Summarizing a pattern involving
many numbers: Generalization,
example, exception (“GEE”)
Jane E. Miller, PhD
The Chicago Guide to Writing about Multivariate Analysis, 2nd edition.
Overview
• Before watching this podcast, watch podcasts on
– Reporting one #
– Comparing two numbers or series of numbers.
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Common problems with summarizing patterns
Systematic approach to summarizing patterns
Illustrative example
Suggestions for wording
The Chicago Guide to Writing about Multivariate Analysis, 2nd edition.
Describing complex patterns
• Relationships among three variables, e.g.,
– Multiple-line or clustered bar chart
– Table with several panels or nested columns
• Want to summarize the big picture
– Convey the pattern that addresses the underlying
research question for which the numbers are
evidence
The Chicago Guide to Writing about Multivariate Analysis, 2nd edition.
Common problems in summarizing
patterns among many numbers
• Two opposite types of problems:
– Report every one of the numeric values in the text.
• Too many numbers!
– Very difficult to organize lots of individual numeric values in
prose.
• Lose the forest for the trees.
– Hard to see the overall pattern.
– Report an arbitrary number from the table or chart.
• Unlikely to be representative of the overall pattern.
• Too few numbers.
The Chicago Guide to Writing about Multivariate Analysis, 2nd edition.
The “Generalization, example,
exception” approach
The Chicago Guide to Writing about Multivariate Analysis, 2nd edition.
Generalization, example, exception (“GEE”)
• Generalization:
– Try to find and describe an overall pattern, rather than
going through every cell in the table or point on the chart.
• The detailed numbers are what the table or chart is for!
• In your write-up, digest and synthesize the patterns.
• Example:
– Give 1 or 2 representative numeric contrasts to illustrate
generalization.
• Exception(s):
– If parts of the table or chart deviate from your
generalization, describe that departure.
– Give a numeric example to illustrate the exception.
The Chicago Guide to Writing about Multivariate Analysis, 2nd edition.
Complementary use of prose
with table or chart
• Create a table or chart to show all of the values
involved in the pattern.
• Provide your readers with the data to verify your
summary.
• Tables and charts are much better than prose for
organizing a lot of individual numbers.
• Write a narrative description of the pattern.
• Refer to the table or chart by name as you report
numeric examples to document that pattern.
The Chicago Guide to Writing about Multivariate Analysis, 2nd edition.
• Topic: prevalence rates
of drug and alcohol
abuse for
– Four substances (x-axis)
– Three age groups (legend)
• Aim to summarize:
– Does the same age pattern
apply to all four substances?
– Is the most common
substance consistent across
age groups?
The Chicago Guide to Writing about Multivariate Analysis, 2nd edition.
“Generalization”
• Point (and paragraph) #1
– “For every type of substance studied,
abuse declined with increasing age.”
– Compare within clusters (type of
substance abuse) across bar colors
(age groups).
• Point (and paragraph) #2
– “For two of the three age groups,
binge alcohol use was most common,
followed by heavy alcohol use,
marijuana use, and cocaine use.”
– Compare across clusters (substance)
within bar color (age groups).
The Chicago Guide to Writing about Multivariate Analysis, 2nd edition.
“Example”
• [Generalization from
previous slide]
– “For every type of substance
studied, abuse declined with
increasing age.”
• Associated example:
– “For example, rates of binge
alcohol use declined from
42% of 18–25 years olds, to
33% of 26–34 year olds, to
18% of those aged 35 and
older.”
The Chicago Guide to Writing about Multivariate Analysis, 2nd edition.
“Exception”
• One age group doesn’t follow
the same rank order of
substances as the other two.
– Look at the blue bars.
• Wording for the exception:
– [Revised from second
generalization] “For the older
two age groups, binge alcohol
use was most common,
followed by heavy alcohol use,
marijuana, and cocaine use.
In the youngest age group, the
pattern was similar except
that marijuana use slightly
exceeded heavy alcohol use.”
The Chicago Guide to Writing about Multivariate Analysis, 2nd edition.
Exceptions in direction
• Generalization: Among
the two oldest age
groups, heavy alcohol
use > marijuana use
• Exception: Among the
youngest adult age
group, marijuana use >
heavy alcohol use
The Chicago Guide to Writing about Multivariate Analysis, 2nd edition.
Exceptions in magnitude
• 4.7 year gap among men
• 3.3 year gap among
women
Life expectancy by race and
gender, US 2010
Black
Life expectancy at birth (years)
• In the US, for both men
and women, whites
outlive blacks
[generalization].
• The black disadvantage
in life expectancy is
larger among men than
among women.
White
100
80
76.5
60
78.0 81.3
71.8
40
20
0
Male
Female
The Chicago Guide to Writing about Multivariate Analysis, 2nd edition.
Exceptions in statistical significance
• Examples:
– All but one association shown in a table are
statistically significant.
– Only one of many associations shown on a chart is
statistically significant
– The association between an independent variable
and a dependent variable is statistically significant
only for some values of a second independent
variable.
The Chicago Guide to Writing about Multivariate Analysis, 2nd edition.
