Bambancin da Daidaitaccen Bambancin Bayanai Guda ɗaya
Kididdiga ita ce kimiyyar da ke nazarin tattara bayanai, bincike, fassara, gabatarwa, da kuma tsara su. Wani muhimmin bangare na kididdiga shine yadda muke aunawa da fahimtar bambancin bayanai. Ma'auni guda biyu masu mahimmanci na bambancin bayanai sune bambancin bayanai da bambancin da aka saba amfani da su. Wannan labarin zai tattauna bambance-bambance da bambancin da aka saba amfani da su a cikin zurfi, yana mai da hankali kan saitin bayanai guda daya, yana bayyana ma'anoni, dabaru, matakan lissafi, da kuma samar da misalai masu amfani don fayyace ra'ayoyin.
Ma'anar Bambancin da Bambancin Daidaitacce
Bambanci
Bambancin bayanai wani ma'auni ne na nisan da bayanai ke yaɗuwa daga matsakaicin bayanai. Yana ba da taƙaitaccen bayani game da bambancin bayanai ko bambancin da ke cikin saitin bayanai. A fannin lissafi, bambancin shine matsakaicin murabba'in da ke cikin karkacewar kowane abu daga matsakaicinsa.
Daidaitaccen Canji
Bambancin daidaito, wanda kuma aka sani da karkacewar daidaito, shine tushen murabba'in bambancin. Yana bayar da bayanai game da rarraba bayanai a cikin raka'o'i iri ɗaya da bayanan asali, wanda hakan ke sauƙaƙa fassara a cikin mahallin rayuwa ta ainihi.
Darussa da Lissafi
Bambanci
Don ƙididdige bambancin (σ^2) na saitin bayanai guda ɗaya na girman n, za mu iya amfani da dabarar mai zuwa:
\[ \sigma^2 = \frac{\sum_{i=1}^{n} (x_i – \bar{x})^2}{n} \]
Ina:
– \( x_i \) shine ƙimar bayanai ta ith.
– \( \bar{x} \) shine matsakaicin ko matsakaicin bayanan.
– n shine adadin bayanai a cikin saitin.
– (x_i – \bar{x})^2 shine murabba'in bambanci tsakanin kowace bayanai da matsakaicinsa.
Daidaitaccen Canji
Bambancin da aka saba amfani da shi (σ) shine tushen murabba'in bambancin, don haka dabarar ita ce:
\[ \sigma = \sqrt{\sigma^2} \]
Matakan Lissafi
Mataki na 1: Lissafa Matsakaicin Bayanan (Matsakaicin)
Mataki na farko shine a ƙididdige matsakaicin (matsakaicin) saitin bayanai. Ana ƙididdige matsakaicin ta hanyar ƙara dukkan ƙimar bayanai da kuma raba su da adadin ƙimar bayanai.
\[ \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} \]
Mataki na 2: Lissafa Bambancin Kowace Saitin Bayanai da Ma'anar
Bayan samun matsakaicin, mataki na gaba shine a ƙididdige bambanci tsakanin kowace ƙimar bayanai da matsakaicin (ɓacewa).
\[ d_i = x_i – \bar{x} \]
Mataki na 3: Sake Daidaita Kowane Bambanci
Na gaba, a yi murabba'i a kowace karkacewa.
\[ d_i^2 = (x_i – \bar{x})^2 \]
Mataki na 4: Tattara Duk Bambancin da Aka Yi a Muƙallai
A tara dukkan sakamakon da aka yi murabba'i daga matakin da ya gabata.
\[ \sum d_i^2 = \sum_{i=1}^{n} (x_i – \bar{x})^2 \]
Mataki na 5: Lissafa Bambancin
Raba jimlar murabba'in karkacewar da adadin bayanai (n) don samun bambancin.
\[ \sigma^2 = \frac{\sum d_i^2}{n} \]
Mataki na 6: Lissafa Bambancin Daidaitacce
A ƙarshe, ƙididdige karkacewar da aka saba ta hanyar ɗaukar tushen murabba'in bambancin.
\[ \sigma = \sqrt{\sigma^2} \]
Misalin Lissafi
Domin fayyace manufar lissafin bambancin da karkacewar misali, bari mu dubi misali mai zuwa. A ce muna da saitin bayanai guda ɗaya: 4, 8, 6, 5, 3.
