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Organizing the Data
Chapter 2
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2. Frequency Distributions of Nominal Data
Formulas and statistical techniques used by social researchers to:
Organize raw data
Test hypotheses
Raw data is often difficult to synthesize
Frequency tables make raw data easier to understand
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3. Frequency Distribution of Nominal Data
Frequency distribution of nominal data consists of two columns:
Left column has characteristics (e.g., Response of Child)
Right column has frequency (f)
Table Responses of Young Boys to Removal of Toy
Response of Child f
Cry 25
Express Anger 15
Withdraw 5
Ply with another toy
N=50
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4. Comparing Distributions
Comparisons clarify and add information
Table Response to Removal of Toy by Gender of Child
Gender of Child
Response of Child
Male
Female
Cry 25 28
Express Anger 15 3
Withdraw 5 4
Play with another toy
Total 50
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5. Proportions and Percentages
Can we standardize frequency distributions for size?
Proportion and percentage – two most popular methods
Proportion – number of cases compared to the total size of distribution
Most prefer percentages to show relative size.
Percentage – the frequency per 100 cases
Formula for proportion
Formula for percentage
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6. Use Percentages for Unequal Populations
Table 2.3 Gender of Students Majoring in Engineering at Colleges A and B
Gender of Student
Engineering Majors
College A
College B f %
Male
1,082 80
146
Female
270 20 37
Total
1,352
100
183
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7. Copyright © Pearson Education, Inc., Allyn & Bacon 2010
Ratios and Rates
Ratios also standardize for size.
Ratio – comparison one category to another
Rate usually preferred by social researchers
Rate – comparison between actual and potential cases
Base terms in rates may vary
Formula for ratio
Formula for rate
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8. Some Common Rate Calculations
Suppose 500 births occur among 4,000 women of childbearing age. This would be a rate of 125 live births for every 1,000 women of childbearing age.
Suppose 562 suicides occur in a state with 4.6 million residents. The suicide rate would be 12.2 suicides per 100,000 residents.
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9. Distributions of Ordinal and Interval Data
Nominal distributions can be listed in any order.
Table 2.4 The Distribution of Marital Status Shown Three Ways
Marital Status f
Married 30
Single 20
Previously married 10
Total 60
Ordinal and interval level must be in order.
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10. Distributions of Ordinal and Interval Data
Ordinal and interval level must be in sequence order.
Table 2.5 A Frequency Distribution of Attitudes toward a Proposed Tuition Hike on a College Campus: Incorrect and Correct Presentations
Attitude toward
a Tuition Hike f
Slightly favorable 2
Strongly favorable
Somewhat unfavorable 21
Somewhat favorable 1
Slightly unfavorable 4
Strongly unfavorable 10
Total 38
INCORRECT
CORRECT
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11. Grouped Frequency Distributions of Interval Data
Grouped frequency distribution used to clarify presentation of data.
Categories or groups referred to a class intervals
Class interval size determined by the number of values
Table 2.6 shows how difficult it is to read ungrouped frequency distributions.
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Table 2.6 Frequency Distribution of Final-Examination Grades
for 71 Students
Grade f 99 85 2 71 4 57 98 1 84 70 9 56 97 83 69 3 55 96 82 68 5 54 95 81 67 53 94 80 66 52 93 79 8 65 51 92 78 64 50 91 77 63
N = 71 90 76 62 89 75 61 88 74 60 87 73 59 86 72 58
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13. Grouped Frequency Distributions of Interval Data
Table 2.7 Grouped Frequency Distribution of Final-Examination Grades for 71 Students
Class Interval f %
95-99 3
4.23
90-94 2
2.82
85-89 4
5.63
80-84 7
9.86
75-79 12
16.90
70-74 17
23.94
65-69
60-64 5
7.04
55-59
50-54 71
100a
a The percentages as they appear add to only 99.99%. We write the sum as 100% instead, because we know that .01% was lost in rounding.
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Class Limits
Class limits include consideration of “rounding”
Class limits are at the point halfway between adjacent intervals
Upper and lower limits
Distance from upper to lower limit determines the size of class interval
where i = size of a class interval
U = upper limit of a class interval
L = lower limit of a class interval
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The Midpoint
Class intervals also have a midpoint.
