Tātaritanga Tohatoha Raraunga mā te Whakamahi i te Paerewa Rerekētanga
I roto i ngā tatauranga, kāore e ranea te mārama noa ki te "pokapū" o tētahi huinga raraunga. E rua ngā huinga raraunga ka taea te whai i te toharite kotahi, engari he rerekē rawa ō rāua āhuatanga nā te nui o te horapa. Koinei te wāhi e hira ai te ariā o te horapa raraunga. Ko tētahi o ngā mehua horapa rongonui, pakari, me te whakamahia pinepine i roto i ngā momo mara - mai i te mātauranga me te ōhanga ki te hauora me te pūtaiao raraunga - ko te paerewa rerekētanga. E matapakihia ana e tēnei tuhinga te ariā, te tatau, te whakamārama, me te whakamahinga o te paerewa rerekētanga hei tātari i te horapa o ngā raraunga mai i tōna uara pokapū.
1. He aha i tika ai kia tātarihia te tohatoha raraunga?
Whakaarohia ngā akomanga e rua me te kaute pāngarau toharite o te 80. I te akomanga A, tata ki te katoa o ngā ākonga i whiwhi i waenga i te 78 me te 82. I te akomanga B, ko ētahi ākonga i whiwhi i te 50, ko ētahi i whiwhi i te 100. He rite tonu ngā toharite, engari he tino rerekē ngā āhuatanga i roto i ngā akomanga e rua. He rite tonu te mahi a te akomanga A, ko te akomanga B ia he rerekētanga nui.
Mā te tātari i te tohatoha, ka taea e tātou te:
– Aromatawai i te ōritetanga, i te rerekētanga rānei o tētahi āhuatanga.
– Te ine i te mōrearea (hei tauira, te rerekētanga o ngā hokinga mai o ngā haumitanga).
– Te whakatairite i te pumau o te tukanga (hei tauira, te kounga o te whakaputanga).
– Te kite i ngā āhuatanga rerekē pea, i ngā raraunga tino nui rānei.
Ko te paerewa rerekētanga te taputapu matua mō tēnei kaupapa, nā te mea e ine ana i te tawhiti o te horapa o ngā raraunga mai i te toharite.
2. Te Whakamāramatanga o te Paerewa Rerekētanga
Ko te paerewa rerekētanga te pūtake tapawhā o te rerekētanga. Ahakoa e ine ana te rerekētanga i te toharite o ngā tapawhā o ngā rerekētanga i waenga i ngā raraunga me te toharite, ka whakahokia e te paerewa rerekētanga ngā waeine ine ki tō rātou tauine taketake (hei tauira, ngā kaute whakamātautau, kirokaramu, rūpia, me ētahi atu). Mā tēnei ka māmā ake te whakamārama i te paerewa rerekētanga.
Mā te whakaaro noa:
– He iti te paerewa rerekētanga → he tata ngā raraunga kua kohia ki te toharite (he ōrite ake).
– Te paerewa rerekētanga nui → kua marara tawhiti ngā raraunga i te toharite (he rerekē ake).
3. Tātai Paerewa Rerekētanga: Taupori vs Tauira
I roto i ngā tatauranga, ka wehewehea te tatau i te paerewa rerekētanga mō ngā taupori me te tatau i ngā tauira.
a) Te Paerewa Rerekētanga o te Taupori (σ)
Mena ko ngā raraunga e tātarihia ana ko ngā mema katoa o te taupori, ko te tātai koia tēnei:
\[
\sigma = \sqrt{\frac{\sum (x_i – \mu)^2}{N}}
\]
Ngā Mōhiohio:
– \(x_i\) = uara raraunga tua-i
– \(\mu\) = te toharite o te taupori
– \(N\) = te maha o ngā raraunga taupori
b) Tauira Paerewa Rerekētanga (ngā)
Mena he wāhanga noa iho o te taupori (tauira) ngā raraunga e tātarihia ana, ko te tātai koia tēnei:
\[
s = \sqrt{\frac{\sum (x_i – \bar{x})^2}{n-1}}
\]
Ngā Mōhiohio:
– \(\bar{x}\) = toharite tauira
– \(n\) = te maha o ngā raraunga tauira
– Ko te \(n-1\) e kiia ana ko ngā nekehanga herekoretanga (te whakatikatika a Bessel), e whakamahia ana kia kore ai e pāngia te whakatau tataitanga rerekētanga/paerewa rerekētanga.
I roto i ngā mahi o ia rā, ko ngā raraunga kei a tātou he tauira te nuinga, nō reira ko te tātai \(n-1\) te mea e whakamahia whānuitia ana.
4. Ngā Hipanga hei Tātai i te Paerewa Rerekētanga
Hei mārama ki te tukanga, anei ngā mahi whānui mō te tatau i te paerewa rerekētanga tauira:
1. Tātaihia te toharite (\(\bar{x}\)).
2. Tātaihia te rerekētanga i waenga i ia raraunga me te toharite (\(x_i – \bar{x}\)).
3. Tātaitia te rerekētanga tapawhā \((x_i – \bar{x})^2\).
4. Tāpirihia ngā tapawhā katoa.
5. Wehea mā te \(n-1\) hei tiki i te rerekētanga tauira.
6. Tātaitia te pūtake tapawhā o te hua hei whiwhi i te paerewa rerekētanga (s).
Tauira Māmā
Me kī ko ngā uara raraunga ko: 70, 75, 80, 85, 90 (n = 5)
– Toharite: \(\bar{x} = (70+75+80+85+90)/5 = 80\)
– Rerekētanga: -10, -5, 0, 5, 10
– Rerekētanga tapawhā: 100, 25, 0, 25, 100
– Te maha o ngā tapawhā: 250
– Tauira rerekētanga: \(250/(5-1)=62,5\)
– Paerewa rerekētanga: \(s=\sqrt{62,5}\approx 7,91\)
Ko te whakamaoritanga māmā: ka rerekē ngā uara mā te toharite o te 7,91 ngā tohu mai i te toharite o te 80.
