Contoh Soal 5 Statistika

$\begin{array}{l}\\ 21.& \text{Simpangan kuartil dari data}\\ & 71,70,68,40,45,48,52,53,53,67,62\\ & \text{adalah}....\\ & \begin{array}{l}\\ \text{a}.& 8\\ \text{b}.& 10\\ \text{c}.& 15\\ \text{d}.& 18\\ \text{e}.& 20\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{b}}\\ & \begin{array}{rl}\text{Data}& \text{mula-mula}(\text{degan total datum ganjil})\\ :& 71,70,68,40,45,48,52,53,53,67,62\\ \text{Sete}& \text{lah data diurutkan menjadi}\\ :& 40,45,48,52,53,53,62,67,68,70,71\\ \text{Dike}& \text{tahui}n=11\mathbf{\text{ganjil}}\\ Q_1& =x_{\frac{1}{4}(n+1)}=x_{\frac{1}{4}.12}=x_3=48\\ Q_2& =x_{\frac{2}{4}(n+1)}=x_{\frac{2}{4}.12}=x_6=53\\ Q_3& =x_{\frac{3}{4}(n+1)}=x_{\frac{3}{4}.12}=x_9=68\\ \text{Sela}& \text{njutnya data dapat dituliskan}\\ & 40,45,\underset{\begin{array}{c}\Downarrow \\ Q_1\end{array}}{\underbrace{48}},52,53,53,62,67,\underset{\begin{array}{c}\Downarrow \\ Q_3\end{array}}{\underbrace{68}},70,71\\ \text{Simp}& \text{angan kuartil data tunggal adalah}:\\ & ={\displaystyle \frac{1}{2}(Q_3-Q_1)}\\ & ={\displaystyle \frac{1}{2}(68-48)}\\ & ={\displaystyle \frac{1}{2}.20=10}\end{array}\end{array}$

$\begin{array}{l}\\ 22.& \text{Data penjualan suatu barang setiap bulan}\\ & \text{di sebuah toko pada tahun 2019 adalah}:\\ & 20,3,9,11,4,12,1,9,9,12,8,10.\\ & \text{Median, kuartil bawah, dan kuartil atasnya}\\ & \text{berturut-turut adalah}....\\ & \begin{array}{l}\\ \text{a}.& 6{\displaystyle \frac{1}{2},3\frac{1}{2},\text{dan}9\frac{1}{2}}\\ \text{b}.& 9,6,\text{dan}11{\displaystyle \frac{1}{2}}\\ \text{c}.& 6{\displaystyle \frac{1}{2},9,\text{dan}12}\\ \text{d}.& 9,4,\text{dan}12\\ \text{e}.& 9,3{\displaystyle \frac{1}{2},\text{dan}12}\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{b}}\\ & \begin{array}{rl}\text{Data}& \text{mula-mula}\\ :& 20,3,9,11,4,12,1,9,9,12,8,10\\ \text{Sete}& \text{lah data diurutkan}\\ :& 1,3,4,8,9,9,9,10,11,12,12,20\\ \text{Dike}& \text{tahui}n=12\mathbf{\text{genap}}\\ Q_1& =x_{\frac{1}{4}n+\frac{1}{2}}=x_{\frac{1}{4}.12+\frac{1}{2}}=x_{3,5}=6\\ Q_2& =x_{\frac{2}{4}n+\frac{1}{2}}=x_{\frac{2}{4}.12+\frac{1}{2}}=x_{6,5}=9=M_e\\ Q_3& =x_{\frac{3}{4}n+\frac{1}{2}}=x_{\frac{3}{4}.12+\frac{1}{2}}=x_{9,5}=11{\displaystyle \frac{1}{2}}\\ \text{Sela}& \text{njutnya data dapat dituliskan}\\ & 1,3,\underset{\begin{array}{c}\Downarrow \\ Q_1\end{array}}{\underbrace{4,8}},9,\underset{\begin{array}{c}\Downarrow \\ Q_2=M_e\end{array}}{\underbrace{9,9}},10,\underset{\begin{array}{c}\Downarrow \\ Q_3\end{array}}{\underbrace{11,12}},12,20\end{array}\end{array}$

