Matekiniki ekuverenga Median yedata rimwe chete uye rakabatanidzwa
Mupakati inzira yekuyera tsika yepakati inowanzo shandiswa muzviverengero. Kusiyana neavhareji (avhareji), iyo inowedzera kukosha kwese uye yobva yapatsanurwa nenhamba yemitengo, mupakati anosimbisa "kukosha kwepakati" kweseti yedata yakarongeka. Nekuda kwekutarisa kwayo panzvimbo, mupakati haakwanise kukundwa nemitengo yakanyanyisa (kunze), senge kana imwe kukosha yakakura kwazvo kana kuti diki kwazvo kana ichienzaniswa nedzimwe. Ndokusaka mupakati uchishandiswa zvakanyanya mukuongorora data rehupfumi, dzidzo, tsvakiridzo yemagariro evanhu, uye kunyange kuongororwa kwemamaki ebvunzo.
Muchinyorwa chino, tichakurukura matekiniki ekuverenga midhiya yemhando mbiri dzedata: data rimwe chete (risina kuunganidzwa) uye data rakaunganidzwa (rinoratidzwa mutafura yekugoverwa kwema frequency). Pamusoro pefomura, hurukuro iyi ichasanganisira matanho anoshanda ekuita kuti zvive nyore kushandisa.
-
1. Pfungwa Yekutanga yeMedian
Mupakati (median) inhamba yepakati (mid-value) mushure mekunge data rarongwa kubva padiki kusvika pakuru. Kana nhamba yemapoinzi edata isina kuenzana (odd), mupakati (median) ndiyo nhamba chaiyo yepakati (mid-value). Kana nhamba yemapoinzi edata yakaenzana, mupakati (median) inhamba yepakati (mid-values) mbiri.
Nenzira yekufungidzira, iyo median inoparadzanisa data muzvikamu zviviri:
- 50% yedata iri pasi (kana kuti yakaenzana ne) avhareji
- 50% yedata iri pamusoro (kana kuti yakaenzana ne) avhareji
Nekuti median inoenderana nekurongeka, danho rekutanga rinowanzo dikanwa kuronga data.
-
2. Kuverenga Median yedata rimwe chete
Data rimwe chete idata rinopihwa sezvariri (semuenzaniso runyorwa rwemagiredhi evadzidzi), risingapfupikidzwe mumakirasi epakati seari mudata remapoka.
A. Matanho Akajairika
1. Ronga data kubva padiki kusvika pahukuru.
2. Sarudza huwandu hwedata, semuenzaniso n.
3. Sarudza nzvimbo yepakati:
– Kana n iri odd, iyo median iri panzvimbo \((n+1)/2\).
– Kana n yakaenzana, muyero wepakati ndiwo avhareji yedata panzvimbo \(n/2\) uye \((n/2)+1\).
B. Fomura yepakati yedata rimwe chete
- Kana n isingawanzoitiki:
\[
Ini = x_{(n+1)/2}
\]
Izvi zvinoreva kuti median ndiyo data value iri mu \((n+1)/2\)th order.
- Kana n yakaenzana:
\[
Ini = \frac{x_{n/2} + x_{(n/2)+1}}{2}
\]
C. Muenzaniso weData Rimwe Chete (n Odd)
Data: 7, 2, 9, 4, 3
1) Kuronga: 2, 3, 4, 7, 9
2) n = 5 (zvisinganzwisisike)
3) Nzvimbo yepakati = \((5+1)/2 = 3\)
Pakati = data rechitatu = 4
Saka avhareji yedata iri 4.
D. Muenzaniso weData Rimwe Chete (n Even)
Data: 10, 4, 6, 8
1) Kuronga: 4, 6, 8, 10
2) n = 4 (yakaenzana)
3) Nzvimbo yepakati ndiyo data rechipiri nerechitatu
Pakati = \((6 + 8)/2 = 7\)
Saka avhareji yedata iri 7.
E. Chinyorwa Chinokosha: Data Rine Kakawanda
Dzimwe nguva seti imwe chete yedata inogona kupihwa sekukosha uye frequency (semuenzaniso, 60 inoonekwa kaviri, 70 inoonekwa kashanu). Muchiitiko ichi, median ichiri kuwanikwa zvichibva pa "kurongeka" kwedata, asi tinogona kushandisa cumulative frequency kuti tione nzvimbo yepakati tisinganyore mapoinzi edata rimwe nerimwe. Nheyo yacho yakafanana: tsvaga nzvimbo ye (n+1)/2nd (odd) kana nzvimbo ye (n/2) uye (n/2)+1st (even), wobva watarisa kukosha kunofukidza nzvimbo iyoyo zvichibva pahuwandu hwakaunganidzwa.
-
3. Kuverenga Median yeData Rakaunganidzwa
Data rakarongwa idata rakapfupikiswa muzvikamu zvekirasi uye mafrequency aro. Semuenzaniso: vanhu vatatu vane hurefu hwe150–154 cm, vanhu vasere vane hurefu hwe155–159 cm, zvichingodaro. Kusiyana nedata rimwe chete, avhareji yedata rakarongwa kazhinji hainyatso tsanangurwa nekuti hatizivi kukosha kwega kwega mukati menguva yacho. Nokudaro, avhareji inoverengwa uchishandisa approximation (kufungidzira) uchishandisa fomura yepakati yekugoverwa kwemapoka.
