Contoh Soal 5 Statistika

$\begin{array}{ll}\\ 21.&\textrm{Simpangan kuartil dari data}\\ &71,70,68,40,45,48,52,53,53,67,62\\ &\textrm{adalah}\: ....\\ &\begin{array}{llll}\\ \textrm{a}.&8\\ \color{red}\textrm{b}.&10\\ \textrm{c}.&15\\ \textrm{d}.&18\\ \textrm{e}.&20 \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{b}\\ &\color{blue}\begin{aligned}\textrm{Data}&\: \textrm{mula-mula}\: (\color{red}\textrm{degan total datum ganjil})\\ :\: &71,70,68,40,45,48,52,53,53,67,62\\ \textrm{Sete}&\textrm{lah data diurutkan menjadi}\\ :\: &\color{cyan}40,45,48,52,53,53,62,67,68,70,71\\ \color{black}\textrm{Dike}&\color{black}\textrm{tahui}\: \: \color{red}n=11\: \: \color{black}\textbf{ganjil}\\ Q_{1}&=x_{\frac{1}{4}(n+1)}=x_{\frac{1}{4}.12}=x_{3}=\color{red}48\\ Q_{2}&=x_{\frac{2}{4}(n+1)}=x_{\frac{2}{4}.12}=x_{6}=53\\ Q_{3}&=x_{\frac{3}{4}(n+1)}=x_{\frac{3}{4}.12}=x_{9}=\color{red}68\\ \textrm{Sela}&\textrm{njutnya data dapat dituliskan}\\ &40,45,\underset{\begin{matrix} \Downarrow\\ \color{black}Q_{1} \end{matrix}}{\underbrace{\color{red}48}},52,53,53,62,67,\underset{\begin{matrix} \Downarrow\\ \color{black}Q_{3} \end{matrix}}{\underbrace{\color{red}68}},70,71\\ \textrm{Simp}&\textrm{angan kuartil data tunggal adalah}:\\ &=\displaystyle \frac{1}{2}\left ( Q_{3}-Q_{1} \right )\\ &=\displaystyle \frac{1}{2}(68-48)\\ &=\displaystyle \frac{1}{2}.20=\color{red}10 \end{aligned} \end{array}$

$\begin{array}{ll}\\ 22.&\textrm{Data penjualan suatu barang setiap bulan}\\ &\textrm{di sebuah toko pada tahun 2019 adalah}:\\ &20,3,9,11,4,12,1,9,9,12,8,10.\\ &\textrm{Median, kuartil bawah, dan kuartil atasnya}\\ &\textrm{berturut-turut adalah}\: ....\\ &\begin{array}{llll}\\ \textrm{a}.&6\displaystyle \frac{1}{2},3\frac{1}{2},\: \textrm{dan}\: \: 9\frac{1}{2}\\ \color{red}\textrm{b}.&9,6,\: \textrm{dan}\: \: 11\displaystyle \frac{1}{2}\\ \textrm{c}.&6\displaystyle \frac{1}{2},9,\: \textrm{dan}\: \: 12\\ \textrm{d}.&9,4,\: \textrm{dan}\: \: 12\\ \textrm{e}.&9,3\displaystyle \frac{1}{2},\: \textrm{dan}\: \: 12 \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{b}\\ &\color{blue}\begin{aligned}\textrm{Data}&\: \textrm{mula-mula}\\ :\: &20,3,9,11,4,12,1,9,9,12,8,10\\ \textrm{Sete}&\textrm{lah data diurutkan}\\ :\: &1,3,4,8,9,9,9,10,11,12,12,20\\ \color{black}\textrm{Dike}&\color{black}\textrm{tahui}\: \: \color{red}n=12\: \: \color{black}\textbf{genap}\\ Q_{1}&=x_{\frac{1}{4}n+\frac{1}{2}}=x_{\frac{1}{4}.12+\frac{1}{2}}=x_{3,5}=\color{red}6\\ Q_{2}&=x_{\frac{2}{4}n+\frac{1}{2}}=x_{\frac{2}{4}.12+\frac{1}{2}}=x_{6,5}=\color{red}9=\color{black}M_{e}\\ Q_{3}&=x_{\frac{3}{4}n+\frac{1}{2}}=x_{\frac{3}{4}.12+\frac{1}{2}}=x_{9,5}=\color{red}11\displaystyle \frac{1}{2}\\ \textrm{Sela}&\textrm{njutnya data dapat dituliskan}\\ &1,3,\underset{\begin{matrix} \Downarrow\\ \color{black}Q_{1} \end{matrix}}{\underbrace{\color{red}4,8}},9,\underset{\begin{matrix} \Downarrow\\ \color{black}Q_{2}=M_{e} \end{matrix}}{\underbrace{\color{red}9,9}},10,\underset{\begin{matrix} \Downarrow\\ \color{black}Q_{3} \end{matrix}}{\underbrace{\color{red}11,12}},12,20\\ \end{aligned} \end{array}$

