Matakuliah
Tahun
Versi
: I0134 – Metoda Statistika
: 2005
: Revisi
Pertemuan 05
Ukuran Deskriptif Lain
1
Learning Outcomes
Pada akhir pertemuan ini, diharapkan mahasiswa
akan mampu :
• Mahasiswa dapat memberikan contoh
tentang penggunaan ukuran deskriptif lain.
2
Outline Materi
•
•
•
•
Koefisien variasi
Skor Z
Skewness
Kurtosis
3
Skewness and Kurtosis

Skewness
– Measure of asymmetry of a frequency distribution
• Skewed to left
• Symmetric or unskewed
• Skewed to right

Kurtosis
– Measure of flatness or peakedness of a frequency
distribution
• Platykurtic (relatively flat)
• Mesokurtic (normal)
• Leptokurtic (relatively peaked)
4
Skewness
Skewed to left
Mean < median < mode
F re q ue nc y
3 0
2 0
1 0
0
1 0 0
2 0 0
3 0 0
4 0 0
5 0 0
6 0 0
x
5
Skewness
Symmetric
Mean = median = mode
F re q ue nc y
3 0
2 0
1 0
0
1 0 0
2 0 0
3 0 0
4 0 0
5 0 0
6 0 0
x
6
Skewness
Skewed to right
Mode > median > mean
F re q ue nc y
3 0
2 0
1 0
0
1 0 0
2 0 0
3 0 0
4 0 0
5 0 0
6 0 0
x
7
Kurtosis
Platykurtic - flat distribution
7 0 0
6 0 0
F re q u e n c y
5 0 0
4 0 0
3 0 0
2 0 0
1 0 0
0
- 3 .5
- 2 .7
- 1 .9
- 1 .1
- 0 .3
0 .5
1 .3
2 .1
2 .9
3 .7
X
8
Kurtosis
Mesokurtic - not too flat and not too peaked
5 0 0
F re q u e n c y
4 0 0
3 0 0
2 0 0
1 0 0
0
-4
-3
-2
-1
0
1
2
3
4
X
9
Kurtosis
Leptokurtic - peaked distribution
F re q u e n c y
2 0 0 0
1 0 0 0
0
-1 0
0
1 0
Y
10
Relations between the Mean
and Standard Deviation

Chebyshev’s Theorem
– Applies to any distribution, regardless of shape
– Places lower limits on the percentages of
observations within a given number of standard
deviations from the mean

Empirical Rule
– Applies only to roughly mound-shaped and
symmetric distributions
– Specifies approximate percentages of
observations within a given number of standard
deviations from the mean
11
Chebyshev’s Theorem

1
At
least
1
At least k of the elements of any
distribution lie within k standard
deviations of the mean
1
2
1
1 3

1

  75%
2
4 4
2
1
1 8
1  2  1    89%
9 9
3
1
1 15
1 2  1

 94%
16
16
4
2
Lie
within
3
Standard
deviations
of the mean
4
12
Empirical Rule

For roughly mound-shaped and symmetric
distributions, approximately:
68%
95%
All
1 standard deviation
of the mean
Lie
within
2 standard deviations
of the mean
3 standard deviations
of the mean
13
• Selamat Belajar Semoga Sukses.
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