GEE for other disciplines
• Interpretations of historical events or scientific
concepts.
– Is there consensus in the literature, or are there
conflicting views?
– Consensus = generalization.
– Conflicting views = exceptions.
• Narrative perspective in literary works.
– Were most of an author’s pieces written in the first
person? Third person?
– Quotations = examples.
• Want representative examples, not arbitrary choices!
The Chicago Guide to Writing about Multivariate Analysis, 2nd edition.
Wording for a GEE
The Chicago Guide to Writing about Multivariate Analysis, 2nd edition.
Wording for a generalization
• If a pattern characterizes many of the numbers
you are summarizing, introduce the
generalization with phrases such as:
– “In general, . . .”
– “Typically, . . .”
– “By and large . . .”
– If the pattern characterizes all groups, say so!
– “In every region studied, . . .”
– “Throughout the entire period of observation . . .”
– “Regardless of [characteristic], . . .”
The Chicago Guide to Writing about Multivariate Analysis, 2nd edition.
Including measures of size
in the generalization
• Convey approximate share of values encompassed by
the generalization using phrases such as
– “Virtually all”
– “In the majority of cases,”
– “Roughly three-quarters of . . .”
• Can include the numeric illustration in the sentence
with the generalization by placing the numeric value in
parentheses after the pertinent phrase. E.g.,
– “Virtually all respondents (98%) . . .”
– “In a majority of cases (59%) . . .”
The Chicago Guide to Writing about Multivariate Analysis, 2nd edition.
Wording to present the example
• To document the pattern characterized by a
generalization or an exception,
– Introduce numeric evidence.
– Refer to the table or chart where all of the numbers
in the pattern can be found.
• E.g.,
– “For example, as shown in Figure X . . . [numeric fact,
with its Ws attached].”
– “For instance, . . . [numeric contrast, with the W’s
specified].”
The Chicago Guide to Writing about Multivariate Analysis, 2nd edition.
Wording to introduce exception(s)
• To introduce exceptions to the general pattern
you have just described, use phrases such as
– “An exception [to that pattern] is . . . ”
– “On the other hand, . . .”
– “However, . . .”
• E.g., “In seven out of 10 years studied, [general
pattern and example]. However, in the other
three years, [contrasting pattern = exception].”
The Chicago Guide to Writing about Multivariate Analysis, 2nd edition.
Wording for exceptions in direction
• Some exceptions are literally the opposite of the
pattern described in the generalization. E.g.,
– A falling rather than rising trend.
– A deficit rather than an excess for one group compared to
another.
• Having already described the direction of the general
pattern, introduce such exceptions using expressions
such as
– “On the contrary, . . .”
– “Conversely, . . .”
The Chicago Guide to Writing about Multivariate Analysis, 2nd edition.
Wording for exceptions in direction
Labor force participation by race
and gender, US
80
Labor force participation, %
• E.g., “Among males, the
labor force participation
rate for whites
exceeded that for
blacks. In contrast,
among females, blacks
had a higher labor force
participation rate than
whites.”
Black
White
70
60
50
40
30
20
10
0
Male
Female
The Chicago Guide to Writing about Multivariate Analysis, 2nd edition.
Summary: Advantages of a GEE
• Avoids reporting every number from a
complex pattern in the text.
– That is what the table or chart is for.
• Summarizes the patterns.
– How do numbers relate to one another?
– How do patterns relate to the original question?
– The forest, not the trees!
The Chicago Guide to Writing about Multivariate Analysis, 2nd edition.
Suggested resources
• Chapter 2 and Appendix A in
– Miller, J. E. 2004. The Chicago Guide to Writing about
Numbers (“WA#”)
OR
– Miller, J. E. 2013. The Chicago Guide to Writing about
Multivariate Analysis, 2nd Edition. (“WAMA”)
• Chapter 9 in WA# or chapter 14 in WAMA, section on
“generalization, example, exceptions revisited”
• Miller, J. E. 2006. “How to Communicate Statistical
Findings: An Expository Writing Approach.” Chance 19
(4): 43–49.
The Chicago Guide to Writing about Multivariate Analysis, 2nd edition.
Suggested online resources
• Podcasts on
– Reporting one number
– Comparing two numbers or series of numbers
– Choosing tools to present numeric information
– Steps for conducting a GEE
The Chicago Guide to Writing about Multivariate Analysis, 2nd edition.
Suggested practice exercises
• Study guide to The Chicago Guide to Writing about
Multivariate Analysis, 2nd Edition.
– Question #9 in problem set for chapter 2
– Question #9 in problem set for chapter 6
– Suggested course extensions for
• chapter 2
– “Reviewing” exercise #7
– “Writing” and “revising” exercise #4
• chapter 14
– “Applying statistics and writing” exercises #3 and 4
– “Revising” exercise #3
The Chicago Guide to Writing about Multivariate Analysis, 2nd edition.
Contact information
Jane E. Miller, PhD
jmiller@ifh.rutgers.edu
Online materials available at
http://press.uchicago.edu/books/miller/multivariate/index.html
The Chicago Guide to Writing about Multivariate Analysis, 2nd edition.