Matakuliah
Tahun
Versi
: I0284 - Statistika
: 2008
: Revisi
Pertemuan 02
Ukuran Pemusatan dan Lokasi
1
Learning Outcomes
Pada akhir pertemuan ini, diharapkan mahasiswa
akan mampu :
• Mahasiswa akan dapat menghitung
ukuran-ukuran pemusatan dan lokasi.
2
Outline Materi
•
•
•
•
•
•
Rata-rata
Median
Modus
Kuartil
Desil
persentil
3
Measures of Center
• A measure along the horizontal axis of
the data distribution that locates the
center of the distribution.
4
1. Arithmetic Mean or Average
• The mean of a set of measurements is
the sum of the measurements divided
by the total number of measurements.
 xi
x
n
where n = number of measurements
 xi  sumof all the measurements
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Example
•The set: 2, 9, 1, 5, 6
 xi
2

9

11

5

6
33
x


 6.6
n
5
5
If we were able to enumerate the whole
population, the population mean would be
called m (the Greek letter “mu”).
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Mean (Arithmetic Mean)
• Approximating the Arithmetic Mean
– Used cwhen raw data are not available
mj f j

j 1
– X
n
n  sample size
c  number of classes in the frequency distribution
m j  midpoint of the jth class
f j  frequencies of the jth class
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2. Median
• The median of a set of measurements is
the middle measurement when the
measurements are ranked from smallest
to largest.
• The position of the median is
.5(n + 1)
once the measurements have been ordered.
a. Md = X
(n+1)/2
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Example
• The set: 2, 4, 9, 8, 6, 5, 3 n = 7
• Sort:
2, 3, 4, 5, 6, 8, 9
• Position: .5(n + 1) = .5(7 + 1) = 4th
Median = 4th largest measurement
• The set: 2, 4, 9, 8, 6, 5
n=6
• Sort: 2, 4, 5, 6, 8, 9
• Position: .5(n + 1) = .5(6 + 1) = 3.5th
Median = (5 + 6)/2 = 5.5 — average of the 3rd and 4th
measurements
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b. Median Data Berkelompok
 1

n  F 


Me  b  p  2
f






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3. Mode
Untuk menyatakan fenomena yang paling
banyak terjadi, juga untuk menentukan “ratarata” dari data kualitatif.
a. Data tak berkelompok : Modus (Mo) dilihat dari
data yang memiliki frekuensi terbanyak
• The set: 2, 4, 9, 8, 8, 5, 3
– The mode is 8, which occurs twice
• The set: 2, 2, 9, 8, 8, 5, 3
– There are two modes—8 and 2 (bimodal)
• The set: 2, 4, 9, 8, 5, 3
– There is no mode (each value is unique).
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b. Modus Data Berkelompok
 b1 

Mo  b  p 
 b1  b2 
4. Kuartil (Q )
i
Membagi kelompok data yang telah terurut menjadi 4
bagian yang sama besar.
a. Data tak berkelompok
i (n  1)
Qi  X
; i  1, 2, 3
4
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b. Kuartil Data Berkelompok
 in


F


 ; i 1, 2, 3
Qi  b  p  4
 f 




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Example
The number of quarts of milk
purchased by 25 households:
0 0 1 1 1 1 1 2 2 2 2 2 2
2 2 2 3 3 3 3 3 4 4 4 5
• Mean?
 xi 55
x

 2.2
n
25
• Median?
m2
• Mode? (Highest peak)
mode  2
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Extreme Values
• The mean is more easily affected by
extremely large or small values than the
median.
Applet
•The median is often used as a measure
of center when the distribution is
skewed.
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Extreme Values
Symmetric: Mean = Median
Skewed right: Mean > Median
Skewed left: Mean < Median
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5. Desil
• Membagi kelompok data yang telah terurut
menjadi 10 bagian yang sama besar.
a. Data tak berkelompok
Di  X i ( n101) ; i 1, 2, ..., 9
b. Data berkelompok
 in
F

10
Di  b  p 
f





 ; i  1, 2, ... , 9



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6. Persentil (Pi)
• Membagi kelompok data yang telah terurut
menjadi 100 bagian yang sama besar.
a. Data tak berkelompok
Pi  X i ( n101) ; i 1, 2, ..., 9
b. Data berkelompok
 in
F

100
Pi  b  p 
f





 ; i 1, 2, ... , 9



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• Selamat Belajar Semoga Sukses.
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