Tambayoyi da Tattaunawa Misali: Bambancin da Bambancin Bayanai Guda ɗaya
Kididdiga wani reshe ne na lissafi wanda ya shafi tattarawa, nazari, fassara, gabatarwa, da kuma tsara bayanai. Muhimman ra'ayoyi guda biyu a cikin kididdiga sune bambance-bambance da karkacewar daidaito. Wannan labarin zai tattauna dalla-dalla yadda ake ƙididdige bambancin da karkacewar daidaito na bayanai guda ɗaya da aka saita ta hanyar misalai da dama.
Fahimtar Bambancin Bambancin da Bambancin Daidaitacce
Bambanci wani ma'auni ne da ke auna nisan da lambobin da ke cikin saitin bayanai suka bazu daga matsakaici ko matsakaici. Bambancin yana bayyana a cikin raka'a murabba'i na bayanan asali, wanda hakan ke sa sau da yawa yana da wahalar fassarawa.
Bambancin da aka saba gani shine tushen murabba'in bambancin. Yana ba da ma'auni mafi fahimta na yadda nisan da bayanai ke karkata daga matsakaicin yake saboda raka'o'insa iri ɗaya ne da raka'o'in asali na bayanan.
Tsarin Janar
Ga bayanai guda ɗaya, dabarar bambancin \( \sigma^2 \) da karkacewar daidaito \( \sigma \) na yawan jama'a kamar haka:
1. Bambanci (σ²):
\( \sigma^2 = \frac{1}{N} \sum_{i=1}^{N} (x_i – \mu)^2 \)
2. Daidaitaccen Canji (σ):
\( \sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (x_i – \mu)^2} \)
Ina:
– \( N \) shine adadin bayanai a cikin al'umma.
– \( x_i \) shine ƙimar bayanai ta ith.
– \( \mu \) shine matsakaicin ko matsakaicin bayanai.
Idan muka ƙididdige bambancin da daidaitaccen karkacewar samfurin, to dabarar da ke sama za ta ɗan yi sauƙi:
1. Bambancin Samfura (s²):
\( s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i – \bar{x})^2 \)
2. Samfurin Bambancin Daidaitacce (s):
\( s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i – \bar{x})^2} \)
Ina:
– \( n \) shine adadin bayanai a cikin samfurin.
– \( \bar{x} \) shine matsakaicin ko matsakaicin samfurin.
Tambayoyi da Tattaunawa Samfura
Misali Tambaya ta 1:
Dangane da waɗannan bayanai:
8, 10, 10, 10, 12, 14
Lissafa bambancin da daidaitaccen karkacewar bayanai!
Matakan Magani:
1. Lissafin Matsakaicin (Matsakaicin) \( \mu \):
\[
\mu = \frac{8 + 10 + 10 + 10 + 12 + 14}{6} = \frac{64}{6} = 10.67
\]
2. Lissafa Bambancin Tsakanin Bayanai da Matsakaicin, Sannan a Yi Murabba'i:
\[
(8 – 10.67)^2 = 7.1289
\]
\[
(10 – 10.67)^2 = 0.4489
\]
\[
(10 – 10.67)^2 = 0.4489
\]
\[
(10 – 10.67)^2 = 0.4489
\]
\[
(12 – 10.67)^2 = 1.7689
\]
\[
(14 – 10.67)^2 = 11.1089
\]
3. Ƙara Duk Sakamakon Mai Siffar Kashi:
\[
\sum (x_i – \mu)^2 = 7.1289 + 0.4489 + 0.4489 + 0.4489 + 1.7689 + 11.1089 = 21.3534
\]
4. Lissafin Bambancin (σ²) na Bayanan Yawan Jama'a:
\[
\sigma^2 = \frac{21.3534}{6} = 3.559
\]
Lura: Tunda an ambaci wannan bayanin a matsayin bayanan yawan jama'a, muna raba shi da 6.
5. Lissafin Daidaitaccen Bambanci (σ):
\[
\sigma = \sqrt{3.559} \kimanin 1.886
\]
Don haka, bambancin bayanai shine 3.559 kuma daidaitaccen karkacewar shine 1.886.
Misali Tambaya ta 2:
Ganin waɗannan bayanai na misali:
5, 6, 8, 9, 10, 11
Lissafa bambancin da daidaitaccen karkacewar samfurin!
Matakan Magani:
1. Lissafin Matsakaicin (Matsakaicin) \( \bar{x} \):
\[
\bar{x} = \frac{5 + 6 + 8 + 9 + 10 + 11}{6} = \frac{49}{6} = 8.167
\]
2. Lissafa Bambancin Tsakanin Bayanai da Matsakaicin, Sannan a Yi Murabba'i:
\[
(5 – 8.167)^2 = 10.035
\]
\[
(6 – 8.167)^2 = 4.694
\]
\[
(8 – 8.167)^2 = 0.028
\]
\[
(9 – 8.167)^2 = 0.694
\]
\[
(10 – 8.167)^2 = 3.361
\]
\[
(11 – 8.167)^2 = 7.945
\]
3. Ƙara Duk Sakamakon Mai Siffar Kashi:
\[
\sum (x_i – \bar{x})^2 = 10.035 + 4.694 + 0.028 + 0.694 + 3.361 + 7.945 = 26.757
\]
4. Lissafin Bambancin Samfurin (s²):
\[
s^2 = \frac{26.757}{5} = 5.351
\]
Lura: Tunda wannan samfurin bayanai ne, muna raba shi da 5 (n-1).
5. Lissafin Daidaitaccen Bambanci (s):
\[
s = \sqrt{5.351} \kimanin 2.313
\]
Don haka, bambancin bayanan samfurin shine 5.351 kuma daidaitaccen karkacewar shine 2.313.
Kammalawa
Lissafin bambancin bayanai da karkacewar daidaito yana da matuƙar muhimmanci don fahimtar yadda bayanai ke yaɗuwa a cikin wani tsari da aka bayar. Duk da cewa bambancin yana ba da ma'aunin ka'ida na watsawa a cikin murabba'ai na raka'o'in asali, karkacewar daidaito yana fassara ma'aunin watsawa dangane da raka'o'in asali na bayanan, wanda hakan ke sauƙaƙa fahimta. A cikin nazarin bayanai, ana amfani da waɗannan ma'auni guda biyu don tantance bambancin bayanai da kuma yanke shawara kan ƙididdiga.
Ta hanyar fahimtar matakai da dabarun da suka wajaba, za mu iya ƙididdige bambance-bambance da karkacewar daidaito cikin sauƙi ga yanayi daban-daban da muke fuskanta a cikin tattara bayanai da nazarin bayanai na yau da kullun. Da fatan, wannan labarin zai ba da ƙarin fahimtar ra'ayoyin bambance-bambance da karkacewar daidaito a cikin saitin bayanai guda ɗaya.
