Download Mean Median Mode

STATISTIK DESKRIPTIF I
(Measures of central tendency
and variation / ukuran pemusatan
dan Penyebaran)
Measures of central tendency
(Ukuran pemusatan)
X
Mean
Median
Mode
Measures of Variation
(Ukuran Penyebaran)
X
Range, interquartile range, variance
and standard deviation, coefficient of
variation
Ukuran pemusatan dan Penyebaran
Summary Measures
Central Tendency
Mean
Quartile
Mode
Median
Range
Variation
Coefficient of
Variation
Variance
Geometric Mean
Standard Deviation
Ukuran pemusatan dan Penyebaran
menurut Tingkat Skala Pengukurannya
TINGKAT SKALA
PENGUKURAN
U K U R A N:
Pemusatan
Penyebaran
NOMINAL
Mode
-----
ORDINAL
Median, Mode
Range, interquartile
range,
INTERVAL
Mean, Median, Mode
Range, interquartile
range, variance and
standard deviation
RASIO
Mean, Median, Mode
Range, interquartile
range, variance and
standard deviation,
coefficient of variation
Penyajian Data dan Statistik
1. Tabel Frekuensi utk variable Nominal /
Ordinal
Var: Stat3us Perkawinan WTS (Nominal)
No Ko
de
e
1
1
2
3
Tally
f
cf
f%
Cf
%
f˚
cf˚
Belum
menikah
6
6
24,0
24,0
86,40
86,40
2
Menikah
7
13 28,0
52,0
100,80
187,20
3
Janda
12
25 48,0
100,
0
172,80
360
25
-
-
360
-
∑
Modus = 3
100
Kode dengan f terbanyak (12)
Var: Status Perkawinan WTS (Nominal)
Var: Status Perkawinan WTS (Nominal)
Var: Status Perkawinan WTS (Nominal)
Tabel Frekuensi untuk Variabel Berskala Interval / Rasio yang belum
dikelompokkan
Variabel Umur WTS (Rasio)
No Nilai
(X)
Tally
f
cf
fx
fx²
1
3
8
8
24
72
2
4
4
12
16
64
3
5
7
19
35
175
4
6
2
21
12
72
5
7
2
23
14
98
6
10
1
24
10
100
7
12
1
25
12
144
25
-
∑
Output SPSS

Symmetric or skewed
Shape of a Distribution
Left-Skewed
Mean < Median < Mode
Symmetric
Mean = Median =Mode
Right-Skewed
Mode < Median < Mean
UKURAN KECONDONGAN
Rumus Skewness (Kecondongan) :
Sk =  - Mo atau Sk = 3( - Md)


13
UKURAN Kurtosis (KERUNCINGAN)
Ke r uncingan Kur va
BENTUK KERUNCINGAN
Platy kurtic
Mesokurtic
Leptokurtic
Rumus Keruncingan:
4 = 1/n  (x - )4
4
Output SPSS
© 2002 Prentice-Hall, Inc.
Chap 3-17
© 2002 Prentice-Hall, Inc.
Chap 3-18
Chapter Topics

Measures of central tendency

Mean, median, mode, geometric mean, midrange

Quartile

Measure of variation


Range, interquartile range, variance and standard
deviation, coefficient of variation
Shape

Symmetric, skewed, using box-and-whisker plots
© 2002 Prentice-Hall, Inc.
Chap 3-19
Chapter Topics


(continued)
Coefficient of correlation
Pitfalls in numerical descriptive measures and
ethical considerations
© 2002 Prentice-Hall, Inc.
Chap 3-20
Measures of Central Tendency
Central Tendency
Average
Median
Mode
n
X 
X
i 1
N

i 1
Geometric Mean
X G   X1  X 2 
n
X
i
 Xn 
1/ n
i
N
© 2002 Prentice-Hall, Inc.
Chap 3-21
Mean (Arithmetic Mean)

