Tauira Pātai me te Kōrero: Te Rerekētanga me te Paerewa Rerekētanga o ngā Raraunga Takitahi
Ko te tatauranga he peka o te pāngarau e pā ana ki te kohikohi, te tātari, te whakamārama, te whakaatu, me te whakarite raraunga. E rua ngā ariā nui o te tatauranga ko te rerekētanga me te paerewa rerekētanga. Ka matapakihia e tēnei tuhinga te āhua o te tatau i te rerekētanga me te paerewa rerekētanga o tētahi huinga raraunga kotahi mā roto i ētahi tauira rapanga.
Te Mārama ki te Rerekētanga me te Paerewa Rerekētanga
Ko te rerekētanga he mehua e ine ana i te tawhiti o te horapa o ngā tau i roto i tētahi huinga raraunga mai i te toharite, i te toharite rānei. Ka whakaatuhia te rerekētanga ki ngā waeine tapawhā o ngā raraunga taketake, ā, he uaua ki te whakamārama.
Ko te paerewa rerekētanga te pūtake tapawhā o te rerekētanga. Mā tēnei ka māmā ake te ine i te tawhiti o te rerekētanga o ngā raraunga mai i te toharite nā te mea he rite tonu ōna waeine ki ngā waeine taketake o ngā raraunga.
Tātai Whānui
Mō ngā raraunga takitahi, ko te tātai mō te rerekētanga \( \sigma^2 \) me te paerewa rerekētanga \( \sigma \) o te taupori e whai ake nei:
1. Rerekētanga (σ²):
\( \sigma^2 = \frac{1}{N} \sum_{i=1}^{N} (x_i – \mu)^2 \)
2. Paerewa Rerekētanga (σ):
\( \sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (x_i – \mu)^2} \)
Kei hea:
– Ko te \( N \) te maha o ngā raraunga i roto i te taupori.
– Ko te uara raraunga tuarima ko \( x_i \).
– Ko te \( \mu \) te toharite, te toharite rānei o ngā raraunga.
Ki te tatauhia te rerekētanga me te paerewa rerekētanga o te tauira, ka paku whakarerekētia te tātai i runga ake nei:
1. Tauira Rerekētanga (s²):
\( s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i – \bar{x})^2 \)
2. Tauira Paerewa Rerekētanga (ngā):
\( s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i – \bar{x})^2} \)
Kei hea:
– Ko te \( n \) te maha o ngā raraunga i roto i te tauira.
– Ko te \( \bar{x} \) te toharite, te toharite rānei o te tauira.
Ngā Pātai Tauira me te Kōrero
Tauira Pātai 1:
I runga i ngā raraunga e whai ake nei:
8, 10, 10, 10, 12, 14
Tātaihia te rerekētanga me te paerewa rerekētanga o ngā raraunga!
Ngā Hipanga Whakaoti:
1. Te Tātai i te Toharite (Toharite) \( \mu \):
\[
\mu = \frac{8 + 10 + 10 + 10 + 12 + 14}{6} = \frac{64}{6} = 10.67
\]
2. Tātaihia te Rerekētanga i waenga i te Raraunga me te Toharite, Kātahi ka Tapawhāhia:
\[
(8 – 10.67)^2 = 7.1289
\]
\[
(10 – 10.67)^2 = 0.4489
\]
\[
(10 – 10.67)^2 = 0.4489
\]
\[
(10 – 10.67)^2 = 0.4489
\]
\[
(12 – 10.67)^2 = 1.7689
\]
\[
(14 – 10.67)^2 = 11.1089
\]
3. Te Tāpiri i ngā Hua Tapawhā Katoa:
\[
\jumlah (x_i – \mu)^2 = 7.1289 + 0.4489 + 0.4489 + 0.4489 + 1.7689 + 11.1089 = 21.3534
\]
4. Te Tātai i te Rerekētanga (σ²) o ngā Raraunga Taupori:
\[
\sigma^2 = \frac{21.3534}{6} = 3.559
\]
Kia mōhio: Nā te mea e kīia ana tēnei raraunga hei raraunga taupori, ka wehea e tātou mā te 6.
5. Te Tātai i te Paerewa Rerekētanga (σ):
\[
\sigma = \sqrt{3.559} \approx 1.886
\]
Nō reira, ko te rerekētanga o ngā raraunga ko 3.559, ā, ko te paerewa rerekētanga ko 1.886.
Tauira Pātai 2:
E whai ake nei ngā raraunga tauira:
5, 6, 8, 9, 10, 11
Tātaihia te rerekētanga me te paerewa rerekētanga o te tauira!
Ngā Hipanga Whakaoti:
1. Te Tātai i te Toharite (Toharite) \( \bar{x} \) :
\[
\bar{x} = \frac{5 + 6 + 8 + 9 + 10 + 11}{6} = \frac{49}{6} = 8.167
\]
2. Tātaihia te Rerekētanga i waenga i te Raraunga me te Toharite, Kātahi ka Tapawhāhia:
\[
(5 – 8.167)^2 = 10.035
\]
\[
(6 – 8.167)^2 = 4.694
\]
\[
(8 – 8.167)^2 = 0.028
\]
\[
(9 – 8.167)^2 = 0.694
\]
\[
(10 – 8.167)^2 = 3.361
\]
\[
(11 – 8.167)^2 = 7.945
\]
3. Te Tāpiri i ngā Hua Tapawhā Katoa:
\[
\tapeke (x_i – \bar{x})^2 = 10.035 + 4.694 + 0.028 + 0.694 + 3.361 + 7.945 = 26.757
\]
4. Te Tātai i te Rerekētanga Tauira (s²):
\[
s^2 = \frac{26.757}{5} = 5.351
\]
Kia mōhio: Nā te mea he tauira raraunga tēnei, ka wehea e tātou mā te 5 (n-1).
5. Te Tātai i te Paerewa Rerekētanga (ngā):
\[
s = \sqrt{5.351} \approx 2.313
\]
Nō reira, ko te rerekētanga o ngā raraunga tauira ko 5.351, ā, ko te paerewa rerekētanga ko 2.313.
Whakamutunga
He mea nui te tatau i te rerekētanga me te paerewa rerekētanga hei mārama ki te horapa o ngā raraunga i roto i tētahi huinga kua hoatu. Ahakoa he ine ariā mō te horapa o ngā raraunga e whakaratohia ana e te rerekētanga i roto i ngā tapawhā o ngā waeine taketake, ka whakamārama te paerewa rerekētanga i te ine o te horapa i runga i ngā waeine taketake o ngā raraunga, kia māmā ake ai te mārama. I roto i te tātari raraunga, ka whakamahia ēnei ine e rua hei aromatawai i te rerekētanga o ngā raraunga me te whakatau whakatau tatauranga.
Mā te mārama ki ngā mahi me ngā tātai e tika ana, ka taea e tātou te tatau ngāwari i te rerekētanga me te paerewa rerekētanga mō ngā āhuatanga rerekē e tūtakihia ana e tātou i roto i te kohikohi me te tātari raraunga o ia rā. Ko te tumanako, ka whakaratohia e tēnei tuhinga he māramatanga hohonu ake mō ngā ariā o te rerekētanga me te paerewa rerekētanga i roto i ngā huinga raraunga takitahi.
