + (64 – 68)^2 + (70 – 72)^2 + (85 – 76)^2 + (57 – 64)^2 + (68 – 68)^2 + (72 – 72)^2 + (90 – 76)^2 + (62 – 68)^2 + (78 – 72)^2 \]
\[ SSR = 36 + 16 + 4 + 81 + 49 + 0 + 0 + 196 + 36 + 36 \]
\[ SSR = 454 \]
SST:
\[ SST = \sum{(Y – \bar{Y})^2} \]
Kei hea:
\[ \bar{Y} = \frac{\sum{Y}}{n} = \frac{704}{10} = 70.4 \]
\[ Taumahatanga = (58 – 70.4)^2 + (64 – 70.4)^2 + (70 – 70.4)^2 + (85 – 70.4)^2 + (57 – 70.4)^2 + (68 – 70.4)^2 + (72 – 70.4)^2 + (90 – 70.4)^2 + (62 – 70.4)^2 + (78 – 70.4)^2 \]
\[ Taumahatanga = 153.76 + 40.96 + 0.16 + 213.16 + 178.56 + 5.76 + 2.56 + 384.16 + 70.56 + 57.76 \]
\[ SST = 1107.44 \]
4. Te Tātai i te Tauwehenga Whakatau \( R^2 \):
\[ R^2 = 1 – \frac{SSR}{SST} \]
\[ R^2 = 1 – \frac{454}{1107.44} \]
\[ R^2 = 1 – 0.41 \]
\[ R^2 = 0.59 \]
Whakamutunga
Mai i te tātaitanga i runga ake nei, i whiwhi mātou i te tauwehenga whakatau (R2 = 0.59). E tohu ana tēnei ka taea e te tauira whakatau rārangi i hangaia e mātou te whakamārama i te 59% o te rerekētanga o ngā kaute whakamātautau i runga i ngā hāora ako. Ko te toenga 41% o te rerekētanga ka puta mai i ētahi atu āhuatanga kāore i roto i te tauira.
Mā te mārama ki ngā mahi me ngā tataunga i runga ake nei, ka kitea te hiranga o te tauwehenga whakatau hei aromatawai i te pai o te whakamārama a te tauira whakatauira i te rerekētanga tuturu o ngā raraunga. He taputapu tino whai hua tēnei i roto i te tātari tatauranga me te whakatauira raraunga.
