Amasu Okunquma Ukuphambuka Okumaphakathi Kudatha Yezibalo
Kuzibalo, ukuqonda nje "isikhungo" sedatha—isibonelo, ngokusebenzisa isilinganiso noma isilinganiso—ngokuvamile akwanele. Amasethi amabili edatha angaba nesilinganiso esifanayo, kodwa ahluke kakhulu emazingeni awo "okuguquguquka." Ngakho-ke, izilinganiso zokuhlakazeka zibalulekile. Isilinganiso esisodwa sokuhlakazeka okulula ukuyiqonda nokusebenzisa ukuphambuka okuphakathi. Lesi sihloko sixoxa ngamasu okunquma ukuphambuka okuphakathi ezinhlotsheni ezahlukene zedatha yezibalo, kuphelele ngezinyathelo zokubala nezibonelo.
Ukuqonda Ukuphambuka Okumaphakathi
Ukuphambuka okumaphakathi kuyisilinganiso esisho ibanga elimaphakathi lephuzu ngalinye ledatha kusuka esilinganisweni sokuthambekela okumaphakathi, ngokuvamile isilinganiso se-arithmetic (isilinganiso) noma i-median. Ibanga okukhulunywa ngalo liyinani eliphelele lomehluko phakathi kwedatha nenani elimaphakathi, ukuze kungabikho mehluko ongemuhle "okhansela" umehluko omuhle.
Ngokuvamile, ukuphambuka okumaphakathi kuchaza ukuthi idatha isakazeke kude kangakanani nenani layo eliphakathi. Uma ukuphambuka okumaphakathi kuncane, kulapho idatha ihlanganiswa khona ngokuqinile phakathi nendawo; lapho inani likhulu, kulapho idatha iba nokuguquguquka okukhulu.
Kungani Kufanele Usebenzise I-Absolute Value?
Uma sibala isilinganiso somehluko phakathi kwedatha kanye nesilinganiso ngaphandle kwenani eliphelele, isamba somehluko sizohlala singu-zero (ngoba isilinganiso siyiphuzu lokulingana). Isibonelo, uma kukhona umehluko ka-+5 no--5, bahlanganisa kufike ku-0. Ngakho-ke, amanani aphelele asetshenziswa ukuze ukuphambuka kubonise ngempela ibanga ledatha ukusuka enkabeni.
Ifomula Yokuphambuka Okumaphakathi Yedatha Eyodwa
Kwidatha eyodwa (engaqoqwanga), ukuphambuka okumaphakathi kusuka ku-mean kwakhiwa:
\[
SR = \frac{\sum |x_i – \bar{x}|}{n}
\]
Imininingwane:
– \( SR \): ukuphambuka okumaphakathi
– \( x_i \): idatha ye-i-th
– \( \bar{x} \): isilinganiso sezibalo (isilinganiso)
– \( n \): inani ledatha
Indlela Yokubala Idatha Eyodwa (Izinyathelo)
1. Bala isilinganiso \( \bar{x} \) sayo yonke idatha.
2. Bala umehluko phakathi kwedatha ngayinye kanye nesilinganiso: \( x_i – \bar{x} \).
3. Thatha inani eliphelele lomehluko ngamunye: \( |x_i – \bar{x}| \).
4. Hlanganisa wonke amanani aphelele omehluko.
5. Hlukanisa ngenani ledatha \( n \).
Isibonelo Sedatha Eyodwa
Idatha yenani: 6, 8, 10, 12, 14
1) Isilinganiso:
\[
\bar{x}=\frac{6+8+10+12+14}{5}=\frac{50}{5}=10
\]
2) Inani eliphelele lomehluko:
– |6 − 10| = 4
– |8 − 10| = 2
– |10 − 10| = 0
– |12 − 10| = 2
– |14 − 10| = 4
Isiyonke = 4 + 2 + 0 + 2 + 4 = 12
3) Ukuphambuka okumaphakathi:
\[
SR=\frac{12}{5}=2{,}4
\]
Lokhu kusho ukuthi ngokwesilinganiso inani ngalinye lihluka ngamayunithi angu-2,4 kusuka ku-average (10).
Ukuphambuka Okumaphakathi Kwedatha Evamile (Engacacile)
Uma idatha yethulwa ngesimo samanani nama-frequencies (isb. ithebula), ifomula iba:
\[
SR = \frac{\sum f_i |x_i – \bar{x}|}{\sum f_i}
\]
Imininingwane:
– \( f_i \): imvamisa yedatha \( x_i \)
Amasu Okubala Idatha Ephindaphindayo
1. Bala isilinganiso: \(\bar{x}=\frac{\sum f_i x_i}{\sum f_i}\)
2. Bala \( |x_i-\bar{x}| \)
3. Phindaphinda ngemvamisa: \( f_i |x_i-\bar{x}| \)
4. Hlanganisa yonke imiphumela yesinyathelo sesi-3
5. Hlukanisa ngenani eliphelele
Izibonelo Zedatha Ehlukile
| Inani (x) | Imvamisa (f) |
|—|—|
| 5 | 2 |
| 7 | 3 |
| 9 | 1 |
Imvamisa iyonke: \(2+3+1=6\)
Isho:
\[
\bar{x}=\frac{(5)(2)+(7)(3)+(9)(1)}{6}=\frac{10+21+9}{6}=\frac{40}{6}=6{,}67
\]
Bala ukuphambuka:
– Ku-x=5: |5−6,67|=1,67 → izikhathi f=2 → 3,34
– Ku-x=7: |7−6,67|=0,33 → izikhathi f=3 → 0,99
– Ku-x=9: |9−6,67|=2,33 → izikhathi f=1 → 2,33
Isiyonke: 3,34 + 0,99 + 2,33 = 6,66
Ukuphambuka okumaphakathi:
\[
SR=\frac{6{,}66}{6}=1{,}11
\]
Ukuphambuka Okumaphakathi Kwedatha Eqoqwe (Isikhawu Sekilasi)
Kudatha eqoqwe ngamaqoqo (isibonelo, ukusatshalaliswa kwemvamisa yezikhawu), amanani edatha awaboniswa ngawodwana, kodwa aboniswa emakilasini. Ngale njongo, i-midpoint yekilasi (xi) isetshenziselwa ukumela idatha ngaphakathi kwekilasi.