Midpoint – sum of lowest and highest score value divided by two
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16. Guidelines for Constructing Class Intervals
Categories must be mutually exclusive
Designed to reveal or emphasize patterns
Possible to have too few or too many groups – blurs the data
Whole numbers preferable to decimals
Lowest scores are typically multiples
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17. Cumulative Distributions
Cumulative frequencies – total number of cases having a given score or a score that is lower
Cumulative frequency shown as cf
cf obtained by the sum of frequencies in that category plus all lower category frequencies
Cumulative percentage – percentage of cases having any score or a lower score
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Table 2.8 Cumulative Frequency (cf) Distribution of PSAT Scores for 336 Students
Class Interval f % cf
75-79 4
1.19
336
70-74 24
7.14
332
65-69 28
8.33
308
60-64 30
8.93
280
55-59 35
10.42
250
50-54 55
16.37
215
45-49 61
18.15
160
40-44 48
14.29 99
35-39 51
30-34 12
3.57 21
25-29 6
1.79 9
20-24 3
.89
Total
100
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Table 2.9 Cumulative Percentage (c%) Distribution of PSAT Scores for 336 Students (Based on Table 2.8)
Class Interval f % cf c%
75-79 4
1.19
336
100.00
70-74 24
7.14
332
98.81
65-69 28
8.33
308
91.67
60-64 30
8.93
280
83.33
55-59 35
10.42
250
74.40
50-54 55
16.37
215
63.99
45-49 61
18.15
160
47.62
40-44 48
14.29 99
29.46
35-39 51
15.18
30-34 12
3.57 21
6.25
25-29 6
1.79 9
2.68
20-24 3
.89
Total
100
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Percentile Ranks
Percentile rank – the percentage of the cases falling at or below given score
Deciles – points that divide a distribution into ten equal portions
Quartiles - points that divide a distribution into quarters
Median – the point that divides a distribution into halves
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More on Class Limits
Not always logical to set class limits as defined earlier
Table Two Approaches for Establishing Class Limits
Acceptable Method
Better Method
Score Values
Lower Limit m
90 up to 100
89.5
94.5 90 95
80 up to 90
79.5
84.5 80 85
70 up to 80
69.5
74.5 70 75
60 up to 70
59.5
64.5 60 65
50 up to 60
49.5
54.5 50 55
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22. Flexible Class Intervals
Intervals are not always equal
Can be open-ended on the top of bottom
Midpoint calculations can be tricky
Use common sense
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Cross-Tabulations
Frequency distributions are limited.
Sometimes we want to know how is one variable (usually the dependent variable) distributed across another (usually independent variable).
Cross-tabulations, or cross-tabs, meet this need.
Cross-tabs allow us to consider two or more dimensions of data.
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Cross-tab example
Table 2:16 Frequency Distribution of Seat Belt Use
Use of Seat Belts f %
All the time
499
50.1
Most of the time
176
17.7
Some of the time
124
12.4
Seldom 83
8.3
Never
115
11.5
Total
997
100
Table 2:17 Cross-Tabulation of Seat Belt Use
by Gender
Gender of Respondents
Use of Seat Belts
Male
Female
Total
All the time
144
355
499
Most of the time 66
110
176
Some of the time 58
124
Seldom 39 44 83
Never 60 55
115
367
630
997
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25. (column totals or N column)
Table 2:18 Cross-Tabulation of Seat Belt Use by Gender with Total Percents
Use of Seat Belts
Gender of Respondent
Total
Male
Female
All of the time
144
355
499
14.4%
35.6%
50.1%
Most of the time 66
110
176
6.6%
11.0%
17.7%
Some of the time 58
124
5.8%
12.4%
Seldom 39 44 83
3.9%
4.4%
8.3%
Never 60 55
115
6.0%
5.5%
11.5%
367
630
997
36.8%
63.2%
100.0%
Row marginal
(row totals or N row)
Column marginal
(column totals or N column)
Total sample size
(N total)
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Cross-tab Formulas
Formula for total percents
Formula for row percents
Formula for column percents
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27. Choosing Total, Row, and Column Percents
Which are correct?
Which are preferred?
If the independent variable is on the rows, use row percents.
If the independent variable is on the columns, use column percents.
If the independent variable is unclear – use whichever is most meaningful.
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28. Graphic Presentations
Graphs are useful tools to emphasize certain aspects of data.
Many prefer graphs to tables.
Types of graphs include:
Pie charts, bar graphs, frequency polygons, line charts, and maps
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Pie Charts
Pie chart – a circular chart whose pieces add up to 100%.
Especially good for nominal data.
Possible to highlight or “explode” certain pieces for emphasis
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Pie Chart Example
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31. Exploded Pie Chart Example
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32. Bar Graphs and Histograms
Far more widely used in social research
Can be used with frequencies or percentages
Small difference between bar graphs and histograms
Bar graphs used primarily for discrete variables
Histograms used to show continuity along a scale
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Bar Graph Examples
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Bar Graph Examples
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Frequency Polygons
Frequency polygon best suited to emphasize continuity rather than differences.
Frequency distribution of a single variable
Useful for ordinal and interval data
Shape of frequency distribution is also revealed – kurtosis and skewness
Leptokurtic – peaked distribution
Platykurtic – flat distribution
Mesokurtic – neither peaked or flat
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36. Frequency Polygon Example
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Kurtosis
Leptokurtic
Platykurtic
Mesokurtic
Figure Some Variation in Kurtosis among Symmetrical Distributions
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Skewness
Negatively skewed
Positively skewed
Symmetrical
Figure Three Distributions Representing Direction of Skewness
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Line Charts
Display changes in a variable of variables between groups over time
Time trend data are far more customarily depicted with line charts.
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Maps
As graphics software has matured, maps had become prevalent.
Especially common in government publications
Hot spot crime analysis
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Summary
Organizing raw data is critical
Data can be summarized using frequency distributions.
Comparisons of groups possible through proportions, percentages and rates.
Cross-tabs allow dimensional (and more) analysis
Graphic presentations:
help to emphasize findings
make data more accessible to consumers of research
help researchers identify trends
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