Maitiro Ekuverenga Kutsauka Kwakajairwa
Kutsauka kwakajairwa ndeimwe yenzira dzinoshandiswa zvakanyanya pakuongorora kuti "data rinopararira" sei kubva paavhareji yaro. Muhupenyu chaihwo, kutsauka kwakajairwa kunotibatsira kunzwisisa kugadzikana kana kusiyana kwechiitiko: semuenzaniso, kusiyana kwemapoinzi ebvunzo, kushanduka kwekutengesa, mutsauko muhurefu, uye kunyange njodzi dziri mudata rezvemari. Kutsauka kwakajairwa kudiki, ndiko kuswedera pedyo kweavhareji. Kusiyana neizvi, kutsauka kwakajairwa kukuru kunoratidza kuti data harisi reavhareji uye rine kusiyana kukuru.
Chinyorwa chino chichakurukura zvinoreva kutsauka kwakajairwa, fomura inoshandiswa, uye matanho ekuiverenga nemaoko kuburikidza nemienzaniso iri nyore kunzwisisa.
1. Tsananguro yeKutsauka Kwakajairwa
Kutsauka kwakajairwa ndiko mudzi wechikwere wemusiyano. Kusiyana pachako ndiko avhareji yemakwere emusiyano uripo pakati pepoindi yega yega yedata neavhareji. Sei uchishandisa sikwere? Nekuti musiyano uripo pakati pemapoindi edata neavhareji unogona kuva negative kana kuti positive. Kana ukawedzerwa zvakananga, musiyano uripo neakanaka unogona kukanzura, zvichikonzera mhedzisiro inotsausa. Nekuisa misiyano pamativi ese, kukosha kwese kunova kwakanaka, zvichibvumira kuyerwa kwakarurama kwekugoverwa kwedata.
Zviri nyore:
– Kusiyana = chiyero chekupararira muzvikamu "zvakaenzana".
– Kutsauka kwakajairwa = chiyero chekupararira kwakadzokera kumayuniti ekutanga edata.
Muenzaniso: kana data riri muchimiro chezvibodzwa zvebvunzo (mapoinzi), musiyano uchava mumapoinzi, nepo kutsauka kwakajairwa kuchavazve mumapoinzi, zvichiita kuti zvive nyore kududzira.
2. Kutsauka Kwehuwandu Hwevanhu vs Muenzaniso
Usati waverenga, zvakakosha kuziva kuti une rudzii rwedata:
1. Huwandu hwevanhu: data rinosanganisira nhengo dzese dzinofanirwa kutsvakurudzwa.
Fomura yekutsauka kwehuwandu hwevanhu inoshandisa muganho weN (nhamba yedata).
2. Muenzaniso: data inongova chikamu chehuwandu hwevanhu, chinowanzo shandiswa kumiririra huwandu hwevanhu vakawanda.
Fomura yekutsauka kwemuenzaniso inoshandisa divisor (n − 1) kugadzirisa fungidziro yekufungidzira. Kugadzirisa uku kunonzi kugadziriswa kwaBessel.
Mumabasa ezuva nezuva (semuenzaniso kutsvagisa, ongororo, ongororo yekirasi), data rinowanzoonekwa semuenzaniso, saka mugovanisi ndiye (n - 1).
3. Fomura Yakajairwa Yokutsauka
A. Fomura yekutsauka kweVagari
Ngatitii data: \( x_1, x_2, \dots, x_N \)
Avhareji yevanhu:
\[
\mu = \frac{\sum_{i=1}^{N} x_i}{N}
\]
Kusiyana kwevanhu:
\[
\sigma^2 = \frac{\sum_{i=1}^{N}(x_i – \mu)^2}{N}
\]
Kusiyana kwehuwandu hwevanhu:
\[
\sigma = \sqrt{\sigma^2}
\]
B. Muenzaniso weForamu Yekutsauka Kwakajairwa
Muenzaniso weavhareji:
\[
\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}
\]
Musiyano wemuenzaniso:
\[
s^2 = \frac{\sum_{i=1}^{n}(x_i – \bar{x})^2}{n – 1}
\]
Muenzaniso wekutsauka kwakajairwa:
\[
s = \sqrt{s^2}
\]
4. Matanho Ekuverenga Kutsauka Kwakajairwa Nemaoko
Kuti zvive nyore kunzwisisa, tinoshandisa muenzaniso wedata remapoinzi ebvunzo dzevadzidzi vashanu:
Data: 60, 70, 70, 80, 90
Fungidzira kuti data iri muenzaniso (unowanikwa mumienzaniso yekudzidza).