Mataki na 1: Lissafa Matsakaicin Bayanan
\[ \bar{x} = \frac{4 + 8 + 6 + 5 + 3}{5} = \frac{26}{5} = 5.2 \]
Mataki na 2: Lissafa Bambancin Kowace Saitin Bayanai da Ma'anar
Bambanci tsakanin kowace darajar bayanai da matsakaicin:
– (4 – 5.2) = -1.2
– (8 – 5.2) = 2.8
– (6 – 5.2) = 0.8
– (5 – 5.2) = -0.2
– (3 – 5.2) = -2.2
Mataki na 3: Sake Daidaita Kowane Bambanci
Murabba'in kowane karkacewa:
– (-1.2)^2 = 1.44
– 2.8^2 = 7.84
– 0.8^2 = 0.64
– (-0.2)^2 = 0.04
– (-2.2)^2 = 4.84
Mataki na 4: Tattara Duk Bambancin da Aka Yi a Muƙallai
\[ \sum d_i^2 = 1.44 + 7.84 + 0.64 + 0.04 + 4.84 = 14.8 \]
Mataki na 5: Lissafa Bambancin
\[ \sigma^2 = \frac{14.8}{5} = 2.96 \]
Mataki na 6: Lissafa Bambancin Daidaitacce
\[ \sigma = \sqrt{2.96} \kimanin 1.72 \]
Don haka, a cikin wannan misalin, bambancin bayanai shine 2.96, kuma daidaitaccen karkacewar shine kusan 1.72.
Fassara
Bambancin da karkacewar daidaito suna ba da muhimman bayanai game da yaɗuwar bayanan. A cikin misalin da ke sama, bambancin 2.96 yana nuna cewa matsakaicin karkacewar murabba'i na ƙimar bayanai daga matsakaicin shine 2.96. Bambancin daidaito na 1.72 yana nuna cewa, a matsakaici, ƙimar bayanai suna karkacewa da raka'a 1.72 daga matsakaicin.
Bambancin daidaito ya fi sauƙi a fassara shi saboda yana da raka'o'i iri ɗaya da bayanan asali. Misali, a cikin mahallin ƙididdigar samun kuɗi, idan bayanan suna da bambancin daidaito na $500, yana nufin cewa matsakaicin kuɗin shiga ya karkace da $500 daga matsakaicin kuɗin shiga.
Aikace-aikace a Rayuwa ta Yau da Kullum
Ana iya amfani da fahimtar bambancin da kuma karkacewar daidaito a fannoni daban-daban. A fannin kuɗi, ana iya amfani da karkacewar daidaito don auna haɗarin saka hannun jari. A fannin ilimi, ana iya amfani da shi don kimanta bambancin sakamakon gwaji tsakanin ɗalibai. A fannin masana'antu, yana iya taimakawa wajen sarrafa inganci ta hanyar auna bambancin samarwa.
Kammalawa
Bambanci da karkacewar daidaito kayan aiki ne guda biyu masu mahimmanci na kididdiga don fahimtar bambancin da ke cikin saitin bayanai. Ta hanyar ƙididdige bambancin, za mu iya tantance yadda aka yaɗa bayanan. karkacewar daidaito, wanda shine tushen murabba'in bambancin, yana ba da fassarar da ta fi fahimta kuma ana auna shi a cikin raka'a ɗaya da bayanan asali. Ta hanyar fahimtar da kuma iya ƙididdige waɗannan ma'auni guda biyu, za mu iya yin nazari sosai kan bayanai da kuma yanke shawara mai zurfi.
