Tentukan Q\xb9, Q\xb2, Q\xb3, dan jangkauan inter quartil dari data tersebut

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Tentukan Q¹, Q², Q³, dan jangkauan inter quartil dari data tersebut​

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a.

Data setelah diurutkan :

39 , 48 , 49 , 49 , 50 , 50 , 50 , 51 , 51 , 51 , 52 , 52 , 53 , 53 , 45

n=15~\to~ganjil dan (n+1)~habis dibagi 4 :

\boxed{\huge{\begin{array}{l}\text{Q}_1=x_{\left(\frac{n~+~1}{4}\right)}\\\text{Q}_2=x_{\left(\frac{2(n~+~1)}{4}\right)}\\\text{Q}_3=x_{\left(\frac{3(n~+~1)}{4}\right)}\end{array}}}

\begin{array}{lll}\red{\huge{\text{Q}_1}}&=&x_{\left(\frac{n~+~1}{4}\right)}\\~\\&=&x_{\left(\frac{15~+~1}{4}\right)}\\~\\&=&x_4\\~\\&=&\red{\huge{49}}\end{array}

\begin{array}{lll}\red{\huge{\text{Q}_2}}&=&x_{\left(\frac{2(n~+~1)}{4}\right)}\\~\\&=&x_{\left(\frac{2(15~+~1)}{4}\right)}\\~\\&=&x_8\\~\\&=&\red{\huge{51}}\end{array}

\begin{array}{lll}\red{\huge{\text{Q}_3}}&=&x_{\left(\frac{3(n~+~1)}{4}\right)}\\~\\&=&x_{\left(\frac{3(15~+~1)}{4}\right)}\\~\\&=&x_{12}\\~\\&=&\red{\huge{52}}\end{array}

Jangkauan interkuartil = \text{Q}_3-\text{Q}_1 = 52 – 49 \red{\huge{=3}}

\\

b.

Data setelah diurutkan :

6 , 6 , 6 , 6 , 7 , 7 , 7 , 8 , 8 , 8 , 9 , 9 , 9 , 9 , 9 , 9 , 10 , 10

n=18~\to~genap dan tidak habis dibagi 4 :

\boxed{\huge{\begin{array}{l}\text{Q}_1=x_{\left(\frac{n~+~2}{4}\right)}\\\text{Q}_2=\frac{x_{\left(\frac{n}{2}\right)}~+~x_{\left(\frac{n}{2}+1\right)}}{2}\\\text{Q}_3=x_{\left(\frac{3n~+~2}{4}\right)}\end{array}}}

\begin{array}{lll}\red{\huge{\text{Q}_1}}&=&x_{\left(\frac{n~+~2}{4}\right)}\\~\\&=&x_{\left(\frac{18~+~2}{4}\right)}\\~\\&=&x_5\\~\\&=&\red{\huge{7}}\end{array}

\begin{array}{lll}\red{\huge{\text{Q}_2}}&=&\frac{x_{\left(\frac{n}{2}\right)}~+~x_{\left(\frac{n}{2}+1\right)}}{2}\\~\\&=&\frac{x_{\left(\frac{18}{2}\right)}~+~x_{\left(\frac{18}{2}+1\right)}}{2}\\~\\&=&\frac{x_9~+~x_{10}}{2}\\~\\&=&\frac{8+8}{2}\\~\\&=&\red{\huge{8}}\end{array}

\begin{array}{lll}\red{\huge{\text{Q}_3}}&=&x_{\left(\frac{3n~+~2}{4}\right)}\\~\\&=&x_{\left(\frac{3.(18)~+~2}{4}\right)}\\~\\&=&x_{14}\\~\\&=&\red{\huge{9}}\end{array}

Jangkauan interkuartil = \text{Q}_3-\text{Q}_1 = 9 – 7 \red{\huge{=2}}

\\

c.