$\begin{array}{l}\\ 23.& \text{Ragam(varians) dari data}\\ & 6,8,6,7,8,7,9,7,7,6,7,8,6,5,8,7\\ & \text{adalah}....\\ & \begin{array}{l}\\ \text{a}.& 1\\ \text{b}.& 1{\displaystyle \frac{3}{8}}\\ \text{c}.& 1{\displaystyle \frac{1}{8}}\\ \text{d}.& {\displaystyle \frac{7}{8}}\\ \text{e}.& {\displaystyle \frac{5}{8}}\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{a}}\\ & \begin{array}{rl}\text{Data}& \text{mula-mula}\\ :& 6,8,6,7,8,7,9,7,7,6,7,8,6,5,8,7\\ \text{Sete}& \text{lah data diurutkan(untuk memudahkan)}\\ :& 5,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9\\ \text{Dike}& \text{tahui}n=16.\text{Selanjutnya kita cari}\\ \bar{x}& ={\displaystyle \frac{5.1+6.4+7.6+8.4+9.1}{16}}\\ & =\frac{112}{16}=7\\ \text{Dan}& \text{rumus untuk menghitung ragam adalah}:\\ S^2& ={\displaystyle \frac{{\displaystyle \sum _{i=1}^{16}{(x_i-\bar{x})}^2}}{n}}\\ & ={\displaystyle \frac{(5-7{)}^2+4(6-7{)}^2+6(7-7{)}^2+4(8-7{)}^2+(9-7{)}^2}{16}}\\ & ={\displaystyle \frac{4+4.1+6.0+4.1+4}{16}}\\ & ={\displaystyle \frac{16}{16}}\\ & =1\end{array}\end{array}$

$\begin{array}{l}\\ 24.& \text{Diketahui}{x}_1=2,x_2=3,5,x_3=5,x_4=7,\\ & \text{dan}{x}_5=7,5.\text{Deviasi rata-rata data di atas}\\ & \text{adalah}....\\ & \begin{array}{l}\\ \text{a}.& 0\\ \text{b}.& 1\\ \text{c}.& 1,8\\ \text{d}.& 2,6\\ \text{e}.& 5\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{c}}\\ & \begin{array}{rl}\text{Data}& \text{mula-mula}\\ :& 2,3{\displaystyle \frac{1}{2},5,7,7{\displaystyle \frac{1}{2}}}\\ \text{Dike}& \text{tahui}n=5.\text{Selanjutnya kita cari}\\ \bar{x}& ={\displaystyle \frac{2+3,5+5+7+7,5}{5}=\frac{25}{5}=5}\\ \text{Dan}& \text{rumus simpangan rata-rata adalah}:\\ SR& ={\displaystyle \frac{{\displaystyle \sum _{i=1}^5|x_i-\bar{x}|}}{n}}\\ & ={\displaystyle \frac{|2-5|+|3,5-5|+|5-5|+|7-5|+|7,5-5|}{5}}\\ & ={\displaystyle \frac{|-3|+|-1,5|+\left|0\right|+\left|2\right|+|2,5|}{5}}\\ & ={\displaystyle \frac{3+1,5+0+2+2,5}{5}=\frac{9}{5}}\\ & =1,8\end{array}\end{array}$

$\begin{array}{l}\\ 25.& \text{Jumlah rataan dan median dari}\\ & (x-6),(x+5),(x+4),(x-7),\\ & (x+9),\text{dan}(x-2)\text{adalah}....\\ & \begin{array}{l}\\ \text{a}.& 2x-1,5\\ \text{b}.& 2x-0,5\\ \text{c}.& 2x+1,5\\ \text{d}.& 2x+2,5\\ \text{e}.& 2x+3,5\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{c}}\\ & \begin{array}{rl}\text{Data}& \text{mula-mula}\\ :& (x-6),(x+5),(x+4),(x-7),(x+9),(x-2)\\ \text{Dike}& \text{tahui}n=6.\text{Selanjutnya kita urutkan datanya}\\ & (x-7),(x-6),(x-2),(x+4),(x+5),(x+9)\\ \text{Rata}& \text{annya}:\\ \bar{x}& ={\displaystyle \frac{6x+3}{6}=x+0,5}\\ \text{Medi}& \text{annya}:\\ M_e& =x_{3,4}={\displaystyle \frac{x_3+x_4}{2}=\frac{(x-2)+(x+4)}{2}}\\ & ={\displaystyle \frac{2x+2}{2}=x+1}\\ \text{Rata}& \text{an+median}=x+0,5+x+1=2x+1,5\end{array}\end{array}$