Matekiniki ekuona Avhareji yekutsauka muStatistical Data
Muhuwandu hwezviverengero, kungonzwisisa "pakati" pedata—semuenzaniso, kuburikidza nepakati kana pakati—kazhinji hakukwani. Maseti maviri edata anogona kuva neavhareji yakafanana, asi akasiyana zvakanyanya mumazinga avo e "kushanduka." Nokudaro, zviyero zvekupararira zvakakosha. Imwe chiyero chekupararira chiri nyore kunzwisisa uye kushandisa ndeyepakati pekutsauka. Chinyorwa chino chinokurukura matekiniki ekuona avhareji yekutsauka mumhando dzakasiyana dzedata rezviverengero, pamwe nematanho ekuverenga nemienzaniso.
Kunzwisisa Avhareji Yokutsauka
Kutsauka pakati (mean deviation) chiyero chinoratidza avhareji yedaro repoinzi yega yega yedata kubva pachiyero chetsika yepakati (central tendency), kazhinji arithmetic mean (mean) kana median. Daro riri kutaurwa ndiro kukosha kwakakwana kwemusiyano uripo pakati pedata nehukuru hwepakati (central value), kuitira kuti pasave nemusiyano we "negative" unodzima musiyano wakanaka (positive difference).
Kazhinji, avhareji yekutsauka inotsanangura kuti data rakapararira sei kubva pahukuru hwayo hwepakati. Kana avhareji yekutsauka kudiki, data rinounganidzwa zvakasimba pakati; kana kukosha kukuru, data rinoshanduka-shanduka.
Sei Uchifanira Kushandisa Absolute Value?
Kana tikaverenga avhareji yemusiyano uripo pakati pedata neavhareji pasina kukosha kwakakwana, huwandu hwemusiyano huchagara huri zero (nekuti avhareji ndiyo equilibrium point). Semuenzaniso, kana paine musiyano we +5 ne -5, unowedzera kusvika pa 0. Saka, absolute values dzinoshandiswa kuitira kuti kutsauka kuratidze daro redata kubva pakati.
Fomura Yekutsauka Kwepakati yeData Rimwe Chete
Kune data rimwe chete (risina kuiswa mumapoka), avhareji yekutsauka kubva paavhareji inogadzirwa:
\[
SR = \frac{\sum |x_i – \bar{x}|}{n}
\]
Information:
– \( SR \): kutsauka kwepakati
– \( x_i \): i-th data
– \( \bar{x} \): chiyero chekuverenga masvomhu (pakati)
– \( n \): huwandu hwedata
Maitiro Ekuverenga Data Rimwe Chete (Matanho)
1. Verenga avhareji \( \bar{x} \) yedata rese.
2. Verenga musiyano uripo pakati pedata rega rega neavhareji: \( x_i – \bar{x} \).
3. Tora kukosha kwakakwana kwemusiyano wega wega: \( |x_i – \bar{x}| \).
4. Wedzera zvese zvakakosha zvemusiyano.
5. Govanisa nenhamba yedata \( n \).
Muenzaniso weData Rimwe Chete
Ruzivo rwekukosha: 6, 8, 10, 12, 14
1) Avhareji:
\[
\bar{x}=\frac{6+8+10+12+14}{5}=\frac{50}{5}=10
\]
2) Kukosha kwakakwana kwemusiyano:
– |6 − 10| = 4
– |8 − 10| = 2
– |10 − 10| = 0
– |12 − 10| = 2
– |14 − 10| = 4
Huwandu = 4 + 2 + 0 + 2 + 4 = 12
3) Avhareji yekutsauka:
\[
SR=\frac{12}{5}=2{,}4
\]
Izvi zvinoreva kuti paavhareji kukosha kwega kwega kunosiyana nemayuniti 2,4 kubva paavhareji (10).
Kutsauka Kunopindirana Kwemashoko Anowanzoitwa (Asina Kurongwa)
Kana data racho rikaratidzwa muchimiro chehunhu uye ma frequency (semuenzaniso tafura), fomura inova:
\[
SR = \frac{\sum f_i |x_i – \bar{x}|}{\sum f_i}
\]
Information:
– \( f_i \): maitikiro edata \( x_i \)
Matekiniki eKuverenga Ruzivo rweManguva
1. Verenga avhareji: \(\bar{x}=\frac{\sum f_i x_i}{\sum f_i}\)
2. Verenga \( |x_i-\bar{x}| \)
3. Wedzera ne frequency: \( f_i |x_i-\bar{x}| \)
4. Wedzera zvese zvabuda padanho rechitatu
5. Govanisa nehuwandu hwehuwandu
Mienzaniso yeData Rakasiyana
| Kukosha (x) | Kuwanda (f) |
|—|—|
| 5 | 2 |
| 7 | 3 |
| 9 | 1 |
Kuwanda kwese: \(2+3+1=6\)
Zvinoreva:
\[
\bar{x}=\frac{(5)(2)+(7)(3)+(9)(1)}{6}=\frac{10+21+9}{6}=\frac{40}{6}=6{,}67
\]
Verenga kutsauka:
– Pa x=5: |5−6,67|=1,67 → kakapetwa f=2 → 3,34
– Pa x=7: |7−6,67|=0,33 → kakapetwa f=3 → 0,99
– Pa x=9: |9−6,67|=2,33 → kakapetwa f=1 → 2,33
Huwandu: 3,34 + 0,99 + 2,33 = 6,66
Kutsauka kwepakati:
\[
SR=\frac{6{,}66}{6}=1{,}11
\]
Kutsauka Kunopindirana kweData Rakaunganidzwa (Nguva yeKirasi)
Mudata rakabatanidzwa (semuenzaniso, kugoverwa kwema frequency epakati), kukosha kwedata hakuratidzwi rimwe nerimwe, asi mumakirasi. Nechinangwa ichi, kirasi yepakati (xi) inoshandiswa kumiririra data riri mukirasi.
