Ukuran Letak Data (Materi Kelas XII Matematika Wajib) Bagian 2

 C. Desil

$\begin{aligned}&\textrm{Dilambangkan dengan}\: \: D_{i}\\ &\textrm{dibaca: desil ke}-i\\ &\color{red}\textrm{Rumus data tunggal}:\\ &\textbf{Dengan rumus pendekatan interpolasi linear}\\ &D_{i}=\textrm{datum ke}-\displaystyle \frac{i}{10}(n+1)\\ &\textrm{jika}\: \: \displaystyle \frac{i}{10}(n+1)\: \: \textrm{tidak bulat, gunakan rumus}\\ &\textrm{interpolasi linear, yaitu}:\\ &\qquad\quad D_{i}=x_{k}+d\left ( x_{k+1}-x_{k} \right )\\ &\textrm{dengan}\: \: d\: \: \textrm{adalah nilai desimalnya}\\ &\textbf{Dengan tanpa rumus interpolasi linear}\\ &\begin{array}{|c|c|}\hline n\: \: \textrm{ganjil}&n\: \: \textrm{genap}\\\hline D_{1}=x_{._{\frac{1}{10}(n+1)}}&D_{1}=\displaystyle \frac{1}{2}\left (x_{._{\frac{1}{10}n}}+x_{._{\frac{1}{10}n+1}} \right )\\ D_{2}=x_{._{\frac{2}{10}(n+1)}}&D_{2}=\displaystyle \frac{1}{2}\left (x_{._{\frac{2}{10}n}}+x_{._{\frac{2}{10}n+1}} \right )\\ \vdots &\vdots \\ D_{9}=x_{._{\frac{9}{10}(n+1)}}&D_{9}=\displaystyle \frac{1}{2}\left (x_{._{\frac{9}{10}n}}+x_{._{\frac{9}{10}n+1}} \right )\\\hline \end{array} \\ &\textrm{Catatan: sesuaikan dengan kondisi soal}\\ &\color{red}\textrm{Rumus data berkelompok/berfrekuensi}:\\ &D_{i}=L_{i}+\left ( \displaystyle \frac{\displaystyle \frac{i}{10}n-f_{k}}{f} \right )\times c\\ &\color{blue}\textrm{Penjelasan}\\ &\qquad \begin{aligned}D_{i}&=\textrm{desil ke}-i\\ i&=1,2,3\\ L_{i}&=\textrm{tepi bawah kelas desil ke}-i\\ f_{k}&=\textrm{frekuensi kumulatif sebelum}\\ &\quad\: \, \textrm{sebelum kelas desil ke}-i\\ f&=\textrm{frekuensi kelas desil ke}-i\\ c&=\textrm{panjang kelas interval}\\ n&=\textrm{banyak data/kelas interval} \end{aligned} \end{aligned}$.

