Ukuran Letak Data (Materi Kelas XII Matematika Wajib) Bagian 3

 D. Persentil

$\begin{array}{rl}& \text{Dilambangkan dengan}{P}_i\\ & \text{dibaca: persentil ke}-i\\ & \text{Rumus data tunggal}:\\ & \mathbf{\text{Dengan rumus pendekatan interpolasi linear}}\\ & P_i=\text{datum ke}-{\displaystyle \frac{i}{100}(n+1)}\\ & \text{jika}{\displaystyle \frac{i}{100}(n+1)\text{tidak bulat, gunakan rumus}}\\ & \text{interpolasi linear, yaitu}:\\ & P_i=x_k+d(x_{k+1}-x_k)\\ & \text{dengan}d\text{adalah nilai desimalnya}\\ & \mathbf{\text{Dengan tanpa rumus interpolasi linear}}\\ & \begin{array}{|cc|}\hline n\text{ganjil}& n\text{genap}\\ P_1=x_{{.}_{\frac{1}{100}(n+1)}}& P_1={\displaystyle \frac{1}{2}(x_{{.}_{\frac{1}{100}n}}+x_{{.}_{\frac{1}{100}n+1}})}\\ P_2=x_{{.}_{\frac{2}{100}(n+1)}}& P_2={\displaystyle \frac{1}{2}(x_{{.}_{\frac{2}{100}n}}+x_{{.}_{\frac{2}{100}n+1}})}\\ ⋮& ⋮\\ P_{99}=x_{{.}_{\frac{99}{100}(n+1)}}& P_{99}={\displaystyle \frac{1}{2}(x_{{.}_{\frac{99}{100}n}}+x_{{.}_{\frac{99}{100}n+1}})}\\ \hline\end{array}\\ & \text{Catatan: sesuaikan dengan kondisi soal}\\ & \text{Rumus data berkelompok/berfrekuensi}:\\ & P_i=L_i+\left({\displaystyle \frac{{\displaystyle \frac{i}{100}n-f_k}}{f}}\right)\times c\\ & \text{Penjelasan}\\ & \begin{array}{rl}{D}_i& =\text{persentil ke}-i\\ i& =1,2,3\\ L_i& =\text{tepi bawah kelas persentil ke}-i\\ f_k& =\text{frekuensi kumulatif sebelum}\\ & \text{sebelum kelas persentil ke}-i\\ f& =\text{frekuensi kelas persentil ke}-i\\ c& =\text{panjang kelas interval}\\ n& =\text{banyak data/kelas interval}\end{array}\end{array}$.

Catatan :untuk bahasan interpolasi linear ada di sini

$\text{CONTOH SOAL}$.

$\begin{array}{ll}1.& \text{Tentukan persentil ke-29 dan ke-75 dari data berikut}\\ & 4,7,5,6,6,7,8,4,9,5,2,3,6,4,8\\ \\ & \text{Jawab}:\\ & \text{Banyak datum}=15\\ & \mathbf{\text{dengan rumus pendekatan interpolasi linear}}\\ & \text{Data mula-mula}\\ & :4,7,5,6,6,7,8,4,9,5,2,3,6,4,8\\ & \text{Data setelah diurutkan}\\ & :2,3,4,4,4,5,5,6,6,6,7,7,8,8,9\\ & \begin{array}{rl}{P}_{29}& ={\displaystyle \frac{29}{100}(15+1)}\\ & ={\displaystyle \frac{464}{100}=4,64}\\ & =x_4+0,64(x_5-x_4)\\ & =4+0,64(5-5)=4+0=4\\ P_{75}& ={\displaystyle \frac{75}{100}(15+1)}\\ & ={\displaystyle \frac{1200}{100}=12}\\ & =x_{12}\\ & =7\end{array}\end{array}$.

$\begin{array}{l}\\ 2.& \text{Persentil ke-32}\left(P_{32}\right)\text{dari data berikut adalah}....\\ & \begin{array}{|cc|}\hline \text{Nilai}& f\\ 41-45& 7\\ 46-50& 12\\ 51-55& 9\\ 56-60& 8\\ 61-65& 4\\ \hline\end{array}\\ & \begin{array}{l}\\ \text{a}.& 46\\ \text{b}.& 47\\ \text{c}.& 48\\ \text{d}.& 51\\ \text{e}.& 52\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{a}}\\ & \begin{array}{rl}\text{Diketahui}& \text{persentil ke}-32=P_{32},\\ & \text{dengan}n=\sum f=40\\ P_i& =L_i+c\left({\displaystyle \frac{{\displaystyle \frac{i\times n}{100}-f_k}}{f}}\right)\\ P_{32}& =\text{datum ke}-\left({\displaystyle \frac{32n}{100}}\right)\\ & =x_{\frac{32\times 40}{100}}=x_{12,8}\\ \text{Dan}{x}_{12}& \text{terletak di kelas interval}:46-50\\ P_{32}& =545,5+5\left({\displaystyle \frac{12,8-7}{12}}\right)\\ & =45,5+0,48333...\\ & =45.9833...\approx 46\end{array}\end{array}$