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

 B. 2 Data Berkelompok

$\begin{array}{|c|l|l|}\hline \textrm{No}&\textrm{Data Dispersi}&\textrm{Keterangan}\\\hline 1.&\textrm{Jangkauan}&\begin{aligned}\textrm{a}.\: \: &\textrm{selisih titik tengah}\\ &\textrm{kelas tertinggi dengan}\\ &\textrm{titik tengah kelas}\\ &\textrm{terendah}\\ \textrm{b}.\: \: &\textrm{selisih tepi atas kelas}\\ &\textrm{kelas tertinggi dengan}\\ &\textrm{tepi bawah kelas}\\ &\textrm{terendah} \end{aligned}\\\hline 2.&H&Q_{3}-Q_{1}\\\hline 3.&Q_{d}&\displaystyle \frac{1}{2}\left ( Q_{3}-Q_{1} \right )\\\hline 4.&SR&\displaystyle \frac{\displaystyle \sum_{i=1}^{k}f_{i}\left | x_{i}-\overline{x} \right |}{\displaystyle \sum_{i=1}^{k}f_{i}} \\\hline 5.&S^{2}&\displaystyle \frac{\displaystyle \sum_{i=1}^{k}f_{i} \left (x_{i}-\overline{x} \right )^{2} }{\displaystyle \sum_{i=1}^{k}f_{i}}\\\hline 6.&S&\sqrt{\displaystyle \frac{\displaystyle \sum_{i=1}^{k}f_{i} \left (x_{i}-\overline{x} \right )^{2} }{\displaystyle \sum_{i=1}^{k}f_{i}}}\\\hline \end{array}$.

$\LARGE\colorbox{yellow}{CONTOH SOAL}$.

$\begin{array}{ll} 1.&\textrm{Tentukanlah nilai simpangan rata-ratanya}\\ &\begin{array}{|c|c|c|c|c|c|}\hline \textrm{Nilai}&\colorbox{white}{47-49}&\colorbox{white}{50-52}&\colorbox{white}{53-55}&\colorbox{white}{56-58}&\colorbox{white}{59-61}\\\hline \textrm{Frek}&2&4&6&5&3\\\hline \end{array}\\\\ &\textbf{Jawab}:\\ &\textbf{Alternatif 1}\\ &\begin{array}{ll} &\textrm{Perhatikan tabel berikut}\\ &\begin{array}{|c|c|c|c|c|c|}\hline \textrm{Nilai}&x_{i}&f_{i}&f_{i}.x_{i}&\left | x_{i}-\overline{x} \right |&f_{i}.\left | x_{i}-\overline{x} \right |\\\hline 47-49&48&2&96&6,45&12,49\\\hline 50-52&51&4&204&3,45&13,8\\\hline 53-55&\colorbox{yellow}{54}&6&\colorbox{yellow}{324}&\colorbox{yellow}{0,45}&\colorbox{yellow}{2,7}\\\hline 56-58&57&5&285&2,55&12,75\\\hline 59-61&60&3&180&5,55&16,65\\\hline \textrm{Jumlah}&&20&1089&&58,8\\\hline \end{array}\\ &\textrm{ingat}\: \: x_{i}=\textrm{nilai tengah interval kelas}\\ &\begin{aligned}\overline{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=\color{red}54,45 \end{aligned}\\ &\begin{aligned}SR&=\displaystyle \frac{\displaystyle \sum_{i=1}^{k}f_{i}.\left | x_{i}-\overline{x} \right |}{\displaystyle \sum_{i=1}^{k}f_{i}}\\ &=\displaystyle \frac{58,8}{20}\\ &=\color{red}2,94 \end{aligned}\\ &\textrm{Jadi, simpangan rata-ratanya adalah}\: SR=2,94 \end{array} \\\\ &\textbf{Alternatif 2}\\ &\textrm{Perhatikan tabel berikut}\\ &\begin{array}{|c|c|c|c|c|c|c|}\hline \textrm{Nilai}&x_{i}&f_{i}&d_{i}&f_{i}.d_{i}&\left | x_{i}-\overline{x} \right |&f_{i}.\left | x_{i}-\overline{x} \right |\\\hline \colorbox{white}{47-49}&48&2&-6&-12&6,45&12,49\\\hline \colorbox{white}{50-52}&51&4&-3&-12&3,45&13,8\\\hline \colorbox{yellow}{53-55}&\colorbox{yellow}{54}&6&\colorbox{yellow}0&\colorbox{yellow}0&\colorbox{yellow}{0,45}&\colorbox{yellow}{2,7}\\\hline \colorbox{white}{56-58}&57&5&3&15&2,55&12,75\\\hline \colorbox{white}{59-61}&60&3&6&18&5,55&16,65\\\hline \textrm{Jumlah}&&20&&9&&58,8\\\hline \end{array}\\ &\textrm{ingat}\: \: x_{i}=\textrm{nilai tengah interval kelas}\\ &\begin{aligned}\overline{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=\color{red}54,45 \end{aligned}\\ &\begin{aligned}SR&=\displaystyle \frac{\displaystyle \sum_{i=1}^{k}f_{i}.\left | x_{i}-\overline{x} \right |}{\displaystyle \sum_{i=1}^{k}f_{i}}\\ &=\displaystyle \frac{58,8}{20}\\ &=\color{red}2,94 \end{aligned}\\ &\textrm{Jadi, simpangan rata-ratanya adalah}\: SR=2,94 \end{array}$.

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

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