Roboguru

Berikut merupakan distribusi frekuensi kumulatif data suhu udara tertinggi (dalam derajat Fahrenheit) yang tercatat di 50 kota besar di Indonesia. Tentukan simpangan rata-rata, simpangan baku, dan ragam

Pertanyaan

Berikut merupakan distribusi frekuensi kumulatif data suhu udara tertinggi (dalam derajat Fahrenheit) yang tercatat di 50 kota besar di Indonesia. Tentukan simpangan rata-rata, simpangan baku, dan ragam space 


Pembahasan Soal:

Pertama ubahlah distribusi frekuensi kumulatif tersebut ke distribusi frekuensi seperti berikut:



Ingat nilai rata-rata:

x with bar on top equals fraction numerator begin display style sum from i equals 1 to k of end style x subscript i times f subscript i over denominator begin display style sum from i equals 1 to k of end style space f subscript i end fraction 

dimana x with bar on top rata-rata data kelompok, x subscript i adalah titik tengah kelas ke-i dan f subscript i adalah frekuensi kelas ke-i. Sehingga titik tengah distribusi frekuensi tersebut adalah

table attributes columnalign right center left columnspacing 0px end attributes row cell x subscript 1 end cell equals cell fraction numerator 99 comma 5 plus 104 comma 5 over denominator 2 end fraction equals 102 end cell row cell x subscript 2 end cell equals cell fraction numerator 104 comma 5 plus 109 comma 5 over denominator 2 end fraction equals 107 end cell row cell x subscript 3 end cell equals cell fraction numerator 109 comma 5 plus 114 comma 5 over denominator 2 end fraction equals 112 end cell row cell x subscript 4 end cell equals cell fraction numerator 114 comma 5 plus 119 comma 5 over denominator 2 end fraction equals 117 end cell row cell x subscript 5 end cell equals cell fraction numerator 119 comma 5 plus 124 comma 5 over denominator 2 end fraction equals 122 end cell row cell x subscript 6 end cell equals cell fraction numerator 124 comma 5 plus 129 comma 5 over denominator 2 end fraction equals 127 end cell row cell x subscript 7 end cell equals cell fraction numerator 129 comma 5 plus 134 comma 5 over denominator 2 end fraction equals 132 end cell end table

sehingga

begin mathsize 12px style x with bar on top equals fraction numerator begin display style sum from i equals 1 to 7 of x subscript i times f subscript i end style over denominator begin display style sum from i equals 1 to 7 of space f subscript i end style end fraction equals fraction numerator 102 times 2 plus 107 times 8 plus 112 times 18 plus 117 times 13 plus 122 times 7 plus 127 times 1 plus 132 times 1 over denominator 2 plus 8 plus 18 plus 13 plus 7 plus 1 plus 1 end fraction equals fraction numerator 204 plus 856 plus 2.016 plus 1.521 plus 854 plus 127 plus 132 over denominator 50 end fraction equals fraction numerator 5.710 over denominator 50 end fraction equals 114 comma 2 end style  

Ingat konsep simpangan rata-rata:

SR equals fraction numerator begin display style sum from i equals 1 to k of end style space f subscript i times open vertical bar x subscript i minus x with bar on top close vertical bar over denominator begin display style sum from i equals 1 to k of space f subscript i end style end fraction  

dimana x with bar on top rata-rata data kelompok, x subscript i adalah titik tengah kelas ke-i dan f subscript i adalah frekuensi kelas ke-i. Dari a) nilai rata-rata diperoleh x subscript i dan x with bar on top sehingga:

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar x subscript 1 minus x with bar on top close vertical bar end cell equals cell open vertical bar 102 minus 114 comma 2 close vertical bar equals open vertical bar negative 12 comma 2 close vertical bar equals 12 comma 2 end cell row cell open vertical bar x subscript 2 minus x with bar on top close vertical bar end cell equals cell open vertical bar 107 minus 114 comma 2 close vertical bar equals open vertical bar negative 7 comma 2 close vertical bar equals 7 comma 2 end cell row cell open vertical bar x subscript 3 minus x with bar on top close vertical bar end cell equals cell open vertical bar 112 minus 114 comma 2 close vertical bar equals open vertical bar negative 2 comma 2 close vertical bar equals 2 comma 2 end cell row cell open vertical bar x subscript 4 minus x with bar on top close vertical bar end cell equals cell open vertical bar 117 minus 114 comma 2 close vertical bar equals open vertical bar 2 comma 8 close vertical bar equals 2 comma 8 end cell row cell open vertical bar x subscript 5 minus x with bar on top close vertical bar end cell equals cell open vertical bar 122 minus 114 comma 2 close vertical bar equals open vertical bar 7 comma 8 close vertical bar equals 7 comma 8 end cell row cell open vertical bar x subscript 6 minus x with bar on top close vertical bar end cell equals cell open vertical bar 127 minus 114 comma 2 close vertical bar equals open vertical bar 12 comma 8 close vertical bar equals 12 comma 8 end cell row cell open vertical bar x subscript 7 minus x with bar on top close vertical bar end cell equals cell open vertical bar 132 minus 114 comma 2 close vertical bar equals open vertical bar 17 comma 8 close vertical bar equals 17 comma 8 end cell end table

