Te Tātari i te Rerekētanga me te Paerewa Rerekētanga i roto i te Tohatoha Raraunga
I roto i ngā tatauranga, he rite tonu te hiranga o te mārama ki te tohatoha o ngā raraunga ki te mārama ki ngā uara matua pērā i te toharite, te tau waenga rānei. E rua ngā huinga raraunga, ahakoa he rite tonu te toharite, he rerekē rawa ō rātou tohatoha: tērā pea ka noho pūmau tetahi ki te toharite, ko tetahi ka horapa whānuitia. Koinei te wāhi e uru mai ai te rerekētanga me te paerewa rerekētanga—he mehua matua ēnei mō te nui o ngā raraunga e rerekē ana mai i tōna uara matua. Ka matapakihia e tēnei tuhinga ō rātou ariā, ngā tātai, ngā whakamārama, me ngā tauira o tā rātou whakamahinga i roto i te tātari raraunga.
1. He aha te hiranga o te tohatoha raraunga?
Mā te horapa raraunga ka whakaratohia he mōhiohio mō te ōritetanga me te mōrearea. Hei tauira, i roto i te horopaki o ngā kaute whakamātautau, ko te toharite mō ngā akomanga A me B ka eke ki te 80. Heoi, ki te iti te rerekētanga o ngā kaute o te akomanga A, ka rite te mahi a te nuinga o ngā ākonga. I tetahi atu taha, ki te nui te rerekētanga o ngā kaute o te akomanga B, tera pea he tino teitei ngā kaute o ētahi ākonga, he tino iti ngā kaute o ētahi. I roto i te pakihi, ko te horapa o ngā raraunga hoko e tohu ana i te pumau o te moni whiwhi; i roto i te pūtea, ko te horapa o ngā hokinga mai o te haumitanga e tohu ana i te taumata o te mōrearea.
Mā te mārama ki te rerekētanga me te paerewa rerekētanga, ka taea e ngā kaiwhakatau te:
– Aromatawaihia mēnā he pumau te tukanga, kāore rānei (hei tauira, te hanga wheketere).
– Te whakatairite i te ōritetanga i waenga i ngā rōpū (hei tauira, ngā tikanga ako e rua).
– Te tautuhi i ngā raraunga tauhou e tika ana kia arotakehia.
– Te whakatau tata i te koretake o ngā matapae me ngā tauira.
2. Te Ariā Taketake o te Rerekētanga
Ka ine te rerekētanga i te toharite tapawhā o ia huinga raraunga mai i te toharite. Ko te rerekētanga te rerekētanga i waenga i ngā uara raraunga me te toharite. Mena he maha ngā uara kei tawhiti atu i te toharite, ka nui te rerekētanga. Mena he tata ngā uara ki te toharite, ka iti te rerekētanga.
Mehemea he raraunga kei reira: \(x_1, x_2, …, x_n\) me te toharite o \(\bar{x}\). Ko te rerekētanga o ia raraunga ko \(x_i – \bar{x}\). Heoi, ki te tāpirihia tika ngā rerekētanga, ka kore tonu te hua nā te mea he rerekētanga pai me te rerekētanga kino e whakakore ana tetahi i tetahi. Hei whakakore i tēnei, ka tapawhāhia ngā rerekētanga kia pai katoa. Koinei te wāhi e whānau mai ai te rerekētanga.
a) Te Rerekētanga o te Taupori
Mena ka whakaarohia ngā raraunga hei tohu mō te taupori katoa, ka tuhia te rerekētanga taupori penei:
\[
\sigma^2 = \frac{\sum_{i=1}^{N}(x_i – \mu)^2}{N}
\]
kāore i te mana:
– Ko te \(N\) te maha o ngā raraunga taupori,
– Ko te toharite o te taupori ko \(\mu\),
– Ko te \(\sigma^2\) te rerekētanga taupori.
