writing about ratios jane e. miller, phd the chicago guide to writing about numbers, 2nd edition

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, 2nd 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.


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


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.


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.”


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).”


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.


Writing about ratios >1.0using 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 differencePercentage 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.


Writing about ratios >1.0using 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.”


Calculating percentage difference for ratios >2.0

• Convert the ratio to a percentage differencePercentage 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.


Writing about ratios >2.0using 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.”


Writing about a ratio that isclose 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.”


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


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 valueDON’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.


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.”


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).


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 ratio2.60 > 1.60


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


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 ratioThe Chicago Guide to Writing about Numbers, 2nd Edition.

Suggested resources

• Miller, J. E. 2015. The Chicago Guide to Writing about Numbers, 2nd Edition. University of Chicago Press, Chapter 5.


Suggested online resources

• Podcasts on– Comparing two numbers or series of numbers– Types of quantitative comparisons– Choosing a reference category


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


Contact information

Jane E. Miller, [email protected]

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


mailto:[email protected]

http://press.uchicago.edu/books/miller/numbers/index.html

writing about ratios jane e. miller, phd the chicago guide to writing about numbers, 2nd edition

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