Data setelah diurutkan :

6 , 6 , 6 , 6 , 6 , 7 , 7 , 7 , 8 , 8 , 8 , 8 , 9 , 9 , 9 , 9 , 9 , 9 , 10 , 10 , 10

n=21~\to~ganjil dan (n+1)~tidak habis dibagi 4 :

\boxed{\huge{\begin{array}{l}\text{Q}_1=\frac{x_{\left(\frac{n~-~1}{4}\right)}~+~x_{\left(\frac{n~-~1}{4}+1\right)}}{2}\\\text{Q}_2=x_{\left(\frac{2(n~+~1)}{4}\right)}\\\text{Q}_3=\frac{x_{\left(\frac{3n~+~1}{4}\right)}~+~x_{\left(\frac{3n~+~1}{4}+1\right)}}{2}\end{array}}}

\begin{array}{lll}\red{\huge{\text{Q}_1}}&=&\frac{x_{\left(\frac{n~-~1}{4}\right)}~+~x_{\left(\frac{n~-~1}{4}+1\right)}}{2}\\~\\&=&\frac{x_{\left(\frac{21~-~1}{4}\right)}~+~x_{\left(\frac{21~-~1}{4}+1\right)}}{2}\\~\\&=&\frac{x_5~+~x_6}{2}\\~\\&=&\frac{6~+~7}{2}\\~\\&=&\red{\huge{6,5}}\end{array}

\begin{array}{lll}\red{\huge{\text{Q}_2}}&=&x_{\left(\frac{2(n~+~1)}{4}\right)}\\~\\&=&x_{\left(\frac{2(21~+~1)}{4}\right)}\\~\\&=&x_{11}\\~\\&=&\red{\huge{8}}\end{array}

\begin{array}{lll}\red{\huge{\text{Q}_3}}&=&\frac{x_{\left(\frac{3n~+~1}{4}\right)}~+~x_{\left(\frac{3n~+~1}{4}+1\right)}}{2}\\~\\&=&\frac{x_{\left(\frac{3.(21)~+~1}{4}\right)}~+~x_{\left(\frac{3.(21)~+~1}{4}+1\right)}}{2}\\~\\&=&\frac{x_{16}~+~x_{17}}{2}\\~\\&=&\frac{9~+~9}{2}\\~\\&=&\red{\huge{9}}\end{array}

Jangkauan interkuartil = \text{Q}_3-\text{Q}_1 = 9 – 6,5 \red{\huge{=1,5}}

\\

d.

Data setelah diurutkan :

27 , 27 , 27 , 27 , 28 , 28 , 29 , 29 , 30 , 30 , 31 , 31 , 32 , 32 , 32 , 32

n=16~\to~genap dan habis dibagi 4 :

\boxed{\huge{\begin{array}{l}\text{Q}_1=\frac{x_{\left(\frac{n}{4}\right)}~+~x_{\left(\frac{n}{4}+1\right)}}{2}\\\text{Q}_2=\frac{x_{\left(\frac{n}{2}\right)}~+~x_{\left(\frac{n}{2}+1\right)}}{2}\\\text{Q}_3=\frac{x_{\left(\frac{3n}{4}\right)}~+~x_{\left(\frac{3n}{4}+1\right)}}{2}\end{array}}}

\begin{array}{lll}\red{\huge{\text{Q}_1}}&=&\frac{x_{\left(\frac{n}{4}\right)}~+~x_{\left(\frac{n}{4}+1\right)}}{2}\\~\\&=&\frac{x_{\left(\frac{16}{4}\right)}~+~x_{\left(\frac{16}{4}+1\right)}}{2}\\~\\&=&\frac{x_4~+~x_5}{2}\\~\\&=&\frac{27~+~28}{2}\\~\\&=&\red{\huge{27,5}}\end{array}

\begin{array}{lll}\red{\huge{\text{Q}_2}}&=&\frac{x_{\left(\frac{n}{2}\right)}~+~x_{\left(\frac{n}{2}+1\right)}}{2}\\~\\&=&\frac{x_{\left(\frac{16}{2}\right)}~+~x_{\left(\frac{16}{2}+1\right)}}{2}\\~\\&=&\frac{x_8~+~x_9}{2}\\~\\&=&\frac{29~+~30}{2}\\~\\&=&\red{\huge{29,5}}\end{array}

\begin{array}{lll}\red{\huge{\text{Q}_3}}&=&\frac{x_{\left(\frac{3n}{4}\right)}~+~x_{\left(\frac{3n}{4}+1\right)}}{2}\\~\\&=&\frac{x_{\left(\frac{3.(16)}{4}\right)}~+~x_{\left(\frac{3.(16)}{4}+1\right)}}{2}\\~\\&=&\frac{x_{12}~+~x_{13}}{2}\\~\\&=&\frac{31~+~32}{2}\\~\\&=&\red{\huge{31,5}}\end{array}

\begin{array}{lll}\red{\sf Jangkauan~interkuartil}&=&\text{Q}_3-\text{Q}_1\\~\\&=&31,5-27,5\\~\\&=&\red{\huge{4}}\end{array}


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