Writing about ratios Jane E. Miller, PhD The Chicago Guide to Writing about Numbers, 2nd Edition.

Overview Vocabulary for ratio calculations Phrasing interpretation of different values of ratios  <1.0  ~1.0  >1.0  Integer values Calculating percentage difference from a ratio Common pitfalls in writing about ratios The Chicago Guide to Writing about Numbers, 2 nd edition.

Vocabulary for ratios Ratio = X ÷ Y = X/Y  X is the value in the numerator  Y is the value in the denominator When writing about ratios, phrasing will implicitly compare the value in the numerator to the value in the denominator.  In our behind the scenes work, we’ll refer to the value in the denominator as the reference or comparison value.  However, the most user-friendly presentations of ratios avoid jargon. The Chicago Guide to Writing about Numbers, 2nd Edition.

Objectives for writing about ratios When interpreting the result of a ratio calculation for most audiences – DO want to convey the topic under study the groups, times, or places being compared the direction and magnitude – DON’T want to use jargon like numerator denominator ratio reference value

Example comparison For the examples used in this podcast, all ratios will compare males to females, specifically with – Males in the numerator – Females in the denominator The reference or comparison group E.g., if our topic is unemployment rates, the ratio = Unemployment rate among males Unemployment rate among females The Chicago Guide to Writing about Numbers, 2nd Edition.

For ratios <1.0 The value in the numerator is less than the value in the denominator. Convert the ratio to a percentage difference Percentage difference = ratio × 100 E.g., if ratio = 0.8, the percentage difference is 0.8 × 100 = 80% – Consistent with a lower value in the numerator than in the denominator. In this case, the value in the numerator is 80% of the value in the denominator. The Chicago Guide to Writing about Numbers, 2nd Edition.

Writing about ratios <1.0 The general wording is “[Group] is z% as [fill in adjective or verb related to the topic under study] as [name the comparison group].” – Where “z” is the percentage difference Example: Topic of study = graduation rates Ratio Graduation rate among males/Graduation rate among females = 0.8 “Males were 80% as likely as females to graduate from the program.” The Chicago Guide to Writing about Numbers, 2nd Edition.

For ratios close to 1.0 Use phrasing to convey similarity of the two values. Again, name the groups and the topic being compared. Example: Topic of study = average test scores Ratio Average test score for males/Average test score for females = 1.02 “Average test scores were virtually identical for males and females (ratio = 1.02 for males vs. females).” The Chicago Guide to Writing about Numbers, 2nd Edition.

For ratios >1.0 The value in the numerator is greater than the value in the denominator. Two options for interpreting the ratio: – Express the value in the numerator as a multiple of the value in the denominator. – Convert the ratio to a percentage difference, and convey accordingly. The Chicago Guide to Writing about Numbers, 2nd Edition.

Writing about ratios >1.0 using multiples E.g., if the ratio = 1.2, the value in the numerator is 1.2 times that in the denominator. The general wording is “[Group] is [ratio] times as [fill adjective or verb that conveys the topic] as [name the comparison group].” Example: Topic of study = height Ratio Average height for males/Average height for females = 1.2 “Males were on average 1.2 times as tall as females.” The Chicago Guide to Writing about Numbers, 2nd Edition.

Calculating percentage difference for ratios >1.0 Convert the ratio to a percentage difference Percentage difference = (ratio – 1.0)  100 E.g., if the ratio = 1.2, the percentage difference is (1.2 – 1.0) × 100, or 0.2  100 = 20% The value in the numerator is 20% higher than the value in the denominator. The Chicago Guide to Writing about Numbers, 2nd Edition.

Writing about ratios >1.0 using percentage difference The general wording is “[Group] is z% [fill in adjective that conveys direction, ideally using vocabulary related to the topic] greater than [name the comparison group].” Example: Topic of study = height Ratio Average height for males/Average height for females = 1.2 “Males were on average 20% taller than females.” The Chicago Guide to Writing about Numbers, 2nd Edition.

Calculating percentage difference for ratios >2.0 Convert the ratio to a percentage difference Percentage difference = (ratio – 1.0)  100 E.g., if the ratio = 2.34, the percentage difference is (2.34 – 1.0) × 100, or 1.34  100 = 134% The value in the numerator is 134% higher than the value in the denominator. The Chicago Guide to Writing about Numbers, 2nd Edition.

Writing about ratios >2.0 using percentage difference The general wording is “[Group] is z% [fill in vocabulary related to the topic] greater than [name the comparison group].” Example: Topic of study = income Ratio Average income for males/Average income for females = 2.34 “Males earned on average 134% more than females.” The Chicago Guide to Writing about Numbers, 2nd Edition.

