Dabaru don Tantance Matsakaicin Bambanci a Bayanan Kididdiga
A cikin kididdiga, kawai fahimtar "tsakiyar" bayanan - misali, ta hanyar matsakaici ko matsakaici - sau da yawa ba shi da isasshen. Saitin bayanai guda biyu na iya samun matsakaicin iri ɗaya, amma sun bambanta sosai a matakan "bambancinsu." Saboda haka, ma'aunin watsawa yana da mahimmanci. Ma'aunin watsawa wanda yake da sauƙin fahimta da amfani shine matsakaicin karkacewa. Wannan labarin ya tattauna dabarun tantance matsakaicin karkacewa a cikin nau'ikan bayanan kididdiga daban-daban, tare da matakan lissafi da misalai.
Fahimtar Matsakaicin Bambanci
Matsakaicin karkacewa ma'auni ne da ke nuna matsakaicin nisan kowace ma'aunin bayanai daga ma'aunin yanayin tsakiya, yawanci matsakaicin lissafi (ma'ana) ko matsakaici. Nisan da ake magana a kai shine cikakken ƙimar bambanci tsakanin bayanan da ƙimar tsakiya, don haka babu wani bambanci mara kyau da ke "sake" bambanci mai kyau.
Gabaɗaya, matsakaicin karkacewa yana bayyana nisan da aka bazu daga ƙimar tsakiya. Yayin da ƙaramin karkacewar matsakaici, haka bayanan ke taruwa a tsakiya; girman ƙimar, haka ma yawan canjin da bayanai ke samu.
Me Yasa Ake Amfani da Cikakken Darajar?
Idan muka ƙididdige matsakaicin bambance-bambancen da ke tsakanin bayanai da matsakaicin ba tare da cikakkiyar ƙima ba, jimlar bambance-bambancen za ta kasance sifili koyaushe (saboda matsakaicin shine ma'aunin daidaito). Misali, idan akwai bambance-bambancen +5 da -5, suna ƙaruwa zuwa 0. Saboda haka, ana amfani da cikakkun ƙima don bambance-bambancen su nuna nisan bayanai daga tsakiya da gaske.
Tsarin Ma'anar Bambancin Bayanai Guda ɗaya
Ga bayanai guda ɗaya (ba a haɗa su cikin rukuni ba), an tsara matsakaicin karkacewa daga matsakaicin:
\[
SR = \frac{\sum |x_i – \bar{x}|}{n}
\]
Bayani:
– \( SR \): matsakaicin karkacewa
– \( x_i \): bayanai na i-th
– \( \bar{x} \): ma'anar lissafi (matsakaici)
– \( n \): adadin bayanai
Dabaru na Lissafin Bayanai Guda ɗaya (Matakai)
1. Lissafa matsakaicin \( \bar{x} \) na duk bayanan.
2. Lissafa bambanci tsakanin kowace bayanai da matsakaicin: \( x_i – \bar{x} \).
3. Ɗauki cikakkiyar ƙimar kowane bambanci: \( |x_i – \bar{x}| \).
4. Ƙara dukkan cikakkun dabi'un bambance-bambancen.
5. Raba ta da adadin bayanai \( n \).
Misalin Bayanai Guda Ɗaya
Bayanan ƙima: 6, 8, 10, 12, 14
1) Ma'ana:
\[
\bar{x}=\frac{6+8+10+12+14}{5}=\frac{50}{5}=10
\]
2) Cikakken ƙimar bambancin:
– |6 − 10| = 4
– |8 − 10| = 2
– |10 − 10| = 0
– |12 − 10| = 2
– |14 − 10| = 4
Jimilla = 4 + 2 + 0 + 2 + 4 = 12
3) Matsakaicin karkacewa:
\[
SR=\frac{12}{5}=2{,}4
\]
Wannan yana nufin cewa a matsakaici kowace ƙima tana karkata da raka'a 2,4 daga matsakaicin (10).
Matsakaicin Bambanci ga Bayanan da Aka Saba (Rarraba)
Idan aka gabatar da bayanan a cikin nau'in dabi'u da mitoci (misali tebur), dabarar za ta zama:
\[
SR = \frac{\sum f_i |x_i – \bar{x}|}{\sum f_i}
\]
Bayani:
– \( f_i \): mitar bayanai \( x_i \)
Dabaru na Lissafin Bayanai na Mita
1. Lissafa matsakaicin: \(\bar{x}=\frac{\sum f_i x_i}{\sum f_i}\)
2. Lissafi \( |x_i-\bar{x}| \)
3. Ninkuwa da mitar: \( f_i |x_i-\bar{x}| \)
4. A tattara dukkan sakamakon mataki na 3
5. Raba ta da jimillar mita
Misalan Bayanan da ba a iya tantancewa ba
| Darajar (x) | Mita (f) |
|—|—|
| 5 | 2 |
| 7 | 3 |
| 9 | 1 |
Jimlar mita: \(2+3+1=6\)
Nufin:
\[
\bar{x}=\frac{(5)(2)+(7)(3)+(9)(1)}{6}=\frac{10+21+9}{6}=\frac{40}{6}=6{,}67
\]
Lissafa karkacewar:
– Ga x=5: |5−6,67|=1,67 → sau f=2 → 3,34
– Ga x=7: |7−6,67|=0,33 → sau f=3 → 0,99
– Ga x=9: |9−6,67|=2,33 → sau f=1 → 2,33
Jimilla: 3,34 + 0,99 + 2,33 = 6,66
Matsakaicin karkacewa:
\[
SR=\frac{6{,}66}{6}=1{,}11
\]
Matsakaicin Ragewar Bayanai ga Rukunin Bayanai (Tazarar Aji)
A cikin bayanan da aka haɗa a rukuni (misali, rarrabawar mita tazara), ba a nuna ƙimar bayanai daban-daban ba, amma a cikin azuzuwan. Don wannan dalili, ana amfani da tsakiyar aji (xi) don wakiltar bayanai a cikin aji.
