Ukuran Penyebaran Data Berkelompok (Materi Kelas XII Matematika Wajib) (Bagian 2)

 B. 2 Data Berkelompok

$\begin{array}{|cll|}\hline \text{No}& \text{Data Dispersi}& \text{Keterangan}\\ 1.& \text{Jangkauan}& \begin{array}{rl}\text{a}.& \text{selisih titik tengah}\\ & \text{kelas tertinggi dengan}\\ & \text{titik tengah kelas}\\ & \text{terendah}\\ \text{b}.& \text{selisih tepi atas kelas}\\ & \text{kelas tertinggi dengan}\\ & \text{tepi bawah kelas}\\ & \text{terendah}\end{array}\\ 2.& H& Q_3-Q_1\\ 3.& Q_d& {\displaystyle \frac{1}{2}(Q_3-Q_1)}\\ 4.& SR& {\displaystyle \frac{{\displaystyle \sum _{i=1}^k{f}_i|x_i-\bar{x}|}}{{\displaystyle \sum _{i=1}^k{f}_i}}}\\ 5.& S^2& {\displaystyle \frac{{\displaystyle \sum _{i=1}^k{f}_i{(x_i-\bar{x})}^2}}{{\displaystyle \sum _{i=1}^k{f}_i}}}\\ 6.& S& \sqrt{{\displaystyle \frac{{\displaystyle \sum _{i=1}^k{f}_i{(x_i-\bar{x})}^2}}{{\displaystyle \sum _{i=1}^k{f}_i}}}}\\ \hline\end{array}$.

$\text{CONTOH SOAL}$.

$\begin{array}{ll}1.& \text{Tentukanlah nilai simpangan rata-ratanya}\\ & \begin{array}{|cccccc|}\hline \text{Nilai}& \text{47-49}& \text{50-52}& \text{53-55}& \text{56-58}& \text{59-61}\\ \text{Frek}& 2& 4& 6& 5& 3\\ \hline\end{array}\\ \\ & \mathbf{\text{Jawab}}:\\ & \mathbf{\text{Alternatif 1}}\\ & \begin{array}{ll}& \text{Perhatikan tabel berikut}\\ & \begin{array}{|cccccc|}\hline \text{Nilai}& x_i& f_i& f_i.x_i& |x_i-\bar{x}|& f_i.|x_i-\bar{x}|\\ 47-49& 48& 2& 96& 6,45& 12,49\\ 50-52& 51& 4& 204& 3,45& 13,8\\ 53-55& \text{54}& 6& \text{324}& \text{0,45}& \text{2,7}\\ 56-58& 57& 5& 285& 2,55& 12,75\\ 59-61& 60& 3& 180& 5,55& 16,65\\ \text{Jumlah}& & 20& 1089& & 58,8\\ \hline\end{array}\\ & \text{ingat}{x}_i=\text{nilai tengah interval kelas}\\ & \begin{array}{rl}\bar{x}& ={\displaystyle \frac{{\displaystyle \sum _{i=1}^k{f}_i.x_i}}{{\displaystyle \sum _{i=1}^k{f}_i}}}\\ & =54+{\displaystyle \frac{1089}{20}=54+0,45=54,45}\end{array}\\ & \begin{array}{rl}SR& ={\displaystyle \frac{{\displaystyle \sum _{i=1}^k{f}_i.|x_i-\bar{x}|}}{{\displaystyle \sum _{i=1}^k{f}_i}}}\\ & ={\displaystyle \frac{58,8}{20}}\\ & =2,94\end{array}\\ & \text{Jadi, simpangan rata-ratanya adalah}SR=2,94\end{array}\\ \\ & \mathbf{\text{Alternatif 2}}\\ & \text{Perhatikan tabel berikut}\\ & \begin{array}{|ccccccc|}\hline \text{Nilai}& x_i& f_i& d_i& f_i.d_i& |x_i-\bar{x}|& f_i.|x_i-\bar{x}|\\ \text{47-49}& 48& 2& -6& -12& 6,45& 12,49\\ \text{50-52}& 51& 4& -3& -12& 3,45& 13,8\\ \text{53-55}& \text{54}& 6& \text{0}& \text{0}& \text{0,45}& \text{2,7}\\ \text{56-58}& 57& 5& 3& 15& 2,55& 12,75\\ \text{59-61}& 60& 3& 6& 18& 5,55& 16,65\\ \text{Jumlah}& & 20& & 9& & 58,8\\ \hline\end{array}\\ & \text{ingat}{x}_i=\text{nilai tengah interval kelas}\\ & \begin{array}{rl}\bar{x}& =x_s+{\displaystyle \frac{{\displaystyle \sum _{i=1}^k{f}_i.d_i}}{{\displaystyle \sum _{i=1}^k{f}_i}}}\\ & =54+{\displaystyle \frac{9}{20}=54+0,45=54,45}\end{array}\\ & \begin{array}{rl}SR& ={\displaystyle \frac{{\displaystyle \sum _{i=1}^k{f}_i.|x_i-\bar{x}|}}{{\displaystyle \sum _{i=1}^k{f}_i}}}\\ & ={\displaystyle \frac{58,8}{20}}\\ & =2,94\end{array}\\ & \text{Jadi, simpangan rata-ratanya adalah}SR=2,94\end{array}$.

$\begin{array}{ll}2.& \text{Tentukanlah nilai varian/ragamnya}\\ & \text{dari data soal no.1 di atas}\\ \\ & \mathbf{\text{Jawab}}:\\ & \text{Perhatikan tabel berikut}\\ & \begin{array}{|cccccc|}\hline \text{Nilai}& x_i& f_i& |x_i-\bar{x}|& (x_i-\bar{x}{)}^2& f_i.(x_i-\bar{x}{)}^2\\ 47-49& 48& 2& 6,45& 41,6025& 83,205\\ 50-52& 51& 4& 3,45& 11,9025& 47,61\\ 53-55& 54& 6& 324& 0,2025& 1,215\\ 56-58& 57& 5& 285& 6,5025& 32,5125\\ 59-61& 60& 3& 180& 30,8025& 92,4075\\ \text{Jumlah}& & 20& & & 256,95\\ \hline\end{array}\\ & \text{ingat}{x}_i=\text{nilai tengah interval kelas}\\ & \text{dan}\bar{x}=54,45(\text{lihat soal no.1})\\ & \text{maka}\\ & \begin{array}{rl}{S}^2& ={\displaystyle \frac{{\displaystyle \sum _{i=1}^k{f}_i.(x_i-\bar{x}{)}^2}}{{\displaystyle \sum _{i=1}^k{f}_i}}}\\ & ={\displaystyle \frac{256,95}{20}}\\ & =12,8475\end{array}\\ & \text{Jadi, varian/ragamnya adalah}\\ & S^2=12,8475\end{array}$.

$\begin{array}{ll}3.& \text{Tentukanlah nilai simpangan baku dari}\\ & \text{dari data soal no.1 di atas}\\ \\ & \mathbf{\text{Jawab}}:\\ & S=\sqrt{S^2}=\sqrt{12,8475}\approx 3,58\end{array}$.