Ifomula:
\[
SR=\frac{\sum f_i |x_i-\bar{x}|}{\sum f_i}
\]
Noma kunjalo, i-\(x_i\) iyindawo ephakathi yekilasi.
Amasu Okubala Idatha Yeqembu
1. Thola indawo ephakathi yekilasi ngalinye:
\[
x_i=\frac{\text{umkhawulo ophansi + umkhawulo ophezulu}}{2}
\]
2. Bala isilinganiso seqembu:
\[
\bar{x}=\frac{\sum f_i x_i}{\sum f_i}
\]
3. Bala \( |x_i-\bar{x}| \)
4. Phindaphinda ngemvamisa \( f_i \)
5. Hlanganisa yonke imiphumela, bese uhlukanisa ngenani eliphelele.
Isibonelo Sedatha Yeqembu
| Ikilasi | f |
|—|—|
| 10–14 | 3 |
| 15–19 | 5 |
| 20–24 | 2 |
Iphuzu eliphakathi:
– 10–14 → 12
– 15–19 → 17
– 20–24 → 22
Isamba f = 10
Isho:
\[
\bar{x}=\frac{(12)(3)+(17)(5)+(22)(2)}{10}=\frac{36+85+44}{10}=\frac{165}{10}=16{,}5
\]
Ukuphambuka:
– |12−16,5|=4,5 → ×3 = 13,5
– |17−16,5|=0,5 → ×5 = 2,5
– |22−16,5|=5,5 → ×2 = 11
Isamba = 13,5 + 2,5 + 11 = 27
Ukuphambuka okumaphakathi:
\[
SR=\frac{27}{10}=2{,}7
\]
Ukuphambuka Okumaphakathi Kusuka Ku-Median
Ngaphezu kwesilinganiso, ukuphambuka okumaphakathi nakho kungabalwa kusukela ku-median. Isimiso siyafana, inani elimaphakathi kuphela elihlukile. Lokhu kuyasiza uma idatha iqukethe izinto ezingaphandle ngoba i-median iyakwazi ukumelana namanani adlulele.
Ngedatha eyodwa:
\[
SR_{Me}=\frac{\sum |x_i-Me|}{n}
\]
Ngedatha yemvamisa:
\[
SR_{Me}=\frac{\sum f_i|x_i-Me|}{\sum f_i}
\]
Izinzuzo kanye nemikhawulo yokuphambuka okuphakathi
Izinzuzo:
1. Kulula ukuyiqonda ngoba isebenzisa "ibanga elimaphakathi" ukusuka esikhungweni sedatha.
2. Sebenzisa yonke idatha (hhayi idatha ethile kuphela).
3. Kungabalwa ngezindlela ezahlukene zokwethulwa kwedatha.
Imikhawulo:
1. Okungadumile kangako kunokuphambuka okujwayelekile ekuhlaziyweni kwezibalo okuthuthukisiwe.
2. Ukusetshenziswa kwamanani aphelele kwenza kube lula kakhulu kwezinye izindlela zokuphatha ze-algebra.
3. Akunamandla njengokuphambuka okujwayelekile ezindleleni eziningi zokuphetha.
I-Penutup
Indlela yokunquma ukuphambuka okumaphakathi kudatha yezibalo ilandela iphethini ehambisanayo: thola inani elimaphakathi (elimaphakathi noma elimaphakathi), bala ibanga eliphelele lephuzu ngalinye ledatha (noma iphuzu elimaphakathi lekilasi) ukusuka enkabeni, bese ulilinganisa—ucabangela imvamisa uma idatha yethulwa etafuleni. Ukuphambuka okumaphakathi kuyindlela elula yokukala ukuhlakazeka yokuchaza ngokunembile ukuhlukahluka kwedatha. Ngokuqonda izinyathelo, ungaqhathanisa ukuhlukahluka kuwo wonke amasethi edatha bese uhlola ngokugcwele ukuzinza kwesethi yedatha.
Uma uthanda, ngingakusiza ukuthi wenze inguqulo yalesi sihloko ngendlela yesabelo sesikole (ngesingeniso-ingxoxo-isiphetho) noma ungeze imibuzo yokuzijwayeza kanye nengxoxo.
