3/26/2011
Framework
Frequency
distribution
characteristics
Dispersion, Skewness and
Kurtosis
Presented by:
Mahendra AN
Dispersion
Sources:
http://business.clayton.edu/arjomand/business/stat%20presentations/busa310
1.html
Djarwanto, Statistik Sosial Ekonomi Bagian pertama edisi 3, BPFE, 2001
Dispersion
Dispersion
Relative
dispersion
Absolute
dispersion
• Dispersion is separate
measures of values
among its central
tendency.
Range
quartile
deviation
Percentile
deviation
Absolute
dispersion
Skewness
Kurtosis
Relative
dispersion
Absolute Dispersion
Range
• Simplest dispersion measure
• Effected by extreme values
Quartile Deviation
• Dispersion of inter quartile range
Average
deviation
Standard
deviation
Absolute Dispersion (cont.)
Absolute Dispersion (cont.)
Percentile Deviation
• Dispersion of inter percentile range (P10 and
P90)
Standard deviation
• Ungrouped data
Average deviation (mean deviation)
• Ungrouped data
• Grouped data
• Grouped data
© Mahendra Adhi Nugroho, M.Sc, Accounting Program Study of Yogyakarta State University
For internal use only!
1
3/26/2011
Standard Units
• To compare two or more distributions
• Standard unit show deviation of a variable value
(X) on mean ( ) in standard deviation unit (s)
• Commonly base on zero value (Z=0)
• Example: Z=1.2 is better than z=1.0
Skewness
• An important measure of the shape of a
distribution is called skewness
• The formula for computing skewness for a data
set is somewhat complex
Relative Dispersion
• To know smallest variation in a distribution
• To compare two or more frequency distribution
• All of standard dispersion measurement can be
used.
• Stated in coefficient of variation
Skewness (cont.)
Karl Pearson method
• Base on mean and median values
Bowley method
• Base on quartile values
Skewness (cont.)
10 – 90 percentile's method
• Base on percentile
Skewness (cont.)
• Grouped data skewness
• Third moment method is used
• The better measurement for skewness base on
the third moment.
© Mahendra Adhi Nugroho, M.Sc, Accounting Program Study of Yogyakarta State University
For internal use only!
2
3/26/2011
Distribution Shape: Skewness
„
Deciding skewness
Symmetric (not skewed, SK = 0)
0)
• If skewness is zero, then
• Mean and median are equal.
Left-Skewed
Mean Median Mode
Symmetric
Mean = Median = Mode
Relative Frequency
y
.35
Right-Skewed
Mode Median Mean
Skewness = 0
30
.30
.25
.20
.15
.10
.05
0
Slide 14
Distribution Shape: Skewness
Distribution Shape: Skewness
Moderately Skewed Left
• Is skewness is negative (left skewed SK >= -3), then
• Mean will usually be less than the median.
Relative Frequency
y
.35
„
Moderately Skewed Right
• If skewness is positive (right skewed, SK >= +3), then
• Mean will usually be more than the median.
Skewness = − .31
.35
30
.30
Relative Frequency
y
„
.25
.20
.15
.10
.05
Skewness = .31
30
.30
.25
.20
.15
.10
.05
0
0
Slide 15
Slide 16
Distribution Shape: Skewness
„
Highly Skewed Right
• Skewness is positive (often above 1.0).
• Mean will usually be more than the median.
Relative Frequency
y
.35
Kurtosis
• Measurement of peakedness
• Difficult to interpret
• Only in theoretic not in practice
Skewness = 1.25
.30
30
.25
.20
.15
.10
.05
Mesokurtic
0
Leptokurtic
Slide 17
Plakurtic
© Mahendra Adhi Nugroho, M.Sc, Accounting Program Study of Yogyakarta State University
For internal use only!
3