5. Te Whakamārama i te Paerewa Rerekētanga i roto i te Tātari Raraunga
Kāore te paerewa rerekētanga e tū takitahi; ko tōna tikanga e whakawhirinaki ana ki te horopaki. Heoi anō, he āwhina ētahi aratohu whānui:
– Mena he tata te paerewa rerekētanga ki te 0, ka tino kukū ngā raraunga ki te toharite.
– Ki te nui te paerewa rerekētanga, he rerekē ake ngā raraunga, e tohu ana i te kore ōritetanga.
Ka whakamahia hoki te paerewa rerekētanga mō:
– Te whakatairite i ngā rōpū e rua: hei tauira, e rua ngā akomanga he ōrite te toharite, engari he rerekē ngā paerewa rerekētanga.
– Te aromatawai i te pumau o te tukanga: ko te hanga wheketere me te iti o te paerewa rerekētanga o te rahi o te hua he tikanga kia pumau ake te kounga.
– Te ine i te piki me te heke o te uara: i roto i te pūtea, ko te paerewa rerekētanga o ngā hokinga mai o ngā hea ka whakamahia pinepinetia hei tohu mōrearea.
6. Te whanaungatanga i waenga i te Paerewa Rerekētanga me te Tohatoha Noa
I roto i ngā raraunga e whai ana i te tohatoha noa, he tino kaha te whakamaoritanga o te paerewa rerekētanga mā te ture empirika:
– 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 \(\bar{x} \pm 3s\)
He whai hua tēnei ture mō te whakatau tata i te nui o ngā raraunga e "noa" ana i te toharite, ā, ka māmā ake te kite i ngā uara tino nui. Heoi, he mea nui kia mahara ko tēnei ture ka tika anake mēnā he tata tonu ngā raraunga ki te noa.
7. Te Paerewa Rerekētanga me Ētahi Atu Inenga o te Horahanga
Ahakoa he tino rongonui te paerewa rerekētanga, he mea nui anō hoki ētahi atu inenga o te marara:
– Awhe: te rerekētanga i waenga i ngā uara mōrahi me ngā uara mōkito. He māmā engari he tino tairongo ki ngā mea o waho.
– IQR (awhe interquartile): te awhe i waenga i te hauwhā 1 me te hauwhā 3. He ātete ake ki ngā mea o waho atu i te paerewa rerekētanga.
– MAD (te rerekētanga tino waenga): he ine pakari e ahu mai ana i te waenga, e tika ana mō ngā raraunga he maha ngā mea kei waho.
He pai ake te paerewa rerekētanga ina he "ma" te raraunga, ā, kāore te tohatoha i te tino whaiti. Mena he maha ngā mea kei waho o te raraunga, ka nui ake te paerewa rerekētanga, ā, ka iti ake te whakaatu i te nuinga o ngā raraunga.
8. Ngā Painga me ngā Here o te Paerewa Ine
Te taikaha
– Ka whakamahi i ngā raraunga katoa (kaua ko ngā uara tino nui anake).
– He pūtake ariā pakari tōna, ā, e whakamahia pinepinetia ana i roto i ngā tikanga tatauranga matatau maha.
– He māmā te whakamārama nā te mea he rite tonu ngā waeine ki ngā raraunga taketake.
Ngā Here
– He tino tairongo ki ngā mea o waho nā te mea e uru ana te tapawhā o te rerekētanga.
– Ko te whakamārama i te kupu “nui” me te “iti” e whakawhirinaki ana ki te tauine me te horopaki.
– I roto i ngā tohatoha tino kore-noa, tērā pea kāore e tino whakaatu ana te paerewa rerekētanga.
9. Whakamutunga
He taahiraa nui te tātari i te horapa o ngā raraunga hei mārama ki ngā āhuatanga o tētahi huinga raraunga. Mā te paerewa rerekētanga ka mārama te ine i te tawhiti o te horapa o ngā raraunga mai i te toharite, ā, ka āwhina i a tātou ki te aromatawai i te ōritetanga, te mōrearea, me te kounga o tētahi tukanga, o tētahi āhuatanga rānei. Mā te mārama ki te tatau me te whakamārama, ka taea e tātou te whakatau whai whakaaro, ahakoa i roto i te rangahau mātauranga, te aromatawai mahi, te whakahaere kounga, te tātari pakihi rānei.
Hei whakamutunga, ehara te paerewa rerekētanga i te tau noa iho, engari he whakarāpopototanga nui o te koretake me te rerekētanga kei roto i ngā raraunga. Mō te tātari pakari ake, me whakamahi te paerewa rerekētanga me ētahi atu inenga—pērā i te tau waenga, te IQR, te whakaaturanga raraunga rānei—hei whakarato i tētahi pikitia tino tika ake, tino whānui ake hoki o te tohatoha.