Danho 1: Verenga avhareji
\[
\bar{x} = \frac{60 + 70 + 70 + 80 + 90}{5} = \frac{370}{5} = 74
\]
Saka avhareji yemapoinzi i74.
Danho rechipiri: Verenga musiyano uripo pakati pedata rega rega neavhareji
Gadzira koramu yemusiyano \( (x_i – \bar{x}) \):
– 60 − 74 = −14
– 70 − 74 = −4
– 70 − 74 = −4
– 80 − 74 = 6
– 90 − 74 = 16
Danho rechitatu: Sika musiyano pakati pemativi maviri
Verenga \( (x_i – \bar{x})^2 \):
– (−14)² = 196
– (−4)² = 16
– (−4)² = 16
– 6² = 36
– 16² = 256
Danho rechina: Wedzera zvikwere zvemusiyano
\[
\sum (x_i – \bar{x})^2 = 196 + 16 + 16 + 36 + 256 = 520
\]
Danho rechishanu: Verenga musiyano (wemuenzaniso, patsanura na n − 1)
Sezvo n = 5, ipapo n − 1 = 4:
\[
s^2 = \frac{520}{4} = 130
\]
Saka musiyano wemuenzaniso i130.
Nhanho 6: Mudzi wepakati wemusiyano kuti uwane kutsauka kwakajairwa
\[
s = \sqrt{130} \inenge 11{,}40
\]
Saka kutsauka kwakajairika kwedata kunenge 11,40. Izvi zvinoreva kuti, paavhareji, mamakisi evadzidzi anosiyana nemapoinzi angangoita 11 kubva paavhareji ye74.
5. Nzira Yekukurumidza: Fomura Yeimwe Nzira (Kuverenga)
Kunze kwenzira yemaoko iri pamusoro apa, kune fomura yekuverenga inowanzo shandiswa kukurumidzisa kuverenga, kunyanya kana paine data rakawanda:
Musiyano wemuenzaniso:
\[
s^2 = \frac{\sum x_i^2 – \frac{(\sum x_i)^2}{n}}{n – 1}
\]
Fomura iyi inodzivisa kuverenga mutsauko mumwe nemumwe, asi ichiri kuda kunyatsojeka pakuverenga huwandu hwemaseko edata.
Zvisinei, kuti tinzwisise pfungwa, nzira yedanho nedanho (musiyano → sikweya → huwandu) inowanzo kuve nyore uye yakachengeteka kubva pakukanganisa.
6. Kududzirwa kweStandard Deviation
Kutsauka kwakajairika hakungogumiri panhamba chete, asi kunofanira kududzirwa:
- Kutsauka kudiki kwemazinga: data rakaunganidzwa pedyo neavhareji, musiyano wakaderera, mhedzisiro yacho inowirirana zvakanyanya.
- Kuchinja kukuru kwemaitiro: data rakapararira kure neavhareji, musiyano wakakwira, mhedzisiro haina kuenderana zvakanyanya.
Semuenzaniso, ngatitii makirasi maviri ane avhareji yakafanana, 74. Kana kirasi A ine standard deviation ye5 uye kirasi B ine standard deviation ye15, saka mamakisi ari mukirasi A akaenzana uye akagadzikana. Kirasi B ine musiyano mukuru: mamwe akaderera zvikuru uye mamwe akakwira zvikuru.
7. Zvikanganiso Zvakajairika
Zvimwe zvikanganiso zvakajairika pakuverenga standard deviation:
1. Ndakanganwa kusiyanisa pakati pemuenzaniso nenhamba yevanhu, saka mugovanisi haana kururama (N vs n - 1).
2. Kuverenga avhareji zvisirizvo, izvo zvinoita kuti matanho ese anotevera asave akakodzera.
3. Kukanganwa kuyera musiyano, kana kukanganisa pakutora mudzi wepakati pamagumo.
4. Kukanganisa kwekuverenga pakuwedzera kana kuverenga masikweya enhamba.
Kudzivirira zvikanganiso kunogona kuitwa nekugadzira tafura yekuverenga uye kuongorora zvabuda.
Penutup
Kuverenga standard deviation hakusi kwakaoma kana ukatevera matanho nemazvo: kuverenga avhareji, kuwana mutsauko uripo pakati pedata rega rega, kuisa misiyano pakati pemaseti ese, kuabatanidza pamwe chete, kupatsanura (na n kana n - 1), wobva watora square root. Nekunzwisisa standard deviation, unogona kuona kuti data rako rinowirirana sei uye kuti rakasiyana sei.
Kana muchida, ndinogonawo kugadzira mimwe mienzaniso ine data rakawanda, mienzaniso yedata rakarongwa (ma frequency tables), kana kuti maitiro ekuverenga standard deviation ndichishandisa Excel/Google Sheets.
