Comparing two numbers or
series of numbers
Jane E. Miller, PhD
The Chicago Guide to Writing about Numbers, 2nd edition.
Overview
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Build on principles for reporting one number
Report and interpret
Aside on types of variables
Direction and magnitude
– Of a cross-sectional comparison
– Of a trend
• Mathematical and verbal approaches
• Reference groups
The Chicago Guide to Writing about Numbers, 2nd edition.
Apply principles for
reporting a number
• Set the context (W’s) using topic sentence for each
paragraph or section.
– Once W’s are reported, don’t need to repeat them unless
they change.
• E.g., if same place and time pertain to all numbers you
compare, state them in the topic sentence.
• Convey other W’s (e.g., gender, race) in the sentences
reporting those numbers.
• Specify the units as you report the numbers.
– System of measurement.
– Scale and level of aggregation.
The Chicago Guide to Writing about Numbers, 2nd edition.
Report and interpret
• Give both the raw numbers and the result of
the comparison.
• Report means including a number in a
sentence, table, or chart.
• Interpret means explaining to your readers
what that number means in the context of the
question at hand. E.g.,
– Compare it to other numbers.
– Relate it to hypotheses.
– A “naked” number is not informative.
The Chicago Guide to Writing about Numbers, 2nd edition.
Why report and interpret?
• Interpret the comparison to answer your
question.
– Answer the “word problem” behind your analysis.
– Provide a context so readers can understand the
point you are making with the numbers.
• Report numbers to give readers the raw data
to answer other questions. E.g.,
– Compare with other times, places, or groups.
– Conduct additional calculations with your
numbers.
The Chicago Guide to Writing about Numbers, 2nd edition.
Why not just report?
• Often, readers need a context in which to
understand the numbers.
– Is a given number big or small?
– Are things changing? Stable?
– Does a particular value exceed an important cutoff?
• Every number you include should be chosen for
a specific reason.
– Convey that reason (or question) to your audience.
– Write the solution, not the problem set.
The Chicago Guide to Writing about Numbers, 2nd edition.
Why not just interpret?
• Readers should have access to the “raw
numbers” upon which calculations are based.
– Suppose you read that the prevalence of low birth
weight (LBW) is 30% lower than five years ago, but
the LBW rate is not given for either year.
Very low: Almost
– A 30% difference could mean:
eradicated LBW
• 1.0 LBW infant vs. 1.3 LBW infants per 1,000 live births
• 400 LBW infants vs. 520 LBW infants per 1,000 live
births
Very high: Serious LBW problem
The Chicago Guide to Writing about Numbers, 2nd edition.
Specify direction and magnitude
of the association
• Which group is bigger,
faster, smarter?
– Instead of:
• “In Florida, Bush received
2,912,790 votes to Gore’s
2,912,253.”
– Write:
• “In Florida, Bush
defeated Gore.”
• How much bigger, faster,
smarter?
– Instead of:
• “In Florida, Bush defeated
Gore.”
– Write:
• “In the closest election in
US history, Bush won
Florida by a mere 537
votes.”
The Chicago Guide to Writing about Numbers, 2nd edition.
Aside: Types of variables
• Variables come in two broad types, also known
as “levels of measurement”
– Continuous
– Categorical
• Type of variable affects many aspects of writing
about numbers, including pertinent and valid:
– Types of calculations and statistics,
– Types of charts,
– Wording of sentences to present and compare
numbers.
The Chicago Guide to Writing about Numbers, 2nd edition.
Categorical variables
• Categorical variables are those that are classified
into categories.
• There are two types of categorical variables.
– Nominal
• Named categories with no inherent numeric order
– e.g., gender, race, religion
– Ordinal
• Ordered categories
– e.g., Likert scale, income group, letter grade, self-rated health
The Chicago Guide to Writing about Numbers, 2nd edition.
Continuous variables
• Continuous variables
– Measured in numeric units, but not grouped.
• Two types of continuous variables:
– Interval
• Zero is not lowest possible value
• e.g., temperature °Fahrenheit
– Ratio
• Zero is lowest possible value
• e.g., temperature °Kelvin, height, weight
The Chicago Guide to Writing about Numbers, 2nd edition.
Why does type of variable matter?
• Type of variable affects how you write about direction
of association.
– For nominal variables, must name the reference
group.
• Doesn’t make sense to say “as gender increases.”
– For ordinal, interval, or ratio variables, can say “as
variable A increases, variable B
[increases/decreases.]”
• Positive or negative association.
• Type of variable also affects which statistics and types
of chart make sense.
The Chicago Guide to Writing about Numbers, 2nd edition.
Specify direction and magnitude
• For a cross-sectional comparison
– Direction
• Which value is bigger?
– Magnitude
• How much bigger?
• For a trend
– Direction
• Is the trend rising, falling, or level?
– Magnitude
• How steeply?
The Chicago Guide to Writing about Numbers, 2nd edition.
Direction and magnitude:
Cross-section
• E.g., across groups or experimental conditions.
• Of two values:
– Direction: Which is bigger?
– Magnitude: How much bigger?
• “Company B’s policy costs ten dollars per month more than
Company A’s.”
The Chicago Guide to Writing about Numbers, 2nd edition.
Direction and magnitude: Trend
• Direction: Is it level, rising, or falling?
• Magnitude: How rapidly?
• “The population of City A rose gradually over the period,
while the population of City C decreased rapidly and B
remained stable.”
The Chicago Guide to Writing about Numbers, 2nd edition.
Example: Direction and magnitude
• Poor: “Age and income are correlated.”
Or “Age and income are associated.”
– Neither direction nor magnitude is specified.
• Better: “As age increases, income increases.”
– Now we know direction but not magnitude.