b) Tauira Rerekētanga
Mena he tauira te raraunga mai i tētahi taupori nui ake, ka whakamahia te rerekētanga tauira:
\[
s^2 = \frac{\sum_{i=1}^{n}(x_i – \bar{x})^2}{n-1}
\]
Ko te whakawehenga \(n-1\) e kiia ana ko te whakatikatika Bessel, ā, e whakamahia ana hei whakarite kia kore he pānga o te whakatau tata rerekētanga mō te taupori. Ko te tikanga, nā te mea ka tatauhia te toharite tauira mai i ngā raraunga tonu, ka "ngaronga ngā nekehanga herekoretanga," nō reira ka whakatikatikaina te whakawehenga kia rite.
3. Paerewa Rerekētanga: Te Pūtake o te Rerekētanga
Kotahi te ngoikoretanga o te rerekētanga: ko ōna waeine ko te tapawhā o ngā waeine o ngā raraunga. Mena kei roto i te "rupiah" ngā raraunga, kei roto i te "rupiah²" te rerekētanga, he uaua ki te whakamārama tika. Nō reira, ka whakamahia e mātou te paerewa rerekētanga, arā, te pūtake tapawhā o te rerekētanga.
a) Te Paerewa Rerekētanga o te Taupori
\[
\sigma = \sqrt{\sigma^2}
\]
b) Tauira Paerewa Rerekētanga
\[
s = \sqrt{s^2}
\]
He rite tonu ngā waeine o te paerewa rerekētanga ki ngā raraunga taketake, kia māmā ake ai te mārama. Ko te paerewa rerekētanga teitei e tohu ana i te horapa haere o ngā raraunga; ko te paerewa rerekētanga iti e tohu ana i te matotoru ake o te huinga raraunga.
4. Tauira Tātaitanga Māmā
Hei tauira, ko ngā raraunga kaute whakamātautau: 70, 75, 80, 85, 90.
1) Tātaihia te toharite:
\[
\bar{x} = \frac{70+75+80+85+90}{5} = 80
\]
2) Tātaihia te rerekētanga o ia uara mai i te toharite:
– 70: \(70-80=-10\)
– 75: \(75-80=-5\)
– 80: \(80-80=0\)
– 85: \(85-80=5\)
– 90: \(90-80=10\)
3) Tāruatia te rerekētanga:
– 100, 25, 0, 25, 100
4) Tāpirihia:
\[
\tapeke (x_i-\bar{x})^2 = 250
\]
5) Te rerekētanga tauira:
\[
s^2 = \frac{250}{5-1} = 62.5
\]
6) Tauira paerewa rerekētanga:
\[
s = \sqrt{62.5} \approx 7.91
\]
Whakamāoritanga: ko te kaute toharite ko te 80, ā, ko te rerekētanga o ngā kaute "tikanga" he 7–8 ngā piro mai i te toharite.
5. Te Whakamārama i te Rerekētanga me te Paerewa Rerekētanga
Ehara i te mea he tau noa te rerekētanga me te paerewa rerekētanga; me whakamārama i roto i te horopaki.
– He iti te paerewa rerekētanga: he tino ōrite. Hei tauira, ko te tukanga whakaputa me te iti rawa o te paerewa rerekētanga o te rahi o te hua e tohu ana i te kounga pumau.
– Te rerekētanga paerewa nui: he nui te rerekētanga. I roto i te haumi, ko te rerekētanga paerewa teitei o ngā hokinga mai he nui te piki me te heke (he nui ake te mōrearea).
– Te whakatairite i waenga i ngā rōpū: mēnā he ōrite te toharite engari he rerekē ngā paerewa rerekētanga o ngā rōpū e rua, ka ōrite ake te rōpū he iti ake te rerekētanga.
Heoi, he mea nui kia maumahara he tairongo te paerewa rerekētanga ki ngā mea o waho. Ka taea e te uara tino kotahi te whakanui ake i te rerekētanga me te paerewa rerekētanga. Nō reira, he maha ngā wā ka tāpirihia te tātari tohatoha e ngā whakaaturanga (ngā histogram, ngā pouaka kauwhata) me ngā ine pakari pēnei i te IQR (te awhe interquartile).
