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Matakuliah
Tahun
: I0134 -Metode Statistika
: 2007
Pertemuan 03
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
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
5
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”).
6
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.
7
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
8
Mode
• The mode is the measurement which occurs most frequently.
• 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).
9
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
• Median?
n
25
•
m2
Mode? (Highest peak)
mode  2
10
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.
11
Extreme Values
Symmetric: Mean = Median
Skewed right: Mean > Median
Skewed left: Mean < Median
12
Key Concepts
I. Measures of Center
1. Arithmetic mean (mean) or average
a. Population: m
b. Sample of size n:
 xi
x
n
2. Median: position of the median  .5(n 1)
3. Mode
4. The median may preferred to the mean if the data are
highly skewed.
II. Measures of Variability
1. Range: R  largest - smallest
13
Key Concepts
2. Variance
a. Population of N measurements:
b. Sample of n measurements:
2

(
x
m
)
i
2 
N
2
(

x
)
2
i
 xi 2
( xi - x )
2
n
s 

n -1
n -1
3. Standard deviation
Population standard deviation :    2
4. A rough approximation for s can be calculated as s  R / 4.
The divisor can be adjusted depending on the2sample size.
Sample standard deviation : s  s
14
• Selamat Belajar Semoga Sukses.
15
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