Amasu Okubala I-Median Yedatha Eyodwa Neqoqwe
I-median iyisilinganiso sokuthambekela okuphakathi okuvame ukusetshenziswa kuzibalo. Ngokungafani nesilinganiso (isilinganiso), esengeza wonke amanani bese sihlukaniswa ngenani lamanani, i-median igcizelela "inani eliphakathi" lesethi yedatha ehlungiwe. Ngenxa yokugxila kwayo endaweni, i-median imelana kakhulu namanani adlulele (angaphandle), njengalapho inani elilodwa likhulu kakhulu noma lincane kakhulu uma liqhathaniswa namanye. Yingakho i-median isetshenziswa kabanzi ekuhlaziyweni kwedatha yezomnotho, emfundweni, ocwaningweni lwezenhlalo, ngisho nasekuhlolweni kwamaphuzu okuhlola.
Kulesi sihloko, sizoxoxa ngamasu okubala i-median yezinhlobo ezimbili zedatha: idatha eyodwa (engaqoqwanga) kanye nedatha eqoqwe (eyethulwe kuthebula lokusatshalaliswa kwemvamisa). Ngaphezu kwefomula, ingxoxo izofaka izinyathelo ezisebenzayo zokusetshenziswa okulula.
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1. Umqondo Oyisisekelo We-Median
I-median yinani eliphakathi ngemva kokuba idatha ihlungwe kusukela kwencane kuya kwenkulu. Uma inani lamaphuzu edatha lingajwayelekile, i-median yinani eliphakathi ngqo. Uma inani lamaphuzu edatha lilingana, i-median iyisilinganiso samanani amabili aphakathi.
Ngokwemvelo, i-median ihlukanisa idatha ibe izingxenye ezimbili:
– 50% wedatha ingaphansi (noma ilingana) nesilinganiso esimaphakathi
– 50% wedatha ingaphezulu (noma ilingana) nesilinganiso esimaphakathi
Ngenxa yokuthi i-median isekelwe ekuhleleni, isinyathelo sokuqala esidingeka cishe njalo ukuhlunga idatha.
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2. Ukubala i-Median yedatha eyodwa
Idatha eyodwa yidatha eyethulwa njengoba injalo (isibonelo uhlu lwamamaki abafundi), engafinyezwanga emakilasini aphakathi njengasedathani yeqembu.
A. Izinyathelo Ezijwayelekile
1. Hlunga idatha kusukela kwencane kakhulu kuya kwenkulu kakhulu.
2. Nquma inani ledatha, isibonelo n.
3. Thola indawo ye-median:
– Uma u-n engajwayelekile, i-median isendaweni \((n+1)/2\).
– Uma u-n elingana, i-median iyisilinganiso sedatha ezikhundleni \(n/2\) kanye \((n/2)+1\).
B. Ifomula Ephakathi Yedatha Eyodwa
– Uma u-n engajwayelekile:
\[
Mina = x_{(n+1)/2}
\]
Lokhu kusho ukuthi i-median iyinani ledatha ku-oda le-\((n+1)/2\)th.
– Uma u-n elingana:
\[
Mina = \frac{x_{n/2} + x_{(n/2)+1}}{2}
\]
C. Isibonelo Sedatha Eyodwa (n Odd)
Idatha: 7, 2, 9, 4, 3
1) Hlunga: 2, 3, 4, 7, 9
2) n = 5 (okungajwayelekile)
3) Isikhundla esimaphakathi = \((5+1)/2 = 3\)
Isilinganiso = idatha yesithathu = 4
Ngakho-ke i-median yedatha ingu-4.
D. Isibonelo Sedatha Eyodwa (n Even)
Idatha: 10, 4, 6, 8
1) Hlunga: 4, 6, 8, 10
2) n = 4 (okulinganayo)
3) Indawo ephakathi yidatha yesibili neyesithathu
I-Median = \((6 + 8)/2 = 7\)
Ngakho-ke i-median yedatha ingu-7.
E. Inothi Elibalulekile: Idatha Enemvamisa
Ngezinye izikhathi isethi yedatha eyodwa inganikezwa njengenani kanye nemvamisa (isb., u-60 uvela kabili, u-70 uvela kahlanu). Kulesi simo, i-median isatholakala ngokusekelwe "ekuhleleni" kwedatha, kodwa singasebenzisa imvamisa ehlanganisiwe ukunquma indawo ephakathi ngaphandle kokufaka amaphuzu edatha ngokwawo. Isimiso siyafana: thola isikhundla (n+1)/2nd (okungajwayelekile) noma isikhundla (n/2) kanye ne-(n/2)+1st (elinganayo), bese ubheka amanani amboza leso sikhundla ngokusekelwe kumvamisa eqoqwe.
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3. Ukubala i-Median yedatha eqoqwe ngamaqembu
Idatha eqoqwe yidatha efingqiwe ngezikhawu zekilasi kanye nama-frequency azo. Isibonelo: abantu abathathu abanobude obungu-150-154 cm, abantu abangu-8 abanobude obungu-155-159 cm, njalo njalo. Ngokungafani nedatha eyodwa, i-median yedatha eqoqwe ngokuvamile ayinqunywa ngokunembile ngoba asazi amanani ngamanye ngaphakathi kwe-interval. Ngakho-ke, i-median ibalwa kusetshenziswa ukulinganisela (isilinganiso) kusetshenziswa ifomula ye-median yokusabalalisa okuqoqwe.
