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

• 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.

• 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.

• 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

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.”

• 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.

• 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.

• 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.

• 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.

• 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.

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

• 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.

• 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.

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