Mean (arithmetic mean) of data values

Sample mean
Sample Size
n
X

X
i 1
i
n
 Xn
Population mean
Population Size
N

© 2002 Prentice-Hall, Inc.
X1  X 2 

n
X
i 1
N
i
X1  X 2 

N
 XN
Chap 3-22
Mean (Arithmetic Mean)
(continued)


The most common measure of central
tendency
Affected by extreme values (outliers)
0 1 2 3 4 5 6 7 8 9 10
Mean = 5
© 2002 Prentice-Hall, Inc.
0 1 2 3 4 5 6 7 8 9 10 12 14
Mean = 6
Chap 3-23
Median


Robust measure of central tendency
Not affected by extreme values
0 1 2 3 4 5 6 7 8 9 10
Median = 5

0 1 2 3 4 5 6 7 8 9 10 12 14
Median = 5
In an ordered array, the median is the
“middle” number


If n or N is odd, the median is the middle number
If n or N is even, the median is the average of the
two middle numbers
© 2002 Prentice-Hall, Inc.
Chap 3-24
Mode






A measure of central tendency
Value that occurs most often
Not affected by extreme values
Used for either numerical or categorical data
There may may be no mode
There may be several modes
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Mode = 9
© 2002 Prentice-Hall, Inc.
0 1 2 3 4 5 6
No Mode
Chap 3-25
Geometric Mean

Useful in the measure of rate of change of a
variable over time
X G   X1  X 2 

 Xn 
1/ n
Geometric mean rate of return

Measures the status of an investment over time
RG  1  R1   1  R2  
© 2002 Prentice-Hall, Inc.
 1  Rn  
1/ n
1
Chap 3-26
Example
An investment of $100,000 declined to $50,000 at the
end of year one and rebounded to $100,000 at end of
year two:
X1  $100,000
X 2  $50,000
X 3  $100,000
Average rate of return:
(50%)  (100%)
X
 25%
2
Geometric rate of return:
RG  1   50%    1  100%   
1/ 2
  0.50    2  
1/ 2
© 2002 Prentice-Hall, Inc.
1
 1  1  1  0%
1/ 2
Chap 3-27
Quartiles

Split Ordered Data into 4 Quarters
25%
25%
 Q1 

25%
 Q2 
Position of i-th Quartile
25%
Q3 
i  n  1
 Qi  
4
Data in Ordered Array: 11 12 13 16 16 17 18 21 22
1 9  1
Position of Q1 
 2.5
4
Q1
12  13


 12.5
2
Q1 and Q3 Are Measures of Noncentral Location
 Q = Median, A Measure of Central Tendency
2

© 2002 Prentice-Hall, Inc.
Chap 3-28
Measures of Variation
Variation
Variance
Range
Population
Variance
Sample
Variance
Interquartile Range
© 2002 Prentice-Hall, Inc.
Standard Deviation
Coefficient
of Variation
Population
Standard
Deviation
Sample
Standard
Deviation
Chap 3-29
Range


Measure of variation
Difference between the largest and the
smallest observations:
Range  X Largest  X Smallest

Ignores the way in which data are distributed
Range = 12 - 7 = 5
Range = 12 - 7 = 5
7
8
© 2002 Prentice-Hall, Inc.
9
10
11
12
7
8
9
10
11
12
Chap 3-30
Interquartile Range


Measure of variation
Also known as midspread


Spread in the middle 50%
Difference between the first and third
quartiles
Data in Ordered Array: 11 12 13 16 16 17
17 18 21
Interquartile Range  Q3  Q1  17.5  12.5  5

Not affected by extreme values
© 2002 Prentice-Hall, Inc.
Chap 3-31
Variance


Important measure of variation
Shows variation about the mean

Sample variance:
n
S 
2

 X
i 1
X
i
2
n 1
Population variance:
N
 
2
© 2002 Prentice-Hall, Inc.
 X
i 1
i

N
2
Chap 3-32
Standard Deviation



Most important measure of variation
Shows variation about the mean
Has the same units as the original data

