Dabaru don Lissafin Matsakaici ga Bayanan Mutum ɗaya da na Rukuni
Matsakaici ma'auni ne na yanayin tsakiya wanda ake yawan amfani da shi a cikin kididdiga. Ba kamar matsakaicin (matsakaicin) ba, wanda ke ƙara dukkan ƙima sannan ya raba da adadin ƙima, matsakaiciyar tana jaddada "ƙimar tsakiya" na saitin bayanai da aka tsara. Saboda mayar da hankali kan matsayi, matsakaiciyar tana da juriya ga ƙima mai tsauri (masu fita), kamar lokacin da ƙima ɗaya ta yi girma sosai ko ƙarami idan aka kwatanta da sauran. Wannan shine dalilin da ya sa ake amfani da matsakaiciyar sosai a cikin nazarin bayanan tattalin arziki , ilimi, binciken zamantakewa, har ma da kimanta maki na gwaji.
A cikin wannan labarin, za mu tattauna dabarun ƙididdige matsakaicin nau'ikan bayanai guda biyu: bayanai ɗaya (ba a haɗa su ba) da bayanai masu rukuni (wanda aka gabatar a cikin teburin rarraba mita ). Baya ga dabarar, tattaunawar za ta haɗa da matakai masu amfani don sauƙin aiwatarwa.
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1. Asalin Ma'anar Tsakiyar Tsakiya
Matsakaicin shine matsakaicin ƙimar bayan an tsara bayanai daga ƙarami zuwa babba. Idan adadin maki na bayanai ba daidai ba ne, matsakaicin shine ainihin ƙimar tsakiya. Idan adadin maki na bayanai daidai ne, matsakaicin shine matsakaicin ƙimar tsakiya guda biyu.
A hankali, matsakaicin yana raba bayanai zuwa sassa biyu:
– Kashi 50% na bayanai suna ƙasa da (ko daidai yake da) matsakaicin
– Kashi 50% na bayanai sun fi (ko kuma daidai yake da) matsakaicin bayanai
Domin kuwa matsakaicin yana dogara ne akan tsari, matakin farko da ake buƙata kusan koyaushe shine a tsara bayanai.
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2. Lissafin Matsakaici don Bayanai Guda ɗaya
Bayanai ɗaya shine bayanai da aka gabatar kamar yadda yake (misali jerin maki na ɗalibai), ba a taƙaita su a cikin azuzuwan tazara kamar yadda yake a cikin bayanan rukuni ba.
A. Matakai na Gabaɗaya
1. Rarraba bayanai daga mafi ƙanƙanta zuwa mafi girma.
2. Kayyade adadin bayanai, misali n.
3. Kayyade matsayin matsakaicin:
– Idan n ya zama baƙo, matsakaicin yana a matsayi \((n+1)/2\).
– Idan n daidai ne, matsakaicin shine matsakaicin bayanai a matsayi \(n/2\) da \((n/2)+1\).
B. Tsarin Tsakiyar Bayanai na Bayanai Guda ɗaya
- Idan n baƙo ne:
\[
Ni = x_{(n+1)/2}
\]
Wannan yana nufin cewa matsakaicin shine ƙimar bayanai a cikin tsari na \((n+1)/2\).
– Idan n ya kasance daidai:
\[
Ni = \frac{x_{n/2} + x_{(n/2)+1}}{2}
\]
C. Misalin Bayanai Guda Ɗaya (n Odd)
Bayanai: 7, 2, 9, 4, 3
1) Rarraba: 2, 3, 4, 7, 9
2) n = 5 (na daban)
3) Matsakaicin matsayi = \((5+1)/2 = 3\)
Matsakaici = bayanai na 3 = 4
Don haka matsakaicin bayanai shine 4.
D. Misalin Bayanai Guda Ɗaya (n Ko da)
Bayanai: 10, 4, 6, 8
1) Rarraba: 4, 6, 8, 10
2) n = 4 (daidai)
3) Matsayin tsakiya shine bayanai na 2 da 3
Matsakaici = \((6 + 8)/2 = 7\)
Don haka matsakaicin bayanai shine 7.
E. Muhimmin Bayani: Bayanan da ke da Mita
Wani lokaci ana iya bayar da saitin bayanai guda ɗaya a matsayin ƙima da mita (misali, 60 ya bayyana sau biyu, 70 ya bayyana sau biyar). A wannan yanayin, ana samun matsakaiciyar dangane da "tsari" na bayanan, amma za mu iya amfani da mitar tarin bayanai don tantance matsayin matsakaici ba tare da lissafa wuraren bayanai daban-daban ba. Ka'idar iri ɗaya ce: nemo matsayi (n+1)/2 (mara kyau) ko matsayi (n/2) da (n/2)+1 (komai), sannan duba ƙimar da ta rufe wannan matsayi bisa ga mitar da aka tara.
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3. Lissafin Matsakaici don Bayanan Rukunin
Bayanan da aka haɗa rukuni bayanai ne da aka taƙaita su cikin tazara ajin da kuma mitoci. Misali: Mutane 3 masu tsayin 150-154 cm, mutane 8 masu tsayin 155-159 cm, da sauransu. Ba kamar bayanai ɗaya ba, matsakaicin bayanan da aka haɗa rukuni yawanci ba a ƙayyade su daidai ba saboda ba mu san ƙimar mutum ɗaya a cikin tazara ba. Saboda haka, ana ƙididdige matsakaicin ta amfani da kimantawa (ƙididdiga) ta amfani da dabarar matsakaici don rarrabawa rukuni.
