Bambancin da Daidaitaccen Bambancin Bayanan Rukuni
Kididdiga wani reshe ne na lissafi da ake amfani da shi don tattarawa, tsarawa, nazari, fassarawa, da gabatar da bayanai. Wani muhimmin ra'ayi a cikin kididdiga shine auna bambancin bayanai, ko yaduwar bayanai. Manyan ma'auni guda biyu na bambancin sune bambancin bayanai da bambancin da aka saba amfani da su. Wannan labarin zai binciki bambancin bayanai da bambancin da aka saba amfani da su, musamman a cikin mahallin bayanan rukuni.
Ma'anar da Muhimmancin Canji
Bambanci yana auna nisan da bayanai ke yaɗuwa daga matsakaicinsa. Auna bambancin yana da mahimmanci domin yana samar da ƙarin fahimta waɗanda ba za a iya samu kawai daga ma'aunin yanayin tsakiya ba, kamar matsakaici. Ta hanyar sanin ma'aunin bambancin, za mu iya fahimtar yadda bayanan suke da daidaito da kuma gano yiwuwar rashin daidaituwa ko rashin daidaituwa.
Fahimtar Bambancin Bambancin da Bambancin Daidaitacce
Bambanci wani ma'auni ne na rarraba bayanai wanda ke nuna nisan da kowanne ma'aunin bayanai yake da shi daga matsakaicinsa a cikin murabba'in raka'a. Ana bayar da shi ta hanyar alamar \( \sigma^2 \) ga yawan jama'a da \( s^2 \) ga samfuri. Tsarin bambancin bayanai ga yawan jama'a shine:
\[ \sigma^2 = \frac{\sum (X_i – \mu)^2}{N} \]
Dangane da misalin, dabarar ita ce:
\[ s^2 = \frac{\sum (X_i – \bar{X})^2}{n-1} \]
Ina:
– \( X_i \) shine ƙimar bayanai na mutum ɗaya
– \( \mu \) shine matsakaicin yawan jama'a
– \( \bar{X} \) shine matsakaicin samfurin
– \( N \) shine girman yawan jama'a
– \(n \) shine girman samfurin
Bambancin Daidaitacce shine tushen murabba'in bambancin. Ana bayar da shi ta hanyar alamar \( \sigma \) don yawan jama'a da \( s \) don samfurin. Bambancin Daidaitacce yana dawo da raka'o'in bayanai zuwa siffarsu ta asali, wanda hakan ke sauƙaƙa fassara fiye da bambancin.
\[ \sigma = \sqrt{\sigma^2} \]
\[ s = \sqrt{s^2} \]
Bayanan Rukuni
Bayanan da aka haɗa rukuni bayanai ne da aka rarraba zuwa rukuni ko tazara da dama. Misali, tsayin ɗalibai an raba su zuwa tazara tsakanin 150-155 cm, 155-160 cm, da sauransu. Yin nazarin bambance-bambance da karkacewar da aka saba yi akan bayanan da aka haɗa rukuni yana buƙatar wata hanya daban da ta yi nazarin bayanan mutum ɗaya.
Matakai don Lissafin Bambanci da Bambancin Daidaitacce don Bayanan Rukuni
Ga matakan da za a bi don ƙididdige bambancin da kuma karkacewar daidaitattun bayanai na rukuni:
1. Ƙirƙiri Teburin Rarraba Mita
– Ana raba bayanai zuwa azuzuwa ko tazara da dama.
– Ana yin rikodin mitar kowane tazara (adadin bayanai a kowane tazara).
2. Bayyana Tsakiyar Ajin
– Ana ƙididdige tsakiyar kowane tazara kamar haka: \( \text{Midpoint} = \frac{\text{Ƙasa da Haɗi} + \text{Saman Haɗi}}{2} \)
3. Lissafin Matsakaicin Wucin Gadi (\( \bar{X} \))
– Ana ƙididdige matsakaicin ta amfani da dabarar: \( \bar{X} = \frac{\sum f_i x_i}{\sum f_i} \)
– Inda \(f_i \) shine mita da \(x_i \) shine tsakiyar tazara.