Dabarar:
\[
SR=\frac{\sum f_i |x_i-\bar{x}|}{\sum f_i}
\]
Duk da haka, \(x_i\) shine tsakiyar aji.
Dabaru na Lissafin Bayanan Rukuni
1. Kayyade tsakiyar kowane aji:
\[
x_i=\frac{\text{ƙasa iyaka + babba iyaka}}{2}
\]
2. Lissafa matsakaicin rukunin:
\[
\bar{x}=\frac{\sum f_i x_i}{\sum f_i}
\]
3. Lissafi \( |x_i-\bar{x}| \)
4. Ninkuwa da mita \( f_i \)
5. A tara dukkan sakamakon, sannan a raba su da jimillar mitar.
Misalin Bayanan Rukuni
| Aji | f |
|—|—|
| 10–14 | 3 |
| 15–19 | 5 |
| 20–24 | 2 |
Tsakiyar wuri:
– 10–14 → 12
– 15–19 → 17
– 20–24 → 22
Jimilla f = 10
Nufin:
\[
\bar{x}=\frac{(12)(3)+(17)(5)+(22)(2)}{10}=\frac{36+85+44}{10}=\frac{165}{10}=16{,}5
\]
Karkacewa:
– |12−16,5|=4,5 → ×3 = 13,5
– |17−16,5|=0,5 → ×5 = 2,5
– |22−16,5|=5,5 → ×2 = 11
Jimilla = 13,5 + 2,5 + 11 = 27
Matsakaicin karkacewa:
\[
SR=\frac{27}{10}=2{,}7
\]
Matsakaicin karkacewa daga Tsakiyar
Baya ga matsakaicin, ana iya ƙididdige matsakaicin karkacewa daga matsakaiciyar. Ka'idar iri ɗaya ce, ƙimar tsakiya kawai ta bambanta. Wannan yana da amfani lokacin da bayanai suka ƙunshi abubuwan da ba su dace ba saboda matsakaiciyar ta fi juriya ga ƙima mai tsauri.
Don bayanai guda ɗaya:
\[
SR_{Ni}=\frac{\sum |x_i-Ni|}{n}
\]
Don bayanan mita:
\[
SR_{Ni}=\frac{\sum f_i|x_i-Ni|}{\sum f_i}
\]
Fa'idodi da Iyakokin Matsakaicin Canzawa
Kelebihan:
1. Mai sauƙin fahimta saboda yana amfani da "matsakaicin nisa" daga cibiyar bayanai.
2. Yi amfani da duk bayanai (ba kawai takamaiman bayanai ba).
3. Ana iya ƙididdige nau'ikan gabatar da bayanai daban-daban.
Iyakoki:
1. Ba shi da shahara fiye da karkacewar da aka saba gani a cikin ci gaba da nazarin kididdiga.
2. Amfani da cikakkun ƙima yana sa ya zama ƙasa da sauƙi ga wasu sarrafa aljabra.
3. Ba shi da ƙarfi kamar karkacewar da aka saba gani a hanyoyi da yawa na tunani.
Penutup
Dabarar tantance matsakaicin karkacewar bayanai a cikin bayanan kididdiga tana bin tsari mai daidaito: tantance ƙimar tsakiya (matsakaicin ko matsakaici), ƙididdige cikakken nisan kowane ma'aunin bayanai (ko tsakiyar aji) daga tsakiya, sannan a daidaita su - la'akari da mitar idan an gabatar da bayanan a cikin tebur. Matsakaicin karkacewar shine ma'aunin watsawa mai dacewa don bayyana bambancin bayanai cikin fahimta. Ta hanyar fahimtar matakan, zaku iya kwatanta bambancin a cikin saitunan bayanai kuma ku kimanta cikakken daidaiton saitin bayanai.
Idan kana so, zan iya taimaka maka ka ƙirƙiri sigar wannan labarin a cikin tsarin aikin makaranta (tare da gabatarwa-tattaunawa-ƙammalawa) ko kuma ƙara tambayoyin aiki tare da tattaunawar.