Sample standard deviation:
n
S

Population standard deviation:

© 2002 Prentice-Hall, Inc.
 X
i 1
X
i
2
n 1
N
 X
i 1
i

2
N
Chap 3-33
Comparing Standard Deviations
Data A
11 12 13 14 15 16 17 18 19 20 21
Mean = 15.5
s = 3.338
Data B
11 12 13 14 15 16 17 18 19 20 21
Mean = 15.5
s = .9258
Data C
11 12 13 14 15 16 17 18 19 20 21
© 2002 Prentice-Hall, Inc.
Mean = 15.5
s = 4.57
Chap 3-34
Coefficient of Variation

Measures relative variation

Always in percentage (%)

Shows variation relative to mean


Is used to compare two or more sets of data
measured in different units
S
CV  
X
© 2002 Prentice-Hall, Inc.

100%

Chap 3-35
Comparing Coefficient
of Variation

Stock A:



Stock B:



Average price last year = $50
Standard deviation = $5
Average price last year = $100
Standard deviation = $5
Coefficient of variation:

Stock A:

Stock B:
© 2002 Prentice-Hall, Inc.
S
CV  
X

 $5 
100%  
100%  10%

 $50 
S
CV  
X

 $5 
100%  
100%  5%

 $100 
Chap 3-36

Symmetric or skewed
Shape of a Distribution
Left-Skewed
Mean < Median < Mode
Symmetric
Mean = Median =Mode
Right-Skewed
Mode < Median < Mean
Exploratory Data Analysis

Box-and-whisker plot

Graphical display of data using 5-number summary
X smallest Q
1
4
© 2002 Prentice-Hall, Inc.
6
Median( Q2)
8
Q3
10
Xlargest
12
Chap 3-38
Distribution Shape and
Box-and-Whisker Plot
Left-Skewed
Q1
© 2002 Prentice-Hall, Inc.
Q2 Q3
Symmetric
Q1Q2Q3
Right-Skewed
Q1 Q2 Q3
Chap 3-39
Coefficient of Correlation

Measures the strength of the linear
relationship between two quantitative
variables
n
r
 X
i 1
n
 X
i 1
© 2002 Prentice-Hall, Inc.
i
i
 X Yi  Y 
X
2
n
 Y  Y 
i 1
2
i
Chap 3-40
Features of
Correlation Coefficient

Unit free

Ranges between –1 and 1



The closer to –1, the stronger the negative linear
relationship
The closer to 1, the stronger the positive linear
relationship
The closer to 0, the weaker any positive linear
relationship
© 2002 Prentice-Hall, Inc.
Chap 3-41
Scatter Plots of Data with
Various Correlation Coefficients
Y
Y
Y
X
r = -1
X
r = -.6
Y
© 2002 Prentice-Hall, Inc.
X
r=0
Y
r = .6
X
r=1
X
Chap 3-42
Pitfalls in
Numerical Descriptive Measures

Data analysis is objective


Should report the summary measures that best
meet the assumptions about the data set
Data interpretation is subjective

Should be done in fair, neutral and clear manner
© 2002 Prentice-Hall, Inc.
Chap 3-43
Ethical Considerations
Numerical descriptive measures:



Should document both good and bad results
Should be presented in a fair, objective and
neutral manner
Should not use inappropriate summary
measures to distort facts
© 2002 Prentice-Hall, Inc.
Chap 3-44
Chapter Summary

Described measures of central tendency

Mean, median, mode, geometric mean, midrange

Discussed quartile

Described measure of variation


Range, interquartile range, variance and standard
deviation, coefficient of variation
Illustrated shape of distribution

Symmetric, skewed, box-and-whisker plots
© 2002 Prentice-Hall, Inc.
Chap 3-45
Chapter Summary


(continued)
Discussed correlation coefficient
Addressed pitfalls in numerical descriptive
measures and ethical considerations
© 2002 Prentice-Hall, Inc.
Chap 3-46