• Best: “Between the ages of 20 and 49,
income increases roughly 10% for each 10
year increase in age, then levels off through
age 64.”
– Could also include statistical significance.
The Chicago Guide to Writing about Numbers, 2nd edition.
Calculations to express size
• The math is straightforward, e.g.,
– Subtraction
– Division
– Percentage difference or change
• The writing is more challenging
– Often authors focus more on explaining the
arithmetic than the answer to the word problem.
• Use too much jargon
– See suggested resources for references about how to
present the answer to your calculation.
The Chicago Guide to Writing about Numbers, 2nd edition.
Combining calculations and vocabulary
for an effective description
• Start with a verbal description
– Set the context
– Provide a verbal sketch of the shape and size of
the pattern
• Document with numeric evidence
– Specify units
– Continue to use descriptive vocabulary
The Chicago Guide to Writing about Numbers, 2nd edition.
Vocabulary to express
direction and size
• Verbs to describe change
– Mundane: increased, declined
– More interesting: rocketed, plummeted
• Adverbs to modify boring verbs
– Increased dramatically
– Barely budged
• Adjectives
– Mundane: level, rising, smaller
– More interesting: erratic, minuscule
The Chicago Guide to Writing about Numbers, 2nd edition.
Illustrative sentences
• “In the recent election, Candidate Q eked out a
narrow victory over his rival.”
• “Plants fed ‘worm tea’ compost grew rapidly
over the course of the experiment, whereas
those given only water grew more modestly.”
• “Immediately after Hurricane Katrina, gasoline
prices in the US spiked to an all time high. They
remained volatile in the subsequent two
months.”
The Chicago Guide to Writing about Numbers, 2nd edition.
Adding numeric evidence
• “In the recent election, Candidate Q eked out a narrow
victory over his rival, winning by only 231 votes out of a
total of 2 million votes tallied.”
• “Plants fed ‘worm tea’ grew rapidly over the course of the
experiment, whereas those given only water grew more
modestly. The ‘worm tea’ plants averaged 17 cm/week,
versus only 13 cm/week for the water-only plants.”
• “Last week was uniformly hot, with daily high
temperatures within three degrees of one another
[illustrates narrow range] and averaging more than ten
degrees above normal [documents that it was hot].”
The Chicago Guide to Writing about Numbers, 2nd edition.
Comparative writing
• For every comparison specify what is being
compared to what.
• If all you write is “X is 20% higher”, the reader
doesn’t know higher than what?
– Especially if you are comparing several groups or
places or time periods, omission of the referent
can be very confusing.
The Chicago Guide to Writing about Numbers, 2nd edition.
Reference group for multiple
comparisons
• If you are comparing age
distributions for two time
periods in two regions, “The
elderly age group is smaller”
doesn’t tell your reader
whether you mean:
• Smaller than other age groups
in the same region, or
• Smaller than the same age
group in the other region, or
• Smaller than it used to be, in
the same region.
The Chicago Guide to Writing about Numbers, 2nd edition.
Poor comparison statements
• Poor version #1: “The difference was 7.2.”
– Magnitude but not topic, units, or direction.
• Poor version #2: “Insurance and length of stay
were associated.”
– Specifies topic but not direction, size, or units.
The Chicago Guide to Writing about Numbers, 2nd edition.
Improved comparison statements
• Better: “Privately insured children stayed longer
than publicly insured children.”
– Topic, reference group, and direction but not size or
units.
• Best: “Children with private insurance stayed on
average 7.2 days longer than those with public
insurance.”
– Topic, reference group, direction, magnitude, and
units.
The Chicago Guide to Writing about Numbers, 2nd edition.
Summary
• When comparing two numbers or series, always
– Report and interpret values
– Specify direction and magnitude of the association
– Convey reference value
• Use different, complementary approaches
– Results of mathematical calculations to convey size
– Vocabulary to express direction and size
The Chicago Guide to Writing about Numbers, 2nd edition.
Suggested resources: general
• For basic principles, see chapters 1 and 2 in
Miller, J. E. 2015. The Chicago Guide to Writing
about Numbers, 2nd Edition
• For additional examples, see
– Miller, J. E. 2006. “How to Communicate Statistical
Findings: An Expository Writing Approach.” Chance
19 (4): 43–49.
– Miller, J. E. 2010. “Quantitative Literacy across the
Curriculum: Integrating Skills from English
Composition, Mathematics, and the Substantive
Disciplines.” The Educational Forum. 74 (4): 334–46.
The Chicago Guide to Writing about Numbers, 2nd edition.
Suggested resources:
units and calculations
• Units
– Chapter 4
• Types of quantitative comparisons and how to
– Choose which one(s) to use for a specific task
– Write about them
– Chapter 8
The Chicago Guide to Writing about Numbers, 2nd edition.
Suggested online resources
• Podcasts on
– Reporting one number
– Summarizing a pattern involving many numbers
– Choosing a comparison group
– Steps for conducting a GEE
The Chicago Guide to Writing about Numbers, 2nd edition.
Suggested practice exercises
• Study guide to The Chicago Guide to Writing
about Numbers, 2nd Edition.
– Questions #1, 7, and 8 in problem set for chapter 2
– Question #3 in the problem set for chapter 11
– Suggested course extensions for
• Chapter 2
– “Reviewing” exercise #5
– “Writing” exercise #1
• Chapter 9
– “Reviewing” exercises #3, 4, and 5
– “Applying statistics and writing” exercises #1 and 2
– “Revising” exercises #1 and 2
The Chicago Guide to Writing about Numbers, 2nd edition.
Contact information
Jane E. Miller, PhD
jmiller@ifh.rutgers.edu
Online materials available at
http://press.uchicago.edu/books/miller/numbers/index.html
The Chicago Guide to Writing about Numbers, 2nd edition.