4. Lissafin karkacewa daga Ma'auni da kuma murabba'insa
– Ga kowane tazara, ana ƙididdige karkacewar daga matsakaicin kamar haka: \( d_i = x_i – \bar{X} \)
– Sannan a lissafta murabba'in: \( d_i^2 \)
5. Lissafin Bambancin da Daidaitaccen Bambancin
– Ana ƙididdige bambancin ta amfani da dabarar: \( s^2 = \frac{\sum f_i d_i^2}{\sum f_i – 1} \)
– Bambancin da aka saba dashi shine tushen murabba'in bambancin: \( s = \sqrt{s^2} \)
Misalin Lissafi
A ce mun tattara bayanan tsayin ɗalibi kamar haka:
| Tazarar (cm) | Mita (f) |
|—————|—————|
| 150 – 154 | 5 |
| 155 – 159 | 10 |
| 160 – 164 | 15 |
| 165 – 169 | 8 |
| 170 – 174 | 2 |
1. Teburin Rarraba Mita:
| Tazarar (cm) | Mita (f) | Tsakiyar Ma'ana (x) | \( f \cdot x \) | \( d = x – \bar{X} \) | \( d^2 \) | \( f \cdot d^2 \) |
|———————|————————|—————————————–|————————-|
| 150 – 154 | 5 | 152 | 760 | | |
| 155 – 159 | 10 | 157 | 1570 | | |
| 160 – 164 | 15 | 162 | 2430 | | |
| 165 – 169 | 8 | 167 | 1336 | | |
| 170 – 174 | 2 | 172 | 344 | | |
| Jimilla | 40 | | 6440 | | |
2. Lissafin Matsakaicin (\( \bar{X} \)):
\[ \bar{X} = \frac{6440}{40} = 161 \]
3. Lissafin karkacewa daga Ma'auni da kuma murabba'insa:
| Tazarar (cm) | Mita (f) | Tsakiyar maki (x) | \( f \cdot x \) | \( d = x – 161 \) | \( d^2 \) | \( f \cdot d^2 \) |
|———————|————————|—————————————–|—————————-|————————-|
| 150 – 154 | 5 | 152 | 760 | -9 | 81 | 405 |
| 155 – 159 | 10 | 157 | 1570 | -4 | 16 | 160 |
| 160 – 164 | 15 | 162 | 2430 | 1 | 1 | 15 |
| 165 – 169 | 8 | 167 | 1336 | 6 | 36 | 288 |
| 170 – 174 | 2 | 172 | 344 | 11 | 121 | 242 |
| Jimilla | 40 | | 6440 | | | 1110 |
4. Lissafin Bambancin:
\[ s^2 = \frac{1110}{40 – 1} = \frac{1110}{39} \kimanin 28.46 \]
5. Lissafin Daidaitaccen Bambanci:
\[ s = \sqrt{28.46} \kimanin 5.33 \]
Kammalawa
Bambanci da karkacewar daidaito muhimman ma'auni ne a cikin kididdiga waɗanda ke bayyana rarraba bayanai a kusa da ma'auninsa. Duk da cewa ana iya amfani da waɗannan ra'ayoyi ga bayanan mutum ɗaya, tsarin lissafi ya ɗan bambanta ga bayanan da aka haɗa. Daga misalin da ke sama, za mu iya ganin cikakkun matakai don ƙididdige bambanci da karkacewar daidaito don bayanan da aka haɗa. Wannan bayanin yana da amfani a cikin aikace-aikace iri-iri, daga binciken ilimi zuwa nazarin kasuwanci da samarwa.
Da fahimtar bambanci da karkacewar da aka saba da ita, za mu iya fassara da yanke shawara daidai bisa ga bayanan da muke da su. Wannan yana ba mu damar kiyaye sakamakon da ba wai kawai daidai ba ne, har ma da dacewa.
