1. The Chicago Guide to Writing about Numbers, 2 nd edition. Comparing two numbers or series of numbers Jane E. Miller, PhD

2. The Chicago Guide to Writing about Numbers, 2 nd edition. 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

3. The Chicago Guide to Writing about Numbers, 2 nd 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.

4. The Chicago Guide to Writing about Numbers, 2 nd edition. 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

5. The Chicago Guide to Writing about Numbers, 2 nd 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.

6. The Chicago Guide to Writing about Numbers, 2 nd 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.

7. The Chicago Guide to Writing about Numbers, 2 nd edition. 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

8. The Chicago Guide to Writing about Numbers, 2 nd 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.”

9. The Chicago Guide to Writing about Numbers, 2 nd 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.

10. The Chicago Guide to Writing about Numbers, 2 nd 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

11. The Chicago Guide to Writing about Numbers, 2 nd edition. 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

12. The Chicago Guide to Writing about Numbers, 2 nd 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.

13. The Chicago Guide to Writing about Numbers, 2 nd 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?

14. The Chicago Guide to Writing about Numbers, 2 nd 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.”

15. The Chicago Guide to Writing about Numbers, 2 nd 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.”

16. The Chicago Guide to Writing about Numbers, 2 nd 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.

17. The Chicago Guide to Writing about Numbers, 2 nd 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.

18. The Chicago Guide to Writing about Numbers, 2 nd 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

19. The Chicago Guide to Writing about Numbers, 2 nd 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

20. The Chicago Guide to Writing about Numbers, 2 nd 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.”

21. The Chicago Guide to Writing about Numbers, 2 nd 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].”

22. The Chicago Guide to Writing about Numbers, 2 nd 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.

23. The Chicago Guide to Writing about Numbers, 2 nd 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.

24. The Chicago Guide to Writing about Numbers, 2 nd 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.

25. The Chicago Guide to Writing about Numbers, 2 nd 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.

26. The Chicago Guide to Writing about Numbers, 2 nd 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

27. The Chicago Guide to Writing about Numbers, 2 nd 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.

28. The Chicago Guide to Writing about Numbers, 2 nd 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

29. The Chicago Guide to Writing about Numbers, 2 nd edition. Suggested online resources Podcasts on – Reporting one number – Summarizing a pattern involving many numbers – Choosing a comparison group – Steps for conducting a GEE

30. The Chicago Guide to Writing about Numbers, 2 nd 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

31. The Chicago Guide to Writing about Numbers, 2 nd edition. Contact information Jane E. Miller, PhD jmiller@ifh.rutgers.edu Online materials available at http://press.uchicago.edu/books/miller/numbers/index.html