Download Measures of Variability

POLS 7000X
STATISTICS IN POLITICAL SCIENCE
CLASS 3
BROOKLYN COLLEGE – CUNY
SHANG E. HA
Leon-Guerrero and Frankfort-Nachmias,
Essentials of Statistics for a Diverse Society
Chapter 3: Measures of Central Tendency
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The Mode
The Median
The Mean
Finding the Mean in a Frequency Distribution
The Shape of the Distribution
Considerations for Choosing a Measure of Central
Tendency
Statistics in Practice: Representing Income
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
What is a measure of Central
Tendency?
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Numbers that describe what is average or typical of the distribution
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You can think of this value as where the middle of a distribution lies.
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
The Mode
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The category or score with the largest frequency (or
percentage) in the distribution.
•
The mode can be calculated for variables with levels of
measurement that are: nominal, ordinal, or interval-ratio.
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
The Mode: An Example
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Example: Number of Votes for Candidates
for Mayor of Camarillo, California. The
mode, in this case, gives you the
“central” response of the voters: the most
popular candidate.
Sheriff Tupper – 11,769 votes
The Mode:
Jessica Fletcher – 39,443 votes
“Dr. Seth Hazlett”
Dr. Seth Hazlett – 78,331 votes
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
The Median
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The score that divides the distribution into
two equal parts, so that half the cases are
above it and half below it.
The median is the middle score, or
average of middle scores in a distribution.
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
The Mean

The arithmetic average
obtained by adding up all
the scores and dividing by
the total number of scores.
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Formula for the Mean
åY
Y=
N
Where ΣY = sum of all scores
N = the number of scores.
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
An Example
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Annual per capita carbon dioxide emissions (metric tons) for n
= 10 largest nations in population size
 Bangladesh 0.3, Brazil 1.8, China 2.3, India 1.2,
Indonesia 1.4, Pakistan 0.7, Russia 9.9, U.S. 20.1, Japan
1.4, Nigeria 0.6
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Ordered sample: 0.3, 0.6, 0.7, 1.2, 1.4, 1.4, 1.8, 2.3, 9.9,
20.1
Mode = 1.4
Median = (1.4 + 1.4)/2 = 1.4
Mean = (0.3 + 0.6 + 0.7 + …+ 20.1)/10 = 3.97
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Calculating the mean with
grouped scores
Y 
fY
N
where: f Y = a score multiplied by its frequency
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Example: Mean of Grouped Scores
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Grouped Data: the Mean & Median
Calculate the median and mean for the grouped frequency below.
Number of People Age 18 or older living in a U.S. Household in
1996 (GSS 1996)
Number of People
1
2
3
4
5
6
TOTAL
Frequency
190
316
54
17
2
2
581
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Shape of the Distribution
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Symmetrical (mean is about equal to median)
Skewed
 Negatively
(example: years of education)
mean < median
 Positively (example: income)
mean > median
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Bimodal (two distinct modes)
Multi-modal (more than 2 distinct modes)
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Distribution Shape
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Considerations for Choosing a Measure
of Central Tendency
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For a nominal variable, the mode is the only
measure that can be used.
For ordinal variables:
-Use the mode to show what is the most common
value in the distribution.
-Use the median to show which value is located
exactly in the middle of the distribution.
For interval-ratio variables, the mode, median, and
mean may all be calculated. The mean provides
the most information about the distribution, but the
median is preferred if the distribution is skewed.
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Central Tendency
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Chapter 4: Measures of Variability
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The Importance of Measuring Variability
The Range
The Inter-Quartile Range
The Variance and the Standard Deviation
Considerations for Choosing a Measure of Variation
Reading the Research Literature: Differences in
College Aspirations and Expectations Among Latino
Adolescents
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
The Importance of Measuring Variability
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Central tendency - Numbers that describe
what is typical or average (central) in a
distribution
Measures of Variability - Numbers that
describe diversity or variability in the
distribution.
These two types of measures together help us to sum up a distribution
of scores without looking at each and every score. Measures of central
tendency tell you about typical (or central) scores. Measures of
variation reveal how far from the typical or central score that the
distribution tends to vary.
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Notice that both distributions have the same
mean, yet they are shaped differently
The Range
Range = highest score - lowest
score

Range – A measure of variation in
interval-ratio variables. It is the difference
between the highest (maximum) and the
lowest (minimum) scores in the distribution.
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Percentiles
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A score below which a specific percentage
of the distribution falls.
example, the 75th percentile is a score
that divides the distribution so that 75% of
the cases are below it.
 For example, the 25th percentile is a score
that divides the distribution so that 25% of
the cases are below it.
 For
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Inter-Quartile Range
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Inter-Quartile Range (IQR) – A measure of
variation for interval-ratio data. It indicates the
width of the middle 50 percent of the
distribution and is defined as the difference
between the lower and upper quartiles (Q1
and Q3.)
IQR = Q3 – Q1
Q3 = 75th percentile
Q1 = 25th percentile
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
The Difference Between the Range
and IQR
These
values fall
together
closely
Shows greater
variability
Importance of
the IQR
Yet the ranges
are equal!
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Variance
• Variance – A measure of variation for
interval-ratio variables; it is the
average of the squared deviations from
the mean
2
s 
2
Y
 (Y Y )
N 1
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Standard Deviation
• Standard Deviation – A measure of variation
for interval-ratio variables; it is equal to the
square root of the variance.
2
sy  s 
2
Y
 (Y Y )
N 1
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Finding the Mean
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Finding the Standard Deviation
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Considerations for Choosing a Measure
of Variability
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For nominal variables, you can only use IQV
(Index of Qualitative Variation)
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Not discussed in this class!
For ordinal variables, you can calculate the
IQV or the IQR (Inter-Quartile Range). Though,
the IQR provides more information about the
variable.
For interval-ratio variables, you can use IQV,
IQR, or variance/standard deviation. The
standard deviation (also variance) provides the
most information, since it uses all of the values
in the distribution in its calculation.
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications