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Frequency Distribution
I. How Many People Made Each
Possible Score?
A. This is something I show you
for each quiz.
U.S. Distribution of Income Over
Time
Share of Income:
Richest 1%
1920
14.4%

Poorest 20%
NA
1970
7.8%
4.2%
2007
18.0%
3.4%
Measurement I. Quantifying and
Describing Variables -1

Four Levels of Measurement

Measures of Central Tendency
Mode
 Median
 Mean

Measurement I. Quantifying
and Describing Variables - 2
DON’T WRITE – JUST READ!
We need to review the levels of
measurement before continuing. Since
you have definitions for the levels of
measurement on pages 10-12 of the
300Reader, there is NO REASON to
write the following definitions
Four Levels of Precision For
Measuring Variables

Nominal Measure: You can put cases
into a category, but cannot specify an
order or relationship between the
categories.

Example: The variable “religion” can take on
values such as Catholic, Protestant,
Mormon, Jewish, etc.
Four Levels of Precision For
Measuring Variables

Ordinal Measure: You can put cases
into different categories, and order the
categories.

Example: The variable “strength of religious
belief” can take on values such as devoutly
religious, fairly religious, slightly religious,
not religious.
Four Levels of Precision For
Measuring Variables

Interval Measure: Not only can you
order the categories of the variable, you
can specify the difference between any
two categories.

Example. The variable “temperature on the
Fahrenheit scale” can take on values such
as 32 degrees, 74 degrees, 116 degrees.
Four Levels of Precision For
Measuring Variables

Ratio Measure: You can order
categories, specify the difference
between two categories, and the value of
zero on the variable represents the
absence of the variable.

Example. The variable “annual income” can
take on the values of $0, $98,000, or
$694,294,129.
Example of a Ratio Measure:
Income Inequality
The next several slides what groups in the
United States, Japan and Sweden
think is the actual and fair degree of
income inequality between an executive
and an auto worker. Since a score of
zero equals the absence of income, and
the difference between $1 and $2 is the
same as between $1,000 and $1,001,
we have a ratio level measure.
United States
Perceived
Income
Business
15.1/1
Labor
14.8/1
Republicans 13.2/1
Democrats
15.4/1
Youth
13.4/1
Fair
Income
15.6/1
7.2/1
11.3/1
8.2/1
6.0/1
Japan
Business
Labor
Conservative
Party
Left Parties
Perceived
Income
9.1/1
10.1/1
7.1/1
Fair
Income
8.6/1
4.1/1
5.4/1
10.3/1
3.7/1
Sweden
Business
Labor
Conservative/
Center Party
Left Party
Perceived
Income
2.4/1
3.2/1
2.2/1
3.2/1
Fair
Income
3.5/1
1.9/1
2.1/1
1.9/1
Measures of Central Tendency -1
Kobe Bryant
$24.8 million
Paul Gasol
$17.8 million
Andrew Bynum $13.7 million
Lamar Odom
$8.2 million
Ron Artest
$6.3 million
Luke Walton
$5.2 million
Steve Blake
$4.0 million
Derek Fisher
$3.7 million
Shannon Brown $2.1 million
Matt Barnes
$1.7 million
Theo Ratliff
$1.3 million
Joe Smith
$1.3 million
Devin Ebanks
$0.4 million
Derrick Caracter $0.4 million
Measures of Central Tendency - 2

Mode: The most frequently occurring value.


$1.3 million and $0.4 million
Median: The midpoint of the distribution of
cases.




1. Arrange cases in order
2. If the number of cases is odd, median is the
value taken on by the case in the center of the list.
3. If the number of cases is even, median is the
average of the two center values. $3.85 million
(4.0 + 3.7 = 7.7 and 7.7/2 = 3.85)
Measures of Central Tendency - 3
DO NOT WRITE ANY OF THE
FORMULAS THAT APPEAR AHEAD!
THEY ARE IN THE 300READER AND
ONLY APPEAR AHEAD FOR
PURPOSES OF DISCUSSION.
Measures of Central Tendency - 4
Mean is the arithmetic average of the values
that all the cases take on.
Formula: Add up all the values and divide this
sum by the number of cases, N. In our Laker
example, $6.5 million.
X 1  X 2  ...  XN  Xi
Mean  X 

N
N
Measures of Central Tendency - 5
Bush Tax Cut Example
A. The “Mean” household tax cut under the Bush Tax
Cuts is $1,199 while the “Median”
household only receives $217.
1. Quite a Difference!
2. 75% of Households Actually Lose Under
the Bush Tax Cuts
Measures of Central Tendency - 6
Question: Why can’t the mean tell us
everything?
Answer: While the mean tells us the
average, it does NOT tell us how
accurate the mean is when making
predictions.
Measures of Dispersion - 1
Thus, we need to know whether the mean
occurred because many scores were
quite close to the mean or was the mean
an average of scores quite different than
the mean?
This is what measures of dispersion tell us.
DON’T WRITE THE FORMULAS AHEAD!!
Measures of Dispersion - 2
The variance is a measure of how spread
out cases are, calculated by:
Compute the distance from each case to
the mean, then square that distance.
Find the sum of these squared distances,
then divide it by N-1. Lakers: $54.1million.
(X  X )

Variance 
i
N 1
2
Measures of Dispersion - 3
The standard deviation is the square root
of the variance. For the Laker data, $7.4
million.
s  ˆ 
(X  X )
i
N 1
2
Measures of Dispersion - 4
According to page 25 of the 300 Reader,
what conclusion should I draw about the
dispersion of the Laker data when the
mean is 6.5 and the standard deviation is
7.4?
Normally Distributed Curve
Skewed Distributions
Characteristics of the Normal
Distribution -1
It is symmetrical -- Half the cases are to one
side of the center; the other half is on the
other side.
The distribution is single peaked, not bimodal or
multi-modal
Most of the cases will fall in the center portion of
the curve and as values of the variable
become more extreme they become less
frequent, with “outliers” at each of the “tails” of
the distribution few in number.
Characteristics of the Normal
Distribution -2
It is only one of many frequency
distributions but the one we will focus on
for most of this course.
The Mean, Median, and Mode are the
same.
Percentage of cases in any range of the
curve can be calculated.
Family of Normal Curves
Summarizing Distributions
Two key characteristics of a frequency
distribution are especially important when
summarizing data or when making a prediction
from one set of results to another:
 Central Tendency




What is in the “Middle”?
What is most common?
What would we use to predict?
Dispersion


How Spread out is the distribution?
What Shape is it?
Appropriate Measures of
Central Tendency

Nominal variables
Mode

Ordinal variables
Median

Interval level variables
Mean
- If the distribution is normal (median
is better with skewed distribution)
Logic of Measures of Dispersion 1
Why not think of dispersion as the
difference between the highest and
lowest scores?
Logic of Measures of Dispersion 2
If we need a measure based upon all the
scores, why not just subtract the mean
from each score, add up this total and
divide by the total number of scores?
Logic of Measures of Dispersion 3
If the sum of all the deviations from the
mean is zero, we could take the absolute
value of each deviation from the mean
and avoid this problem. So, why don’t
we?
So, what do we do?
Logic of Measures of Dispersion 4
What is the utility of Tchebysheff’s
Theorem?