A. Muhimman Kalmomi a Matsakaici a Bayanan Rukuni
Kafin amfani da dabarar, muna buƙatar fahimtar abubuwa da yawa:
– n = jimlar mita (jimillar adadin bayanai)
– n/2 = matsakaicin matsayi na tara
– Matsakaicin aji = aji na farko na tazara wanda ke samar da tarin mita ≥ n/2
– L = ƙananan gefen matsakaicin aji (ba ƙananan iyaka ba, amma gefen aji; don ci gaba da bayanai yawanci ana amfani da daidaitawa 0,5 idan bayanan lambobi ne)
– F = yawan tarawa kafin matsakaicin aji
– f = matsakaicin mitar aji
– c = tsawon aji (faɗin tazara)
B. Matakai don Tantance Matsakaicin Bayanan Rukuni
1. Ƙirƙiri teburin rarraba mitar kuma ƙara ginshiƙin mitar da aka tara.
2. Lissafa n (adadin mita) kuma a tantance n/2.
3. Kayyade matsakaicin aji, wato aji wanda ya haɗa da matsayi na n/2 bisa ga yawan tarawa.
4. Shigar da ƙimar a cikin dabarar matsakaici don bayanan rukuni.
C. Tsarin Tsakiyar Bayanai na Rukuni
\[
Ni = L + \left(\frac{\frac{n}{2} – F}{f}\right)\times c
\]
Wannan dabarar tana yin haɗin layi a cikin matsakaicin aji, idan aka ɗauka cewa bayanan sun rarraba daidai a cikin tazara ajin.
D. Misali na Matsakaicin Bayanan Rukuni
Misali, bayanan sakamakon gwaji masu zuwa:
| Tazarar Darajar | Mita (f) |
|—|—:|
| 40–49 | 5 |
| 50–59 | 8 |
| 60–69 | 12 |
| 70–79 | 10 |
| 80–89 | 5 |
1) Jimlar mita:
\[
n = 5+8+12+10+5 = 40
\]
2) Lissafa n/2:
\[
n/2 = 20
\]
3) Mita mai tarin yawa:
– 40–49:5
– 50–59: 5+8 = 13
– 60–69: 13+12 = 25
– 70–79:35
– 80–89:40
Matsayi na 20 yana cikin ajin da maki na farko da aka tara ≥ 20, wato 60–69. Don haka wannan shine matsakaicin aji.
4) Ƙayyade abubuwan da aka haɗa:
– L = ƙananan gefen matsakaicin aji. Ga tazara ta 60–69, ƙananan gefen shine 59,5 (idan bayanan ƙimar lamba ce).
– F = yawan tarawa kafin matsakaicin aji = 13
– f = matsakaicin mitar aji = 12
– c = tsawon aji = 10
5) Shigar da dabarar:
\[
Ni = 59,5 + \left(\frac{20 – 13}{12}\right)\times 10
\]
\[
Ni = 59,5 + \left(\frac{7}{12}\right)\times 10
\]
\[
Ni = 59,5 + 5,833… = 65,333…
\]
Don haka matsakaicin bayanan rukuni shine kimanin 65,33.
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4. Kurakurai da Aka Saba Yi
Wasu kurakurai da aka saba yi yayin ƙididdige matsakaicin matsakaici:
1. Rashin rarraba bayanai don bayanai ɗaya, don haka ƙimar tsakiya ba daidai ba ce.
2. Kuskuren tantance matsayin matsakaici lokacin da n ya daidaita (dole ne ya ɗauki matsakaicin ƙimar tsakiya guda biyu).
3. Ga bayanan rukuni, ba daidai ba ne a zaɓi matsakaicin aji domin ba ya haifar da mitar tarin bayanai.
4. Amfani da ƙananan iyaka na ajin gefen ƙasa (L) lokacin da bayanai ke ci gaba/tsawon lokaci lambobi ne.
5. Ba daidai ba ne a tantance tsawon aji (c), musamman idan tazara ba ta daidaita ba.
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5. Kammalawa
Matsakaici ma'auni ne mai sauƙi amma mai ƙarfi na yanayin tsakiya, musamman lokacin da bayanan suka ƙunshi ƙima mai tsanani. Ga saitin bayanai guda ɗaya, ana ƙayyade matsakaicin kai tsaye daga matsayin tsakiya bayan an tsara bayanan, tare da magani daban-daban don lambobin bayanai marasa kyau da ma'ana. A halin yanzu, ga saitin bayanai da aka haɗa, ana ƙididdige matsakaicin ta amfani da dabarar interpolation bisa ga matsakaicin aji, mitar tarin bayanai, da tsawon aji.
Ta hanyar fahimtar ra'ayi da matakai, za ku iya ƙididdige matsakaicin da sauri da daidai, duka akan bayanai masu sauƙi da kuma akan bayanan da aka taƙaita a cikin tebura. A cikin yanayi da yawa na nazari, matsakaicin zaɓi ne mafi wakilci fiye da matsakaicin, musamman lokacin da rarraba bayanai ba ta da daidaito ko kuma ya ƙunshi abubuwan da ba su dace ba.
Idan kana so, zan iya ƙara tambayoyin aiki tare da tattaunawa don ƙarfafa fahimtarka game da matsakaicin bayanai na mutum ɗaya da na rukuni.
