Yadda Ake Lissafin Bambanci: Cikakken Jagora
Bambanci wani muhimmin ƙididdiga ne da ake amfani da shi a fannoni daban-daban, tun daga tattalin arziki da injiniyanci zuwa ilimin halayyar ɗan adam da kididdiga. Yana ba da bayanai game da yadda aka bazu ƙimar da ke cikin saitin bayanai a kusa da matsakaicin. A cikin wannan labarin, za mu bincika yadda ake ƙididdige bambanci a zurfi, daga ma'anar zuwa matakai masu amfani.
Pendahuluan
Domin fahimtar bambancin, muna buƙatar fahimtar wasu muhimman ra'ayoyi a cikin kididdiga. Bambanci ma'auni ne na yadda nisan da ke cikin saitin bayanai ya bambanta da matsakaicin. Ana ƙididdige bambancin a matsayin matsakaicin bambance-bambancen murabba'i tsakanin kowace ƙima da matsakaicin. Bambanci yana ba da alamar "canjin" a cikin bayanan.
Ma'anar Bambanci
A fannin lissafi, bambancin shine:
\[ \text{Bambanci} ( \sigma^2 ) = \frac{1}{N} \sum_{i=1}^{N} (x_i – \mu)^2 \]
Ina:
– \( \sigma^2 \) shine bambancin yawan jama'a.
– \(N\) shine jimlar adadin dabi'u a cikin al'umma.
– \( x_i \) shine ƙimar ith ɗin mutum ɗaya.
– \( \mu \) shine matsakaicin yawan jama'a.
Ga samfurori, dabarar bambancin ta ɗan bambanta:
\[ \text{Samfurin Bambanci} ( s^2 ) = \frac{1}{n-1} \sum_{i=1}^{n} (x_i – \bar{x})^2 \]
Ina:
– \(s^2 \) shine bambancin samfurin.
– \(n \) shine jimlar adadin ƙimomin da ke cikin samfurin.
– \( x_i \) shine ƙimar mutum ɗaya a cikin samfurin.
– \( \bar{x} \) shine matsakaicin samfurin.
Matakai don Lissafin Bambanci
Bari mu sake duba matakan da ake amfani da su wajen ƙididdige bambancin ta hanyar misali mai ma'ana.
Misali: Lissafi Bambancin Yawan Jama'a
A ce muna da ƙaramin bayanai wanda ya ƙunshi waɗannan dabi'u: 2, 4, 6, 8, 10.
1. Mataki na 1: Lissafa Matsakaicin (Matsakaicin)
\[ \mu = \frac{2 + 4 + 6 + 8 + 10}{5} = 6 \]
2. Mataki na 2: Lissafa Bambancin Kowace Ƙima daga Ma'ana sannan a Matsakaita Shi
\[
\begin{daidai}
(2 – 6)^2 &= (-4)^2 = 16 \\
(4 – 6)^2 &= (-2)^2 = 4 \\
(6 – 6)^2 &= 0^2 = 0 \\
(8 – 6)^2 &= 2^2 = 4 \\
(10 – 6)^2 &= 4^2 = 16 \\
\end{daidai}
\]
3. Mataki na 3: Ƙara Duk Muƙallan Bambancin
\[ 16 + 4 + 0 + 4 + 16 = 40 \]
4. Mataki na 4: Raba Jimlar Mudu na Bambanci da Adadin Ƙima (N)
\[ \sigma^2 = \frac{40}{5} = 8 \]
Don haka, bambancin yawan jama'a na wannan bayanin shine 8.
Misali: Lissafin Bambancin Samfura
Yanzu, a ce mun ɗauki ƙaramin samfuri daga bayanan da ke sama: 2, 4, 6.
1. Mataki na 1: Lissafa Ma'aunin Samfurin
\[ \bar{x} = \frac{2 + 4 + 6}{3} = 4 \]
2. Mataki na 2: Lissafa Bambancin Kowace Ƙima daga Ma'ana sannan a Matsakaita Shi
\[
\begin{daidai}
(2 – 4)^2 &= (-2)^2 = 4 \\
(4 – 4)^2 &= 0^2 = 0 \\
(6 – 4)^2 &= 2^2 = 4 \\
\end{daidai}
\]
3. Mataki na 3: Ƙara Duk Muƙallan Bambancin
\[ 4 + 0 + 4 = 8 \]
4. Mataki na 4: Raba Jimillar Muƙallan Bambanci da (n - 1)
\[ s^2 = \frac{8}{3-1} = \frac{8}{2} = 4 \]
Don haka, bambancin samfurin wannan bayanin shine 4.
Bambancin Yawan Jama'a da Samfura
Yana da mahimmanci a fahimci bambanci tsakanin bambancin yawan jama'a da bambancin samfurin. Bambancin yawan jama'a yana auna yaduwar bayanai a cikin dukkan al'umma, yayin da bambancin samfurin ke auna yaduwar da ke cikin wani ƙaramin yanki (samfurin) na al'umma. A lokuta da yawa, ana amfani da bambancin samfurin don kimanta bambancin yawan jama'a. Rarrabawa da \( (n-1) \) a cikin lissafin bambancin samfurin yana rage son zuciya a cikin kimanta bambancin yawan jama'a.
Aikace-aikacen Bambanci
Ana amfani da Variance a fannoni daban-daban, kamar:
1. Binciken Hadarin Kuɗi: A fannin kuɗi, ana amfani da bambancin kuɗi don auna haɗari da kuma sarrafa fayil ɗin saka hannun jari. Babban bambancin yana nufin saka hannun jari mai haɗari.
2. Kimiyyar Zamantakewa: A cikin binciken ilimin halayyar ɗan adam ko ilimin zamantakewa, ana amfani da bambance-bambancen don auna bambance-bambance tsakanin ƙungiyoyin jama'a.
3. Kula da Inganci: A fannin masana'antu, ana amfani da bambance-bambancen abubuwa don sa ido da kuma sarrafa ingancin samfura.
4. Ƙididdigar Gwaji: Ana amfani da shi don nazarin sakamakon gwaji da kuma tantance mahimmancin bambance-bambance.
Bambancin da Daidaitaccen Bambancin
Sau da yawa ana amfani da bambance-bambancen tare da daidaitaccen karkacewa, wanda shine tushen murabba'in bambancin. Daidaitaccen karkacewa yana ba da ma'aunin yaduwa kai tsaye da sauƙin fassara fiye da bambancin. Daidaito tsakanin su biyun shine:
\[ \text{Bambancin Daidaitacce} (\sigma) = \sqrt{\text{Bambancin} (\sigma^2)} \]
Kammalawa
Lissafin bambancin abu muhimmin bangare ne na nazarin kididdiga, yana samar da ma'aunin yaduwar ko yaduwar bayanai a cikin saitin bayanai. Ta hanyar fahimtar mahimman ra'ayoyi da yadda ake lissafin bambancin, za mu iya yin nazari sosai kan bayanai, tantance haɗari, da kuma yanke shawara mai zurfi.
Ko da amfani da bambancin yawan jama'a don ƙarin nazarin kimiyya ko samfurin bambancin don ƙiyasta daga wani ɓangare na bayanai, cikakken fahimtar bambancin yana taimaka mana mu fahimci bambancin bayanai da kuma amfani da shi ga yanayi daban-daban na zahiri. Da fatan, wannan labarin zai ba da jagora mai amfani kuma mai amfani don fahimtar da ƙididdige bambancin.
