comparing two numbers or series of numbers

The Chicago Guide to Writing about Multivariate Analysis, 2nd edition.

Comparing two numbers or series of numbers

Jane E. Miller, PhD


Overview

• 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


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.


• 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.

Report and interpret


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.


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.


• 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.

– A 30% difference could mean: • 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

Why not just interpret?

Very low: Almost eradicated LBW

Very high: Serious LBW problem


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.”


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.


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


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


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.


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?


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.”


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.”


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.


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.


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


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


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.”


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].”


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.


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.


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.


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.


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

• Same principles apply for writing about coefficients from multivariate model


Suggested resources: general• For basic principles, see chapters 1 and 2 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”)

• 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.


Suggested resources: units and calculations

• Units– Chapter 4 of WA# or WAMA

• Types of quantitative comparisons and how to – Choose which one(s) to use for a specific task– Write about them – chapter 8 of WA# or chapter 5 of WAMA


Suggested online resources

• Podcasts on – Reporting one number– Summarizing a pattern involving many numbers– Choosing a reference category– Interpreting multivariate coefficients


Suggested practice exercises

• Study guide to The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.– Questions #1, 7 and 8 in problem set for chapter 2– Suggested course extensions for

• Chapter 2– “Reviewing” exercise #6– “Writing” and “revising” exercise #1

• Chapter 14– “Reviewing” exercise #3– “Applying statistics and writing” exercises #1 and 2– “Revising” exercises #1 and 2


Contact information

Jane E. Miller, [email protected]

Online materials available athttp://press.uchicago.edu/books/miller/multivariate/index.html

comparing two numbers or series of numbers

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