maka

begin mathsize 12px style SR equals fraction numerator begin display style sum from i equals 1 to 7 of space f subscript i times open vertical bar x subscript i minus x with bar on top close vertical bar end style over denominator begin display style sum from i equals 1 to 7 of space f subscript i end style end fraction equals fraction numerator 2 times 12 comma 2 plus 8 times 7 comma 2 plus 18 times 2 comma 2 plus 13 times 2 comma 8 plus 7 times 7 comma 8 plus 1 times 12 comma 8 plus 1 times 17 comma 8 over denominator 2 plus 8 plus 18 plus 13 plus 7 plus 1 plus 1 end fraction equals fraction numerator 24 comma 4 plus 57 comma 6 plus 39 comma 6 plus 36 comma 4 plus 54 comma 6 plus 12 comma 8 plus 17 comma 8 over denominator 50 end fraction equals fraction numerator 243 comma 2 over denominator 20 end fraction equals 4 comma 86 end style      

Ingat konsep varians atau ragam:

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Dimana s squared varians, x subscript i adalah titik tengah kelas ke-i dan f subscript i adalah frekuensi kelas ke-i. Berdasarkan nilai dari simpangan rata-rata maka:

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell sum from i equals 1 to k of space f subscript i open parentheses x subscript i minus x with bar on top close parentheses squared end cell equals cell 2 open parentheses negative 12 comma 2 close parentheses squared plus 8 open parentheses negative 7 comma 2 close parentheses squared plus 18 open parentheses negative 2 comma 2 close parentheses squared plus 13 open parentheses 2 comma 8 close parentheses squared plus end cell row blank blank cell 7 open parentheses 7 comma 8 close parentheses squared plus 1 open parentheses 12 comma 8 close parentheses squared plus 1 open parentheses 17 comma 8 close parentheses squared end cell row blank equals cell 2 open parentheses 148 comma 84 close parentheses plus 8 open parentheses 51 comma 84 close parentheses plus 18 open parentheses 4 comma 84 close parentheses plus 13 open parentheses 7 comma 84 close parentheses plus end cell row blank blank cell 7 open parentheses 60 comma 84 close parentheses plus 1 open parentheses 163 comma 84 close parentheses plus 1 open parentheses 316 comma 84 close parentheses end cell row blank equals cell 297 comma 68 plus 414 comma 72 plus 87 comma 12 plus 101 comma 92 plus end cell row blank blank cell 425 comma 88 plus 163 comma 84 plus 316 comma 84 end cell row blank equals cell 1.808 end cell end table end style  

sehingga

table attributes columnalign right center left columnspacing 0px end attributes row cell s squared end cell equals cell fraction numerator begin display style sum from i equals 1 to k of space f subscript i open parentheses x subscript i minus x with bar on top close parentheses squared end style over denominator sum from i equals 1 to k of space f subscript i end fraction end cell row blank equals cell fraction numerator 1.808 over denominator 50 end fraction end cell row blank equals cell 36 comma 16 end cell end table 

Ingat konsep simpangan baku:

s equals square root of table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator begin display style sum from i equals 1 to k of space f subscript i open parentheses x subscript i minus x with bar on top close parentheses squared end style over denominator sum from i equals 1 to k of space f subscript i end fraction end cell end table end root 

berdasarkan hasil dari varians maka:

table attributes columnalign right center left columnspacing 0px end attributes row s equals cell square root of fraction numerator begin display style sum from i equals 1 to k of space f subscript i open parentheses x subscript i minus x with bar on top close parentheses squared end style over denominator sum from i equals 1 to k of space f subscript i end fraction end root end cell row blank equals cell square root of 36 comma 16 end root end cell row blank equals cell 6 comma 01 end cell end table  

Dengan demikian dari data di atas diperoleh simpangan rata-rata dari 4 comma 86, ragam 36 comma 16 dan simpangan baku 6 comma 01

 

Pembahasan terverifikasi oleh Roboguru

Terakhir diupdate 18 Juli 2021

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