Catatan :untuk bahasan interpolasi linear ada di sini

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

$\begin{array}{ll} 1.&\textrm{Tentukanlah}\: \: \: D_{1},D_{2},D_{3},D_{4},D_{5},D_{6}\\ &D_{7},D_{8},D_{9}\: \: \: \textrm{dari data berikut}\\ & 2,3,8,9,2,4,5,8,9\\\\ &\textrm{Jawab}:\\ &\begin{aligned}&\textrm{Total datum}=9\\ &\textrm{Data mula-mula}:\: 2,3,8,9,2,4,5,8,9\\ &\textrm{Setelah data diurutkan menjadi}\\ &\quad : 2,2,3,4,5,8,8,9,9\\ &D_{i}=\displaystyle \frac{i}{10}(n+1)\: .\: \textrm{Jika hasilnay tidak bulat}\\ &\textrm{maka dihitung dengan}\: \: D_{i}=x_{k}+d.(x_{k+1}-x_{k})\\ &\color{red}\textrm{Sehingga}\\ &\begin{aligned}D_{1}&=\displaystyle \frac{1}{10}(9+1)=\frac{10}{10}=1\\ &=x_{1}=2\\ D_{2}&=\displaystyle \frac{2}{10}(9+1)=\frac{20}{10}=2\\ &=x_{2}=2\\ D_{3}&=\displaystyle \frac{3}{10}(9+1)=\frac{30}{10}=3\\ &=x_{3}=3\\ D_{4}&=\displaystyle \frac{4}{10}(9+1)=\frac{40}{10}=4\\ &=x_{4}=4\\ D_{5}&=\displaystyle \frac{5}{10}(9+1)=\frac{50}{10}=5\\ &=x_{5}=5\\ D_{6}&=\displaystyle \frac{6}{10}(9+1)=\frac{60}{10}=6\\ &=x_{6}=8\\ D_{7}&=\displaystyle \frac{7}{10}(9+1)=\frac{70}{10}=7\\ &=x_{7}=8\\ D_{8}&=\displaystyle \frac{8}{10}(9+1)=\frac{80}{10}=8\\ &=x_{8}=9\\ D_{9}&=\displaystyle \frac{9}{10}(9+1)=\frac{90}{10}=9\\ &=x_{9}=9\\ \end{aligned} \end{aligned} \end{array}$.

$\begin{array}{ll} 2.&\textrm{Tentukan desil ke-4 dan ke-6 dari data berikut}\\ &4,7,5,6,6,7,8,4,9,5,2,3,6,4,8\\\\ &\textrm{Jawab}:\\ &\textrm{Banyak datum}=15\\ &\textbf{dengan rumus pendekatan interpolasi linear}\\ &\textrm{Data mula-mula}\\ &\quad :4,7,5,6,6,7,8,4,9,5,2,3,6,4,8\\ &\textrm{Data setelah diurutkan}\\ &\quad :2,3,4,4,4,5,5,6,6,6,7,7,8,8,9\\ &\begin{aligned}D_{4}&=\displaystyle \frac{4}{10}\left ( 15+1 \right )\\ &=\displaystyle \frac{64}{10}=6,4\\ &=x_{6}+0,4(x_{7}-x_{6})\\ &=5+0,4(5-5)=\color{red}5\\ D_{6}&=\displaystyle \frac{6}{10}\left ( 15+1 \right )\\ &=\displaystyle \frac{96}{10}=9,6\\ &=x_{9}+0,6(x_{10}-x_{9})\\ &=6+0,6(6-6)=\color{red}6\\ \end{aligned} \end{array}$.

$\begin{array}{ll}\\ 3.&\textrm{Desil ke-8}\: \: \left ( D_{8} \right )\: \: \textrm{dari data berikut adalah}\: ....\\ &\begin{array}{|c|c|}\hline \textrm{Nilai}&f\\\hline 41-45&7\\ 46-50&12\\ 51-55&9\\ 56-60&8\\ 61-65&4\\\hline \end{array}\\ &\begin{array}{llll}\\ \color{red}\textrm{a}.&58\\ \textrm{b}.&57,5\\ \textrm{c}.&57\\ \textrm{d}.&56,75\\ \textrm{e}.&56,25 \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{a}\\ &\color{blue}\begin{aligned}\textrm{Diketahui}&\: \textrm{desil ke}-8=\color{black}D_{8},\: \: \textrm{dengan}\: \: n=\sum f=40\\ D_{i}&=\color{black}L_{i}+c\left ( \displaystyle \frac{\displaystyle \frac{i\times n}{10}-f_{k}}{f} \right )\\ D_{8}&=\color{black} \textrm{datum ke}-\left ( \displaystyle \frac{8n}{10} \right )=x_{\frac{8\times 40}{10}}=\color{red}x_{32}\\ \textrm{Dan}\: \: \color{red}x_{32}\: \: &\textrm{terletak di kelas interval}\: :\: \color{red}56-60 \\ D_{8}&=\color{black}55,5+5\left ( \displaystyle \frac{32-28}{8} \right )\\ &=\color{black}55,5+2,5\\ &=\color{red}58 \end{aligned} \end{array}$