A. Mashoko Akakosha muBoka reData Median
Tisati tashandisa fomura iyi, tinofanira kunzwisisa zvinhu zvakawanda:
– n = huwandu hwese (huwandu hwese hwedata)
– n/2 = nzvimbo yepakati yakaunganidzwa
– Kirasi yepakati = kirasi yekutanga yepakati inogadzira cumulative frequency ≥ n/2
– L = mupendero wepasi wepakati pekirasi (kwete muganho wepasi, asi mupendero wekirasi; kune data rinoenderera mberi kazhinji shandisa kugadzirisa kwe0,5 kana data riri nhamba dzese)
– F = kuwanda kwehuwandu hwehuwandu husati hwasvika padanho repakati
– f = maitikiro ekirasi yepakati
– c = urefu hwekirasi (upamhi hwepakati)
B. Matanho Ekuona Pakati Pedata reBoka
1. Gadzira tafura yekugovera mafrequency uye wedzera koramu ye cumulative frequency.
2. Verenga n (nhamba yemafrequencies) uye sarudza n/2.
3. Sarudza kirasi yepakati, kureva kirasi inosanganisira n/2 nzvimbo zvichibva pahuwandu hwehuwandu.
4. Isa ma "values" mu "median formula" yedata reboka.
C. Fomura yepakati yeData reBoka
\[
Ini = L + \left(\frac{\frac{n}{2} – F}{f}\right)\times c
\]
Fomura iyi inoita interpolation yakatwasuka mukati mekirasi yepakati, tichifunga kuti data rakagoverwa zvakaenzana mukati mechikamu chekirasi.
D. Muenzaniso wepakati peData reBoka
Semuenzaniso, data rezvibodzwa zvebvunzo rinotevera:
| Nguva Yekukosha | Kuwanda (f) |
|—|—:|
| 40–49 | 5 |
| 50–59 | 8 |
| 60–69 | 12 |
| 70–79 | 10 |
| 80–89 | 5 |
1) Kuwanda kwese:
\[
n = 5+8+12+10+5 = 40
\]
2) Verenga nhamba n/2:
\[
n/2 = 20
\]
3) Kuwanda kwehuwandu:
– 40–49: 5
– 50–59: 5+8 = 13
– 60–69: 13+12 = 25
– 70–79: 35
– 80–89: 40
Nzvimbo 20 iri mukirasi ine mamakisi ekutanga akabatanidzwa ≥ 20, kureva 60–69. Saka iyi ndiyo kirasi yepakati.
4) Sarudza zvikamu:
– L = mupendero wepasi wepakati pekirasi. Pachikamu che60–69, mupendero wepasi i59,5 (kana data iri nhamba yenhamba yakakwana).
– F = kuwanda kwehuwandu hwehuwandu husati hwasvika padanho repakati = 13
– f = maitikiro epakati pekirasi = 12
– c = kureba kwekirasi = 10
5) Pinda mufomura:
\[
Ini = 59,5 + \left(\frac{20 – 13}{12}\right)\times 10
\]
\[
Ini = 59,5 + \kuruboshwe(\frac{7}{12}\kurudyi)\kawa 10
\]
\[
Ini = 59,5 + 5,833… = 65,333…
\]
Saka avhareji yedata reboka inenge 65,33.
-
4. Zvikanganiso Zvakajairika
Zvimwe zvikanganiso zvakajairika pakuverenga midhiya:
1. Kusaronga data redata rimwe chete, saka kukosha kwepakati hakuna kururama.
2. Kusarudza zvisizvo nzvimbo yepakati kana n yakaenzana (inofanira kutora avhareji yemakoshesi maviri epakati).
3. Padata reboka, hazvina kunaka kusarudza kirasi yepakati nekuti haigadzire cumulative frequency.
4. Kushandisa muganho wepasi wekirasi yemupendero wepasi (L) kana data riri nhamba dzinoenderera mberi/dzinopindirana.
5. Kusaziva urefu hwekirasi (c) zvisizvo, kunyanya kana zvikamu zvisingawirirani.
-
5. Mhedziso
Mutsetse wepakati inzira iri nyore asi ine simba yekuyera mafambiro epakati, kunyanya kana data riine mavalues akanyanya. Kune maseti edata rimwe chete, mutsetse wepakati unotsanangurwa zvakananga kubva panzvimbo yepakati mushure mekunge data rarongwa, nekurapwa kwakasiyana kwenhamba dzisina kuenzana uye dzakafanana dzemaseti edata. Panguva imwecheteyo, kune maseti edata akaunganidzwa, mutsetse wepakati unoverengerwa uchishandisa fomura ye interpolation zvichibva pachikamu chepakati, cumulative frequency, uye urefu hwekirasi.
Nekunzwisisa pfungwa nematanho, unogona kuverenga midhiya nekukurumidza uye nemazvo, zvese padata riri nyore uye padata rakapfupikiswa mumatafura. Muzviitiko zvakawanda zvekuongorora, midhiya isarudzo inomiririra zviri nani pane avhareji, kunyanya kana kugoverwa kwedata kusina kuenzana kana kuti kuine zvinhu zvisingaenzaniswi.
Kana muchida, ndinogonawo kuwedzera mibvunzo yekudzidzira pamwe chete nehurukuro kuti ndisimbise kunzwisisa kwenyu kwepakati pedata remunhu mumwe chete nereboka.