6. Te Hononga ki te Tohatoha Noa me ngā Ture Whakamātautau
I roto i te tohatoha noa (te pihi pere), he tino kaha te hiranga o te paerewa rerekētanga. He ture e whakamahia whānuitia ana:
– Tata ki te 68% o ngā raraunga kei roto i te awhe \(\bar{x} \pm 1s\)
– Tata ki te 95% o ngā raraunga kei roto i te awhe \(\bar{x} \pm 2s\)
– Tata ki te 99,7% o ngā raraunga kei roto i te awhe \(\bar{x} \pm 3s\)
Mā tēnei ture ka āwhina i te tere o te whakamārama, hei tauira, te aromatawai mēnā he "tauhou" te uara, kei roto tonu rānei i te awhe whānui.
7. Ngā tono i roto i ngā mara rerekē
1) Mātauranga: Te aroturuki i te tohatoha o ngā māka a ngā ākonga. Ko ngā rerekētanga iti e tohu ana i ngā putanga ako ōrite, ko ngā rerekētanga nui ia e tohu ana i ngā āputa o te māramatanga.
2) Ahumahi: te whakahaere kounga. Ka whakamahia te rerekētanga hei aromatawai i te ōritetanga o te whakaputanga.
3) Pūtea: e ine ana i te piki me te heke o te utu hea, ngā hokinga mai o te putea, me te mōrearea haumitanga.
4) Hauora: te mātakitaki i ngā rerekētanga o te toto totoka, ngā taumata huka, me ētahi atu tohu haumanu i roto i te taupori tūroro.
5) Rangahau pāpori: aromatawai i te rerekētanga o ngā whakautu rangahau me te kanorau o ngā āhuatanga o ngā kaiwhakautu.
8. Ngā Hapa Noa me ngā Tohutohu Whaihua
Ko ētahi hapa noa:
– Te whakamahi i te rerekētanga tauira (wehewehe \(n-1\)) ahakoa ko te raraunga te taupori katoa, te ritenga kē rānei.
– Te whakamārama i te rerekētanga me te kore e whai whakaaro ki ōna waeine tapawhā; he pai ake te whakamahi i te paerewa rerekētanga hei whakamārama.
– Kaua e aro ki ngā mea o waho; he mea pai kia tirohia ngā raraunga i te tuatahi.
– Whakatauritehia ngā paerewa rerekētanga i waenga i ngā raraunga me ngā tauine rerekē me te kore he whakataurite; i ētahi wā, whakamahia te tauwehenga rerekētanga (CV) arā, \(CV = \frac{s}{\bar{x}}\times 100\%\) mō te whakataurite tika ake.
Te Katinga
He taputapu matua te rerekētanga me te paerewa rerekētanga hei mārama ki te tohatoha raraunga. He pūtake pāngarau pakari te rerekētanga, engari he māmā ake te whakamārama i te paerewa rerekētanga nā te mea he rite ki ngā raraunga taketake. Mā te whakamahi i ēnei inenga e rua, ka taea e tātou te aromatawai mārama ake i te ōritetanga, te mōrearea, me ngā rerekētanga o ngā āhuatanga tohatoha i waenga i ngā huinga raraunga. I roto i te mahi tātari raraunga, he pai ake te whakamahi i te rerekētanga me te paerewa rerekētanga me ngā inenga o te ia matua me te whakaaturanga hei whakarato i tētahi pikitia katoa o ngā raraunga me te whakatau whakatau whai whakaaro.
Ki te hiahia koe, ka taea e au te tāpiri i ngā tauira tatau uaua ake (hei tauira, ngā raraunga rōpū), te whakamārama rānei i te whanaungatanga o te paerewa rerekētanga ki te kaute-z me te kitenga o waho.
