Transcript Levin & Fox, Chapter 2 Review

Organizing Data
Proportions, Percentages, Rates,
and rates of change.
Raw Data
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Often hard to interpret just a bunch of raw
scores
Raw scores can be transformed to show
patterns and trends in the data
Most useful is the frequency distribution or
table
Frequency Tables will have:
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Informative title
Two columns for nominal data:
(1) response and
(2) frequency (How often did certain
responses occur?)
Standardizing data
Proportion: compare the number of cases for
each response (frequency, f) with the total
number of cases (N).
Proportion =
frequency / number = f / N
 In the previous example, 20 out of 45 students
earned a B, so the proportion earning a B is
20/45 = .44444444, which (rounding to 2
decimals more than the original data) = .44
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Percentage is the frequency per 100
cases. (It is a special case of a
proportion.)
Percentage = 100 (f / N)
People are “used” to thinking in
percentages (such as in cents per
dollar....).
Example
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20 our of 45 students earned a B in a
course.
Proportion = f / N = 20/45 = 0.44
Percentage = 100 (20/35) = 44%
(Per cent means per 100, and we write it
0/0. Per thousand would be 0/00)
Ratios
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A ratio of “a” to “b” is the frequency of “a”
compared to the frequency of “b”, with the
frequency of “a” coming first, or in the
numerator, just as it does in the sentence.
a/b or sometimes expressed as a:b
Comparisons using the
Frequency Ratio: f1 / f2
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In a certain class, there were 15 women
and 30 men, in a class of 45. So, in the
class,
Proportion of women = 15/45 = 0.33
Percentage of women = (100).33 = 33%
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(note this is not 0.33%)
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Ratio – depends on how the
question is stated.
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Ratio of women to men = 15/30 = 1/2, or there
was 1 woman for every 2 men.
However, the ratio of men to women would be
30/15 = 2 men for every woman.
Note ratio is used differently than is the
proportion in the class.
Rate
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A rate indicates the number of actual
cases compared to the number of
potential cases. Pretty subtle, eh?
For population studies, these are usually
expressed as the number of actual cases
per 1000 potential cases (usually per 1000
people in the population).
Example
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A town has 5000 people, of whom 450 have
graduated from college.
The town’s college graduation rate is:
450/5000 = .09 = 9% or
90 per thousand.
(Why might I express this a per thousand? I
chose the “per” part so the number was
something easily visualized.)
What denominators to use?
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per 100 = percentage
per 1000 = commonly used for birth and
death rates, divorces, etc.
per 100,000 for lots of things determined
in the U.S. census
per 1,000,000 for things determined
worldwide
Generalization
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Use the denominator that gives you the
simplest whole number, easiest for you to
grasp. Usually this is a number between 1
and 100.
It’s hard for people to visualize the
meaning of very small or large numbers
such as 0.00123, or 132,431,000
Mortality Rates for example
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Mortality Rates per 1000 among blacks &
whites in Baltimore in 1972 were
for whites, 15.2 per 1000 (or 1.52%)
for blacks, 9.8 per 1000 (or 0.98%)
Easier to visualize than .0152 for whites
and .0098 for blacks. Do you agree?
Powers of 10 Review
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Suppose a disease rate of .000567 per
person (per capita).
To convert into something more
comprehensible, move the decimal point
to the right 4 places, to 5.67.
4 places = 10,000 (4 zeroes),
so this becomes 5.67 per 10,000. or go
one step further to 56.7 per 100,000.
Rates of change
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(100) Rate 2 – Rate 1 / Rate 1
then convert into the proper units (per 100, 1000, etc.)