A. Amagama Abalulekile ku-Group Data Median
Ngaphambi kokusebenzisa ifomula, sidinga ukuqonda izingxenye eziningana:
– n = imvamisa iyonke (inani eliphelele ledatha)
– n/2 = isikhundla esimaphakathi esihlanganisiwe
– Isigaba esimaphakathi = isigaba sokuqala sesikhawu esikhiqiza imvamisa ehlanganisiwe ≥ n/2
– L = umkhawulo ophansi wekilasi eliphakathi (hhayi umkhawulo ophansi, kodwa umkhawulo wekilasi; kwedatha eqhubekayo ngokuvamile isebenzisa ukulungiswa okungu-0,5 uma idatha iyizinombolo eziphelele)
– F = imvamisa ehlanganisiwe ngaphambi kwesigaba esimaphakathi
– f = imvamisa yekilasi eliphakathi nendawo
– c = ubude beklasi (ububanzi besikhathi)
B. Izinyathelo Zokunquma Isilinganiso Sedatha Yeqembu
1. Dala ithebula lokusatshalaliswa kwemvamisa bese wengeza ikholomu yemvamisa eqongelelekayo.
2. Bala u-n (inani lamafrikhwensi) bese unquma u-n/2.
3. Nquma isigaba esimaphakathi, okungukuthi isigaba esifaka izikhundla ze-n/2 ngokusekelwe emvamisa eqongelelekayo.
4. Faka amanani kufomula ephakathi yedatha yeqembu.
C. Ifomula Ephakathi Yedatha Yeqembu
\[
Mina = L + \left(\frac{\frac{n}{2} – F}{f}\right)\times c
\]
Le fomula yenza ukuhunyushwa okuqondile ngaphakathi kwekilasi eliphakathi, uma kucatshangwa ukuthi idatha isatshalaliswa ngokulinganayo phakathi kwesikhawu sekilasi.
D. Isibonelo se-Median of Group Data
Isibonelo, idatha elandelayo yamaphuzu okuhlolwa:
| Isikhawu senani | Imvamisa (f) |
|—|—:|
| 40–49 | 5 |
| 50–59 | 8 |
| 60–69 | 12 |
| 70–79 | 10 |
| 80–89 | 5 |
1) Imvamisa ephelele:
\[
n = 5+8+12+10+5 = 40
\]
2) Bala i-n/2:
\[
n/2 = 20
\]
3) Imvamisa eqongelelekayo:
– 40–49: 5
– 50–59: 5+8 = 13
– 60–69: 13+12 = 25
– 70–79: 35
– 80–89: 40
Isikhundla sama-20 sisekilasini elinesilinganiso sokuqala esihlanganisiwe esingu-≥ 20, okungukuthi u-60–69. Ngakho-ke lesi yisigaba esimaphakathi.
4) Thola izingxenye:
– L = unqenqema oluphansi lwesigaba esimaphakathi. Esikhawulweni esingu-60–69, unqenqema oluphansi luyi-59,5 (uma idatha iyinani eliphelele).
– F = imvamisa ehlanganisiwe ngaphambi kwesigaba esimaphakathi = 13
– f = imvamisa yekilasi eliphakathi nendawo = 12
– c = ubude beklasi = 10
5) Faka ifomula:
\[
Mina = 59,5 + \kwesobunxele(\frac{20 – 13}{12}\kwesokudla)\izikhathi eziyi-10
\]
\[
Mina = 59,5 + \kwesobunxele(\frac{7}{12}\kwesokudla)\izikhathi eziyi-10
\]
\[
Mina = 59,5 + 5,833… = 65,333…
\]
Ngakho-ke isilinganiso sedatha yeqembu cishe singama-65,33.
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4. Amaphutha Avamile
Amanye amaphutha avamile lapho kubalwa i-median:
1. Ukungahlungi idatha yedatha eyodwa, ngakho-ke inani eliphakathi alilungile.
2. Ukunquma ngokungalungile indawo ye-median lapho u-n elingana (kumele kuthathwe isilinganiso samanani amabili aphakathi).
3. Ngedatha yeqembu, akulungile ukukhetha isigaba esimaphakathi ngoba asidali imvamisa ehlanganisiwe.
4. Ukusebenzisa umkhawulo ophansi wekilasi lomkhawulo ophansi (L) lapho idatha iyizinombolo eziqhubekayo/zesikhathi.
5. Ukunquma ngokungalungile ubude beklasi (c), ikakhulukazi uma izikhawu zingahambisani.
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5. Isiphetho
I-median iyisilinganiso esilula kodwa esinamandla sokuthambekela okuphakathi, ikakhulukazi lapho idatha iqukethe amanani aqatha. Kumasethi edatha angawodwa, i-median inqunywa ngqo kusuka endaweni ephakathi ngemva kokuba idatha ihlungiwe, ngokuphathwa okuhlukile kwezinombolo ezingalingani nezilinganayo zamasethi edatha. Okwamanje, kumasethi edatha aqoqwe ngamaqoqo, i-median ibalwa kusetshenziswa ifomula yokuhlanganisa ngokusekelwe ekilasini eliphakathi, imvamisa eqongelelekayo, kanye nobude bekilasi.
Ngokuqonda umqondo nezinyathelo, ungakwazi ukubala i-median ngokushesha nangokunembile, kokubili kudatha elula kanye nakudatha efingqiwe kumathebula. Ezimweni eziningi zokuhlaziya, i-median iyindlela emele kakhulu kune-mean, ikakhulukazi lapho ukusatshalaliswa kwedatha kungalingani noma kuqukethe okungaphandle.
Uma uthanda, ngingangeza nemibuzo yokuzijwayeza kanye nezingxoxo ukuze ngiqinise ukuqonda kwakho ngesilinganiso sedatha eyodwa neyeqembu.
