Meeting 05 Describing Data Numeric
3/12/2011
Frequency distribution
Histogram
7
6
F re q u e n c y
Describing Data: Numerical
Presented by:
Mahendra AN
Curve
6
5
5
4
4
3
3
2
1
0
2
0
5
0
15
25
36
45
55 More
Sources:
http://business.clayton.edu/arjomand/business/stat%20presentations/busa3101.html
http://wweb.uta.edu/insyopma/baker/Courses.html
Anderson, Sweeney,wiliams, Statistics for Business and Economics , 10 e, Thomson, 2008
Djarwanto, Statistik Sosial Ekonomi Bagian pertama edisi 3, BPFE, 2001
Chapter 4 (Part A)
Frequency Distribution Basic
Characteristics
A
Descriptive Statistics: Numerical Measures
B
A B
Numerical Data
Properties
XAB
Variability
X
XB
Central
A tendency difference
Central
Tendency
A
B
A
B
XA
Measures of Location
Measures of Variability
XB
Skewness
XA
B
Peakedness of
kurtosis
Measures of Central Tendency
Overview
Variation
Shape
Mean
Range
Skew
Median
Kurtosis
Mode
Interquartile
Range
Variance
Midrange
Standard Deviation
Midhinge
Coeff. of Variation
Slide 4
Notation…
When referring to the number of observations in a
population, we use uppercase letter N
Central Tendency
When referring to the number of observations in a
sample, we use lower case letter n
M
Mean
∑x
M di
Median
M d
Mode
The arithmetic mean for a population is denoted with Greek
letter “mu”:
n
x=
i=1
i
n
Arithmetic
average
Midpoint of
ranked values
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
The arithmetic mean for a sample is denoted with an
“x-bar”:
Most frequently
observed value
Chap 3-5
Copyright © 2005 Brooks/ Cole, a division of Thomson Learning, I nc.
© Mahendra Adhi Nugroho, M.Sc, Accounting Program Study of Yogyakarta State University
For internal use only!
4.6
1
3/12/2011
Arithmetic Mean
Arithmetic Mean
(continued)
The arithmetic mean (mean) is the most
common measure of central tendency
For a population of N values:
∑x
N
μ=
i =1
i
N
=
x1 + x 2 + L + x N
N
Population
values
0 1 2 3 4 5 6 7 8 9 10
∑ xi
n
i=1
n
0 1 2 3 4 5 6 7 8 9 10
Population size
Mean = 3
For a sample of size n:
x=
The most common measure of central tendency
Mean = sum of values divided by the number of values
Affected by extreme values (outliers)
=
x1 + x 2 + L + x n
n
1 + 2 + 3 + 4 + 5 15
=
=3
5
5
Observed
values
Mean = 4
1 + 2 + 3 + 4 + 10 20
=
=4
5
5
Sample size
Chap 3-7
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Statistics is a pattern language…
Size
Population
Sample
N
n
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 3-8
Arithmetic Mean for Grouped Data
z Long method
z Short method
Mean
Copyright © 2005 Brooks/ Cole, a division of Thomson Learning, I nc.
4.9
Weight-Average Mean
Geometric Mean
z Giving weight for each data to make the
data smoother
z This technique is used to fixing
heteroscedacity problem
z Need a rationalization to decide the
“weight”
z Base on all of observation value
z Only for positive values
z GM is used if growth is counted in mean
Or
© Mahendra Adhi Nugroho, M.Sc, Accounting Program Study of Yogyakarta State University
For internal use only!
2
3/12/2011
Harmonic Mean
Median
z Base on all of observation value
z Cannot be used if one of the variables
have zero value
z HM iis used
d if ratio
ti and
d th
the numerator
t h
have
same value
In an ordered list, the median is the “middle”
number (50% above, 50% below)
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10
Median = 3
Median = 3
Not affected by extreme values
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Grouped Data Median
Finding the Median
z GDMe is used for counting grouped data
series
The location of the median:
Median position =
n +1
position in the ordered data
2
If the number of values is odd, the median is the middle number
If the number of values is even, the median is the average of
the two middle numbers
n +1
is not the value of the median, only the
2
position of the median in the ranked data
Note that
Chap 3-15
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Grouped Data Mode
Mode
Chap 3-14
z GDMo is used for counting grouped data
series
z Data present in freq. distribution
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
0 1 2 3 4 5 6
No Mode
Chap 3-17
© Mahendra Adhi Nugroho, M.Sc, Accounting Program Study of Yogyakarta State University
For internal use only!