Ex: a town’s population increases from 20,000 to 30,000
between 1990 and 2005 (note: rate of change can be
positive or negative)
(100) time2f - time1f = (100) 30,000-20,000 = 50%
time 1f
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Increase of 50%
20,000
“Organizing the Data”
Review of: Frequency
Distributions & Histograms
Frequency Distributions
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List or plot data
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Nominal Data -- in any order
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Ordinal & Interval Data – Usually highest
number at top of table to lowest number
at bottom of the table
Statistics Class Height Data
Plotted from shortest to tallest
Height
80
70
60
50
40
30
20
10
0
Perez,
Hood, Carl
Poyer,
Yumol,
Chang,
Hodgson,
Shiroma,
Delosrey,
Companion,
Brown,
Sebastian,
Height
Intervals – Grouping Data
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range of values in the data set
numbers of class intervals desired
size of class interval
upper limit of a class interval
lower limit of a class interval
Statistics Class Height Data
Grouped in 2 inch intervals
8
6
4
Series1
2
0
58- 60- 62- 64- 66- 68- 70- 7259 61 63 65 67 69 71 73
4” intervals
14
12
10
8
6
4
2
0
Series1
58-61
62-65
66-69
70-73
6” intervals
15
10
Series1
5
0
58 - 63
64 - 69
71-73
Cumulative
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Cumulative Frequencies: number of cases
at or below a given score.
Cumulative Percentages: percent of cases
at or below a given score.
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Also = “percentile rank”
Class Limits
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Upper class limit = the highest possible score
which would “round down” to be included in that
class.
Lower class limit = the lowest possible score
which would “round up” to be included in that
class.
Midpoints of Intervals
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Lowest possible score for that interval
plus highest possible score value
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Divided by 2
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Midpoints
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The interval of 58-61” actually has limits from
57.5 to 61.5, so 57.5 + 61.5 = 119
119/2 = 59.5 is the midpoint.
Yes, we’d usually get the same answer by saying
(58 + 61) / 2 however, for irregular classes, it
is better if we get used to the lowest value being
57.5 and the highest being 61.5.
Cumulative Frequency
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To expand our frequency table, add columns for
cumulative frequency, percent, and cumulative
percent.
Arrange your scores from low at the bottom to
high at the top. Then, the Cumulative Frequency
is simply the frequency of scores at or below the
value in question.
Percentile Rank
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= the cumulative percentage
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The % at or below that score
So for a height of 5’4”, or 64”, what is the
percentile rank in our height data?
The following chart shows frequency, cum.
freq., percentage, & cumulative %.
2"
intervals
f
cf
%
cum%
72-73
1
32
3.10%
100.00%
70-71
4
31 12.50%
96.88%
68-69
4
27 12.50%
84.38%
66-67
4
23 12.50%
71.87%
64-65
6
19 18.80%
59.37%
62-63
7
13 21.90%
40.62%
60-61
3
6
9.40%
18.75%
58-59
3
3
9.40%
9.40%
Percentile Rank
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64-65” has a cumulative percent of 59.37%, so
59.37% of class is in this category or shorter
than this category.
62-63 “ has a cumulative percent of 40.62%, so
40.62% of class is in this category or shorter
than this category
So, percentile rank = cumulative percent when
looking at the raw data -- but it is more
complex for grouped data, so be wary.
Cross-tabulations
Cross-Tabulation:
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Cross-tabulation review:
a table which presents the distribution of
one variable (frequency and/or %) across
the categories of one or more additional
variables.
Common Cross-Tab Example
L&F chpt 2
TABLE 2.15
Cross-Tab of Seat Belt Use by Gender
Gender Respondent
Use of
Seat Belt
Male
Female Total
Always
144
355
499
most...
66
110
176
some...
58
66
124
seldom
39
44
83
never
60
55
115
Total
367
630
997
Cross-Tab: Table 2.15
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If asking questions about the differences
between males & females in seat belt use,
use column percents.
If asking questions about different uses of
seat belts by the population as a whole,
use the row percents.
Hint: If totals are not given -- put them in
before you start to evaluate.
Cross-Tab: Table 2.15
L&F chpt 2
TABLE 2.15
Cross-Tab of Seat Belt Use by Gender
Gender Respondent
Use of
Seat Belt
Male
Female Total
Always
144
355
499
most...
66
110
176
some...
58
66
124
seldom
39
44
83
never
60
55
115
Total
367
630
997
Data Format on SPSS
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Note that when you are working with raw
data sets on the computer, you will put
each case in a row, rather than making a
cross-tabulation table. We will do this
when we work with SPSS.