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Describing Data: Numerical
Presented by: Mahendra AN
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 EkonomiBagian pertama edisi 3, BPFE, 2001
Frequency distribution
Histogram
Frequency Distribution Basic Characteristics
A B
X X B
A B
XAB Central tendency differenceA Variability
A Central tendency difference Variability
Skewness Peakedness of
kurtosis
Chapter 4 (Part A) Chapter 4 (Part A)
Descriptive Statistics: Numerical Measures Descriptive Statistics: Numerical Measures
Measures of LocationMeasures of Location
Measures of VariabilityMeasures of Variability
Numerical Data Properties
Standard Deviation
Coeff. of Variation Variation
Skew
Kurtosis Shape
Measures of Central Tendency
Central Tendency
M M di M d
Overview
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 3-5
Mean Median Mode
n
Midpoint of ranked values
Most frequently observed value Arithmetic
average
Notation…
When referring to the number of observations in a
population, we use uppercase letter N
When referring to the number of observations in a
sample, we use lower case letter n
Copyright © 2005 Brooks/ Cole, a division of Thomson Learning, I nc. 4.6 The arithmetic mean for a population is denoted with Greek letter “mu”:
The arithmetic mean for a sample is denoted with an “x-bar”:
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Arithmetic Mean
The arithmetic mean (mean) is the most common measure of central tendency
For a population of N values:
N
∑
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 3-7
For a sample of size n:
Sample size
n
Population size Population values
Arithmetic Mean
The most common measure of central tendency
Mean = sum of values divided by the number of values
Affected by extreme values (outliers)
(continued)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 3-8 0 1 2 3 4 5 6 7 8 9 10
Statistics is a pattern language…
Population Sample
Size
N
n
Copyright © 2005 Brooks/ Cole, a division of Thomson Learning, I nc. 4.9
Mean
Arithmetic Mean for Grouped Data
zLong method
zShort method
Weight-Average Mean
zGiving weight for each data to make the
data smoother
zThis technique is used to fixing
heteroscedacity problem heteroscedacity problem
zNeed a rationalization to decide the
“weight”
Geometric Mean
zBase on all of observation value
zOnly for positive values
zGM is used if growth is counted in mean
Or
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Harmonic Mean
zBase on all of observation value
zCannot be used if one of the variables
have zero value
HM i d if ti d th t h
zHM is used if ratio and the numerator have
same value
Median
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 3-14 Not affected by extreme values
Median = 3 Median = 3
Finding the Median
The location of the median:
data ordered the in position 2
1 n position
Median = +
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 3-15
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
Note that is not the valueof the median, only the
positionof the median in the ranked data 2
1 n+
Grouped Data Median
zGDMe is used for counting grouped data
series
Mode
A measure of central tendency
Value that occurs most often
Not affected by extreme values
Used for either numerical or categorical data
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 3-17 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
0 1 2 3 4 5 6
No Mode
Grouped Data Mode
zGDMo is used for counting grouped data
series
zData present in freq. distribution
