meeting 07 Dispersion Skewness and Kurtosis

3/26/2011

Framework
Frequency
distribution
characteristics

Dispersion, Skewness and
Kurt osis
Presented by:
Mahendra AN

Dispersion

Sources:
http:/ / business.clayton.edu/ arjom and/ business/ stat%20 presentations/ busa310
1.htm l
Djarwanto, Statis tik So s ial Eko n o m i Bagian pertam a edisi 3, BPFE, 20 0 1

Dispersion
Dispersion

Relative
dispersion

Absolute
dispersion

• Dispersion is separate
m easures of values
am ong its central
tendency.

Range
quartile
deviation
Percentile
deviation

Absolute
dispersion

Skewness

Kurtosis

Relative
dispersion

Absolut e Dispersion
Ran ge
• Sim plest dispersion m easure
• Effected by extrem e values
Qu artile D e viatio n
• Dispersion of inter quartile range

Average
deviation
Standard
deviation

Absolut e Dispersion (cont . )

Absolut e Dispersion (cont . )

Pe rce n tile D e viatio n
• Dispersion of inter percentile range (P 10 and
P 90 )

Stan d ard d e viatio n
• Ungrouped data

Ave rage d e viatio n ( m e an d e viatio n )
• Ungrouped data

• Grouped data

• Grouped data

© Mahendra Adhi Nugroho, M.Sc, Accounting Program Study of Yogyakarta State University
For internal use only!

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3/26/2011

St andard Unit s
• To com pare two or m ore distributions
• Standard unit show deviation of a variable value
(X) on m ean ( ) in standard deviation unit (s)
• Com m only base on zero value (Z=0 )
• Exam ple: Z=1.2 is better than z=1.0

Skewness
• An im portant m easure of the shape of a
distribution is called skewness
• The form ula for com puting skewness for a data
set is som ewhat com plex

Relat ive Dispersion
• To know sm allest variation in a distribution
• To com pare two or m ore frequency distribution

• All of standard dispersion m easurem ent can be
used.
• Stated in coefficient of variation

Skewness (cont . )
Karl Pe ars o n m e th o d
• Base on m ean and m edian values

Bo w le y m e th o d
• Base on quartile values

Skewness (cont . )
10 – 9 0 p e rce n tile 's m e th o d
• Base on percentile

Skewness (cont . )
• Grouped data skewness
• Third m om ent m ethod is used

• The better m easurem ent for skewness base on

the third m om ent.

© 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
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

„

Distribution Shape: Skewness

.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

Kurt osis
• Measurem ent 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!

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