3
Frequency distribution
Histogram
7
6
F re q u e n c y
Describing Data: Numerical
Presented by:
Mahendra AN
Curve
6
5
5
4
4
3
3
2
1
0
2
0
5
0
15
25
36
45
55 More
Sources:
http://business.clayton.edu/arjomand/business/stat%20presentations/busa3101.html
http://wweb.uta.edu/insyopma/baker/Courses.html
Anderson, Sweeney,wiliams, Statistics for Business and Economics , 10 e, Thomson, 2008
Djarwanto, Statistik Sosial Ekonomi Bagian pertama edisi 3, BPFE, 2001
Chapter 4 (Part A)
Frequency Distribution Basic
Characteristics
A
Descriptive Statistics: Numerical Measures
B
A B
Numerical Data
Properties
XAB
Variability
X
XB
Central
A tendency difference
Central
Tendency
A
B
A
B
XA
Measures of Location
Measures of Variability
XB
Skewness
XA
B
Peakedness of
kurtosis
Measures of Central Tendency
Overview
Variation
Shape
Mean
Range
Skew
Median
Kurtosis
Mode
Interquartile
Range
Variance
Midrange
Standard Deviation
Midhinge
Coeff. of Variation
Slide 4
Notation…
When referring to the number of observations in a
population, we use uppercase letter N
Central Tendency
When referring to the number of observations in a
sample, we use lower case letter n
M
Mean
∑x
M di
Median
M d
Mode
The arithmetic mean for a population is denoted with Greek
letter “mu”:
n
x=
i=1
i
n
Arithmetic
average
Midpoint of
ranked values
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
The arithmetic mean for a sample is denoted with an
“x-bar”:
Most frequently
observed value
Chap 3-5
Copyright © 2005 Brooks/ Cole, a division of Thomson Learning, I nc.
© Mahendra Adhi Nugroho, M.Sc, Accounting Program Study of Yogyakarta State University
For internal use only!
4.6
1
3/12/2011
Arithmetic Mean
Arithmetic Mean
(continued)
The arithmetic mean (mean) is the most
common measure of central tendency
For a population of N values:
∑x
N
μ=
i =1
i
N
=
x1 + x 2 + L + x N
N
Population
values
0 1 2 3 4 5 6 7 8 9 10
∑ xi
n
i=1
n
0 1 2 3 4 5 6 7 8 9 10
Population size
Mean = 3
For a sample of size n:
x=
The most common measure of central tendency
Mean = sum of values divided by the number of values
Affected by extreme values (outliers)
=
x1 + x 2 + L + x n
n
1 + 2 + 3 + 4 + 5 15
=
=3
5
5
Observed
values
Mean = 4
1 + 2 + 3 + 4 + 10 20
=
=4
5
5
Sample size
Chap 3-7
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Statistics is a pattern language…
Size
Population
Sample
N
n
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 3-8
Arithmetic Mean for Grouped Data
z Long method
z Short method
Mean
Copyright © 2005 Brooks/ Cole, a division of Thomson Learning, I nc.
4.9
Weight-Average Mean
Geometric Mean
z Giving weight for each data to make the
data smoother
z This technique is used to fixing
heteroscedacity problem
z Need a rationalization to decide the
“weight”
z Base on all of observation value
z Only for positive values
z GM is used if growth is counted in mean
Or
© Mahendra Adhi Nugroho, M.Sc, Accounting Program Study of Yogyakarta State University
For internal use only!
2
3/12/2011
Harmonic Mean
Median
z Base on all of observation value
z Cannot be used if one of the variables
have zero value
z HM iis used
d if ratio
ti and
d th
the numerator
t h
have
same value
In an ordered list, the median is the “middle”
number (50% above, 50% below)
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10
Median = 3
Median = 3
Not affected by extreme values
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Grouped Data Median
Finding the Median
z GDMe is used for counting grouped data
series
The location of the median:
Median position =
n +1
position in the ordered data
2
If the number of values is odd, the median is the middle number
If the number of values is even, the median is the average of
the two middle numbers
n +1
is not the value of the median, only the
2
position of the median in the ranked data
Note that
Chap 3-15
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Grouped Data Mode
Mode
Chap 3-14
z GDMo is used for counting grouped data
series
z Data present in freq. distribution
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
0 1 2 3 4 5 6
No Mode
Chap 3-17
© Mahendra Adhi Nugroho, M.Sc, Accounting Program Study of Yogyakarta State University
For internal use only!
3