Tambayoyi Misali Game da Bambancin da Bambancin Bayanai na Rukuni
Pendahuluan
A cikin kididdiga, bambancin da karkacewar daidaito su ne ma'auni guda biyu na kididdiga waɗanda suke da mahimmanci don fahimtar watsawa, ko yaɗuwar bayanai daga matsakaici. Bambancin yana auna nisan da bayanai suka bazu daga matsakaici, yayin da karkacewar daidaito ita ce tushen murabba'in bambancin, yana samar da ma'auni wanda yake cikin raka'a ɗaya da bayanan asali.
Tabbas
– Bambanci (σ² ko S²): Shine matsakaicin murabba'in bambance-bambancen da ke tsakanin kowace ƙimar bayanai da matsakaicin bayanan.
– Daidaitaccen Bambanci (σ ko S): Shine tushen murabba'in bambancin.
Tsarin Bambanci da Bambancin Daidaitaccen Bayanin Rukuni
Don bayanan rukuni, muna amfani da mitar bayanai a cikin kowane aji. Ga dabarar:
Bambanci
\[ S^2 = \frac{ \sum f_i \left( x_i – \bar{x} \right)^2 }{ N-1 } \]
Daidaitaccen Canji
\[ S = \sqrt{S^2} \]
Ina:
– \( f_i \) = yawan kowane aji.
– \( x_i \) = tsakiyar kowane aji.
– \( \bar{x} \) = matsakaicin bayanan rukuni.
– \( N \) = jimlar adadin bayanai.
Tambayoyi da Tattaunawa Samfura
A ɗauka cewa muna da bayanai game da nauyin rukunin mutane da aka haɗa zuwa azuzuwa.
| Tazarar Nauyi (kg) | Mita (f) |
|—————————|————–|
| 50 – 54 | 2 |
| 55 – 59 | 5 |
| 60 – 64 | 8 |
| 65 – 69 | 7 |
| 70 – 74 | 3 |
Mataki na farko shine a tantance tsakiyar kowane aji (\( x_i \) ) sannan a ƙididdige matsakaicin (\( \bar{x} \)).
1. Lissafin Tsakiyar Maki ( \( x_i \) )
\[ \text{Midpoint} = \frac{\text{Ƙaramin Iyaka} + \text{Babban Iyaka}}{2} \]
| Tazarar Nauyi (kg) | Mita (f) | Tsakiyar Ma'auni ( \( x_i \) ) |
|——————|—————–|—————————|
| 50 – 54 | 2 | 52 |
| 55 – 59 | 5 | 57 |
| 60 – 64 | 8 | 62 |
| 65 – 69 | 7 | 67 |
| 70 – 74 | 3 | 72 |
2. Lissafin Matsakaicin ( \( \bar{x} \) )
\[ \bar{x} = \frac{ \sum f_i x_i }{ N } \]
Jimlar adadin bayanai \( N \):
\[ N = 2 + 5 + 8 + 7 + 3 = 25 \]
\[ \sum f_i x_i = (2 \sau 52) + (5 \sau 57) + (8 \sau 62) + (7 \sau 67) + (3 \sau 72) \]
\[ = 104 + 285 + 496 + 469 + 216 = 1570 \]
Don haka, matsakaicin (\( \bar{x} \)):
\[ \bar{x} = \frac{ 1570 }{ 25 } = 62.8 \]
3. Lissafin Bambancin ( \( S^2 \) )
Muna buƙatar yin lissafi \( \sum f_i ( x_i – \bar{x} )^2 \):
\[
\begin{daidai}
(x_i – \bar{x})^2: & (52 – 62.8)^2 = 118.84 \\
& (57 – 62.8)^2 = 33.64 \\
& (62 – 62.8)^2 = 0.64 \\
& (67 – 62.8)^2 = 17.64 \\
& (72 – 62.8)^2 = 84.64
\end{daidai}
\]
Ninkuwa ta mita:
\[
\begin{daidai}
f_i (x_i – \bar{x})^2: & sau 2 118.84 = 237.68 \\
& sau 5 33.64 = 168.2 \\
& sau 8 0.64 = 5.12 \\
& sau 7 17.64 = 123.48 \\
& sau 3 84.64 = 253.92
\end{daidai}
\]
\[
\sum f_i (x_i – \bar{x})^2 = 237.68 + 168.2 + 5.12 + 123.48 + 253.92 = 788.4
\]
Yanzu za mu iya ƙididdige bambancin (\( S^2 \)):
\[ S^2 = \frac{ 788.4 }{ 25 – 1 } = \frac{ 788.4 }{ 24 } \kimanin 32.85 \]
4. Lissafin Daidaitaccen Ragewar Daidaito ( \( S \) )
Bambancin da aka saba ( \( S \)):
\[ S = \sqrt{ S^2 } \]
\[ S = \sqrt{ 32.85 } \kimanin 5.73 \]
Kammalawa
Daga misalin bayanai da ke sama, muna da:
– Lissafin matsakaicin nauyin jiki: 62.8 kg
– Bambancin lissafin: 32.85 kg²
– Lissafin karkacewar da aka saba: 5.73 kg
Fassarar karkacewar da aka saba yi ita ce matsakaicin karkacewar bayanai daga matsakaicin shine kimanin kilogiram 5.73. Wannan yana nuna yaduwar bayanai dangane da matsakaicin, wanda zai iya taimakawa wajen fahimtar yadda bayananmu suke canzawa.
Fahimtar bambanci da karkacewar tsari yana da matuƙar muhimmanci, musamman ga waɗanda ke aiki a kididdiga, bincike, da gwaji, yayin da suke ƙoƙarin fahimtar bayanai a cikin nau'ikan ƙungiyoyi ko rarrabawa. Sanin yadda ake ƙididdigewa da fassara waɗannan ma'auni guda biyu na iya taimakawa wajen yanke shawara mafi kyau bisa ga bayanan da ke hannunsu.
