Comparing two numbers or series of numbers

1. Comparing two numbers or series of numbers Jane E. Miller, PhD

2. 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. 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. 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.

5. Why report and interpret? • Interpret the comparison to answeryourquestion. • 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 otherquestions. E.g., • Compare with other times, places, or groups. • Conduct additional calculations with your numbers.

6. 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. 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. • 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 Very low: Almost eradicated LBW Very high: Serious LBW problem

8. 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 wonFlorida by a mere 537 votes.” Specify direction and magnitude of the association

9. 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. 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. 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

12. 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. 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 • Howsteeply?

14. 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. Direction and magnitude: Trend • Direction: Is it level, rising, or falling? • Magnitude: How rapidly? • “The population of City A rosegraduallyover the period, while the population of City C decreasedrapidly and B remained stable.”

16. 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. 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. 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. Vocabulary to express direction and size • Verbsto describe change • Mundane: increased, declined • More interesting: rocketed, plummeted • Adverbsto modify boring verbs • Increased dramatically • Barely budged • Adjectives • Mundane: level, rising, smaller • More interesting: erratic, minuscule

20. 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. 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] andaveraging more than ten degrees above normal[documents that it was hot].”

22. 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. 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. 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, orunits.

25. Improved comparison statements • Better: “Privately insured children stayed longer than publicly insured children.” • Topic, reference group, and direction but not size orunits. • Best: “Children with private insurance stayed on average 7.2 dayslonger thanthose with public insurance.” • Topic, reference group, direction, magnitude, and units.

26. 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

27. 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.

28. 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

29. Suggested online resources • Podcasts on • Reporting one number • Summarizing a pattern involving many numbers • Choosing a reference category • Interpreting multivariate coefficients

30. 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

31. Contact information Jane E. Miller, PhD jmiller@ifh.rutgers.edu Online materials available at http://press.uchicago.edu/books/miller/multivariate/index.html