$\begin{array}{ll}\\ 23.&\textrm{Ragam(varians) dari data}\\ &6,8,6,7,8,7,9,7,7,6,7,8,6,5,8,7\\ &\textrm{adalah}\: ....\\ &\begin{array}{llll}\\ \color{red}\textrm{a}.&1\\ \textrm{b}.&1\displaystyle \frac{3}{8}\\ \textrm{c}.&1\displaystyle \frac{1}{8}\\ \textrm{d}.&\displaystyle \frac{7}{8}\\ \textrm{e}.&\displaystyle \frac{5}{8} \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{a}\\ &\color{blue}\begin{aligned}\textrm{Data}&\: \textrm{mula-mula}\\ :\: &6,8,6,7,8,7,9,7,7,6,7,8,6,5,8,7\\ \textrm{Sete}&\textrm{lah data diurutkan(untuk memudahkan)}\\ :\: &5,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9\\ \color{black}\textrm{Dike}&\color{black}\textrm{tahui}\: \: \color{red}n=16.\: \: \color{black}\textrm{Selanjutnya kita cari}\\ \overline{x}&=\displaystyle \frac{5.1+6.4+7.6+8.4+9.1}{16}\\ &=\frac{112}{16}=7\\ \textrm{Dan}\: &\textrm{rumus untuk menghitung ragam adalah}:\\ S^{2}&=\displaystyle \frac{\displaystyle \sum_{i=1}^{16}\left ( x_{i}-\overline{x} \right )^{2}}{n}\\ &=\displaystyle \frac{(5-7)^{2}+4(6-7)^{2}+6(7-7)^{2}+4(8-7)^{2}+(9-7)^{2}}{16}\\ &=\displaystyle \frac{4+4.1+6.0+4.1+4}{16}\\ &=\displaystyle \frac{16}{16}\\ &=\color{red}1 \end{aligned} \end{array}$

$\begin{array}{ll}\\ 24.&\textrm{Diketahui}\: \: x_{1}=2,\: x_{2}=3,5,\: x_{3}=5,\: x_{4}=7,\\ &\textrm{dan}\: \: x_{5}=7,5.\: \textrm{Deviasi rata-rata data di atas}\\ &\textrm{adalah}\: ....\\ &\begin{array}{llll}\\ \textrm{a}.&0\\ \textrm{b}.&1\\ \color{red}\textrm{c}.&1,8\\ \textrm{d}.&2,6\\ \textrm{e}.&5 \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{c}\\ &\color{blue}\begin{aligned}\textrm{Data}&\: \textrm{mula-mula}\\ :\: &2,3\displaystyle \frac{1}{2},5,7,7\displaystyle \frac{1}{2}\\ \color{black}\textrm{Dike}&\color{black}\textrm{tahui}\: \: \color{red}n=5.\: \: \color{black}\textrm{Selanjutnya kita cari}\\ \overline{x}&=\displaystyle \frac{2+3,5+5+7+7,5}{5}=\frac{25}{5}=5\\ \textrm{Dan}\: &\textrm{rumus simpangan rata-rata adalah}:\\ SR&=\displaystyle \frac{\displaystyle \sum_{i=1}^{5} \left |x_{i}-\overline{x} \right |}{n}\\ &=\displaystyle \frac{\left | 2-5 \right |+\left | 3,5-5 \right |+\left | 5-5 \right |+\left | 7-5 \right |+\left | 7,5-5 \right |}{5}\\ &=\displaystyle \frac{\left | -3 \right |+\left | -1,5 \right |+\left | 0 \right |+\left | 2 \right |+\left | 2,5 \right |}{5}\\ &=\displaystyle \frac{3+1,5+0+2+2,5}{5}=\frac{9}{5}\\ &=\color{red}1,8 \end{aligned} \end{array}$

$\begin{array}{ll}\\ 25.&\textrm{Jumlah rataan dan median dari}\\ &(x-6),(x+5),(x+4),(x-7),\\ &(x+9),\: \textrm{dan}\: \: (x-2)\: \: \textrm{adalah}\: ....\\ &\begin{array}{llll}\\ \textrm{a}.&2x-1,5\\ \textrm{b}.&2x-0,5\\ \color{red}\textrm{c}.&2x+1,5\\ \textrm{d}.&2x+2,5\\ \textrm{e}.&2x+3,5 \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{c}\\ &\color{blue}\begin{aligned}\textrm{Data}&\: \textrm{mula-mula}\\ :\: &(x-6),(x+5),(x+4),(x-7),(x+9),(x-2)\\ \color{black}\textrm{Dike}&\color{black}\textrm{tahui}\: \: \color{red}n=6.\: \: \color{black}\textrm{Selanjutnya kita urutkan datanya}\\ &(x-7),(x-6),(x-2),(x+4),(x+5),(x+9)\\ \textrm{Rata}&\textrm{annya}:\\ \overline{x}&=\displaystyle \frac{6x+3}{6}=\color{red}x+0,5\\ \textrm{Medi}&\textrm{annya}:\\ M_{e}&=x_{3,4}=\displaystyle \frac{x_{3}+x_{4}}{2}=\frac{(x-2)+(x+4)}{2}\\ &=\displaystyle \frac{2x+2}{2}=\color{red}x+1\\ \textrm{Rata}&\textrm{an+median}=\color{black}x+0,5+x+1=\color{red}2x+1,5 \end{aligned} \end{array}$