Fomura yacho:
\[
SR=\frac{\sum f_i |x_i-\bar{x}|}{\sum f_i}
\]
Zvisinei, \(x_i\) ndiyo nzvimbo iri pakati pekirasi.
Matekiniki ekuverenga data reboka
1. Sarudza pakati pekirasi yega yega:
\[
x_i=\frac{\text{muganhu wepasi + muganho wepamusoro}}{2}
\]
2. Verenga huwandu hweboka:
\[
\bar{x}=\frac{\sum f_i x_i}{\sum f_i}
\]
3. Verenga \( |x_i-\bar{x}| \)
4. Wedzera ne frequency \( f_i \)
5. Wedzera mhinduro dzese, wobva wadzipatsanura nehuwandu hwese.
Muenzaniso weData reBoka
Kirasi | f |
|—|—|
| 10–14 | 3 |
| 15–19 | 5 |
| 20–24 | 2 |
Pakati:
– 10–14 → 12
– 15–19 → 17
– 20–24 → 22
Huwandu hwese f = 10
Zvinoreva:
\[
\bar{x}=\frac{(12)(3)+(17)(5)+(22)(2)}{10}=\frac{36+85+44}{10}=\frac{165}{10}=16{,}5
\]
Kutsauka:
– |12−16,5|=4,5 → ×3 = 13,5
– |17−16,5|=0,5 → ×5 = 2,5
– |22−16,5|=5,5 → ×2 = 11
Huwandu = 13,5 + 2,5 + 11 = 27
Kutsauka kwepakati:
\[
SR=\frac{27}{10}=2{,}7
\]
Kutsauka Kunobva Papakati
Pamusoro peavhareji, avhareji yekutsauka inogonawo kuverengerwa kubva papakati. Nheyo yacho yakafanana, kukosha kwepakati chete ndiko kwakasiyana. Izvi zvinobatsira kana data riine zvinhu zvisingawanzoitiki nekuti pakati payo inotsungirira zvakanyanya kune zvakawandisa.
Kune data rimwe chete:
\[
SR_{Me}=\frac{\sum |x_i-Me|}{n}
\]
Nezve data rema frequency:
\[
SR_{Me}=\frac{\sum f_i|x_i-Me|}{\sum f_i}
\]
Zvakanakira uye Zvisingakwanisike zveKutsauka Kwakaenzana
Zvakawandisa:
1. Zviri nyore kunzwisisa nekuti inoshandisa "avhareji daro" kubva panzvimbo yedata.
2. Shandisa ruzivo rwese (kwete ruzivo rwakati chete).
3. Inogona kuverengerwa mhando dzakasiyana dzekuratidza data.
Keterbatasan:
1. Kusanyanya kufarirwa kupfuura kutsauka kwakajairwa mukuongorora kwehuwandu hwepamusoro.
2. Kushandiswa kwe absolute values kunoita kuti zvisanyanya kuoma kune mamwe ma algebraic manipulations.
3. Hazvina kusimba sekuchinja-chinja kwakajairika munzira dzakawanda dzekufungidzira.
Penutup
Nzira yekuona avhareji yekutsauka mu data rehuwandu hwedata inotevera maitiro akafanana: kuona kukosha kwepakati (avhareji kana pakati), kuverenga daro rakazara repoindi yega yega yedata (kana kirasi yepakati) kubva pakati, wobva wava neavhareji yavo—uchifunga nezvekuwanda kwedata kana data iri mutafura. Avhareji yekutsauka inzira iri nyore yekuyera kupararira kwekutsanangura musiyano wedata. Nekunzwisisa matanho, unogona kuenzanisa kusiyana pakati pemaseti edata uye kuongorora zvizere kugadzikana kweseti yedata.
Kana muchida, ndinogona kukubatsirai kugadzira chinyorwa ichi nenzira yebasa rechikoro (nesumo-hurukuro-mhedziso) kana kuwedzera mibvunzo yekudzidzira pamwe chete nehurukuro.