Writing about a ratio that is close to an integer E.g., if the ratio = 3.02, the value in the numerator is just over three times that in the denominator. – Write about it in terms of a multiple, rounded to the nearest integer The general wording is “[Group] is about [integer] times as [fill adjective or verb that conveys the topic] as [name the comparison group].” Example: Topic of study = crime rates Ratio Crime rate for males/Crime rate for females = 3.02 “Males were about three times as likely as females to commit a crime.” The Chicago Guide to Writing about Numbers, 2nd Edition.

Pitfalls in writing about ratios Writing about the ratio as if it were calculated using subtraction Interpreting units of the ratio incorrectly Writing about the ratio “upside-down” Using wording that conveys an incorrect direction or magnitude of the difference between the values in the numerator and denominator The Chicago Guide to Writing about Numbers, 2nd Edition.

Explain the result of the correct type of calculation Ratios are calculated by dividing one number by another, not by subtracting. E.g., if A/B = 1.5, – Do not explain it as a “A is 1.5 units higher than B.” That implies that you subtracted B from A – Instead, explain it as “A is 1.5 times as high as B.”

Watch your units when interpreting ratios Explain ratios in terms of multiples of the reference value, not multiples of the original units. E.g., if population size was originally measured in millions of persons, a ratio of 1.43 for Region A compared to Region B – Does not mean there were 1.43 million times as many people in Region A compared to Region B. – During the division calculation, the units (millions of persons) “cancel,” so there were 1.43 times as many people in Region A as in Region B.

Avoid describing a ratio “upside-down” If you calculate and report the ratio in a table as Male value/Female value DON’T interpret it in the text as if Female value/Male value E.g., if in a table or chart you report Crime rate for males/Crime rate for females = 3.02 AVOID writing “Females were only about 1/3 as likely to commit a crime as males.” – The math is correct, but your readers will have to stop and calculate the reciprocal of the ratio in the table to verify that. The Chicago Guide to Writing about Numbers, 2nd Edition.

Choosing a reference value for ratios Before you choose a reference value within your own data, anticipate how you will word the description. – E.g., if you naturally want to compare all the other regions to the Midwest, make it the reference, then calculate and describe accordingly Ratio = Value for other region/value for Midwest – “The Northeast is [measure of difference] larger (or smaller) than the Midwest.” The Chicago Guide to Writing about Numbers, 2nd Edition.

Conveying direction and magnitude of a comparison based on a ratio Do not confuse the phrases “A is 60% as high as B” and “A is 60% higher than B.” – The first phrase suggests that A is lower than B (i.e., that the ratio A/B = 0.60) Equivalent to “A is 60% of B.” – The second suggests that A is higher than B (i.e., A/B = 1.60). The Chicago Guide to Writing about Numbers, 2nd Edition.

Ratios with same direction but different magnitude Do not confuse the phrases “A is 60% higher than B” and “A is 160% higher than B.” – The first phrase corresponds with ratio A/B = 1.60 – The second phrase corresponds with ratio A/B = 2.60 The direction of both of those ratios is the same (A>B) But the magnitude of the difference between values A and B is bigger for the second ratio 2.60 > 1.60 The Chicago Guide to Writing about Numbers, 2nd Edition.

Check your wording against the numeric value of a ratio After you calculate a ratio or percentage difference – Describe the difference between the values in the numerator and denominator direction size Check your description against the original numbers – Make sure you have correctly communicated which is bigger, the value in the numerator or the value in the denominator The Chicago Guide to Writing about Numbers, 2nd Edition.

Summary Presenting the results of ratio calculations does not need to involve jargon such as “numerator,” “denominator,” or even “ratio.” Instead, interpret ratios in prose either – As multiples of the reference value – As a % difference compared to the reference value Use language to convey direction of association, e.g., which is bigger – The value in the numerator? – The value in the denominator? Watch that you correctly interpret units of the ratio The Chicago Guide to Writing about Numbers, 2nd Edition.

Suggested resources Miller, J. E The Chicago Guide to Writing about Numbers, 2nd Edition. University of Chicago Press, Chapter 5. The Chicago Guide to Writing about Numbers, 2nd Edition.

Suggested online resources Podcasts on – Comparing two numbers or series of numbers – Types of quantitative comparisons – Choosing a reference category The Chicago Guide to Writing about Numbers, 2nd Edition.

Suggested practice exercises Study guide to The Chicago Guide to Writing about Numbers, 2nd Edition. – Problem sets Chapter 4: Question #13 Chapter 5: Questions #3, 7, 8, and 10a, d and e The Chicago Guide to Writing about Numbers, 2nd Edition.

Contact information Jane E. Miller, PhD Online materials available at The Chicago Guide to Writing about Numbers, 2nd Edition.