Types of quantitative comparisons
Jane E. Miller, PhD
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Types of quantitative comparisons
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Level
Rank
Difference
Ratio
Percentage difference
Percentage change
The Chicago Guide to Writing about Numbers, 2nd edition.
Overview
• For each type of quantitative comparison:
– How it is calculated
– How to write clear sentences to report and
interpret results of that calculation
– Advantages: what questions it can answer
– Disadvantages: what questions it can’t answer
• How to chose among the ways to compare
numbers, based on the question you seek to
address
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Coordinating calculations with your writing
• An important aspect of working with quantitative
comparisons is coordinating them with the ways you
report and interpret them in prose form.
• Before you calculate, think about how you prefer to
phrase your comparisons, then perform the
corresponding calculations accordingly.
• Doing so will avert the confusion of interpreting
– Ratios that you have accidentally described “upside down.”
– Subtraction you have inadvertently explained “backward.”
Level
• Level = value of the measure for one observation.
• Example sentences:
– “Michael Phelps’ time in the 200 meter individual
medley (IM) at the 2012 Olympic Games was 1:54.27.”
– “The average price of a gallon of gasoline in New
Brunswick, New Jersey, in January, 2015 was $1.99.”
– “In 2011, the infant mortality rate in the U.S. was 6.1
deaths per 1000 live births.”
• Checklist for reporting the level
– Context (W’s)
– Units
– See podcast on reporting one number
The Chicago Guide to Writing about Numbers, 2nd edition.
Disadvantages of reporting level alone
• Difficult to assess what the number means in the
context of the topic under study.
– Is it high or low?
– Fast or slow?
– Cheap or expensive?
• To assess those aspects, need a comparison.
The Chicago Guide to Writing about Numbers, 2nd edition.
Rank
• Rank = order, compared to other values in a list, e.g.,
– A series of other values observed in the same time or the
same place
– A well-established standard
– An historic high or low value
• Examples:
– “Michael Phelps placed first in the 200 meter IM at the 2012
London Olympic Games.”
– “The cost of gasoline is the lowest observed in several
years.”
– “In 2009, the U.S. ranked 29th in the world in terms of
infant mortality.”
The Chicago Guide to Writing about Numbers, 2nd edition.
When is rank useful?
• When all that matters is the order, not the size of the
difference between groups.
– In an election, all that matters is who came in first.
• It doesn’t matter whether the margin of victory is 1 vote or 1
million votes!
The Chicago Guide to Writing about Numbers, 2nd edition.
Disadvantages of using rank alone
• Rank tells you where the value falls relative to other
values.
• Doesn’t tell you how close the other values are.
– Could be the fastest, but only by some microscopic amount.
– Could be the least expensive, but with 15 other brands each
costing only a few cents more.
• Need other measures to tell you how big a difference,
which is often critical for interpreting what that value
means in context.
The Chicago Guide to Writing about Numbers, 2nd edition.
Difference
• The value of interest (X), subtracted from some other
value (Y), or Y minus X.
• Difference = Y - X
• Examples
– “Michael Phelps’ 200 meter IM time was 0.63 seconds faster
than that of the second place swimmer.”
– “In January 2015, gasoline cost about $2.00 per gallon less
than it had a month earlier.”
• Checklist for comparing two numbers
– Report and interpret.
– Specify direction and magnitude.
The Chicago Guide to Writing about Numbers, 2nd edition.
When is the difference useful?
• When the difference itself is of interest.
– How much more will something cost?
• Is that amount within your budget?
– How much time will be saved if a certain change is made?
• When you do not need to assess size of difference
compared to anything else.
The Chicago Guide to Writing about Numbers, 2nd edition.
Disadvantages of difference alone
• Is a difference of $2.00 a lot or a little?
– It depends on the level.
• If the price of a gallon of gasoline decreased by $2.00, you would
consider that a big change.
• If the price of a car decreased by $2.00, you probably wouldn’t
even notice!
– Why?
• Because the gallon of gas cost about $3.75 before the change,
whereas the car probably cost at least $15,000.
• To address the question of whether a given difference is
big or small, you need a measure that looks at the
difference or change relative to the initial value.
The Chicago Guide to Writing about Numbers, 2nd edition.
Ratio
• The value of interest (X) divided by some other value
(Y).
• Ratio = X/Y
– If X and Y are measures of risk, the ratio is the relative risk.
• Examples
– “Infant mortality among black infants is more than twice as
high as among white infants.”
– “The cost of gasoline is less than half of what it was last year.”
The Chicago Guide to Writing about Numbers, 2nd edition.
Avoid use of jargon
• When interpreting a ratio, use intuitive phrasing that a
layperson can understand.
• Poor: “The ratio is 2.25.”
– Can’t tell
• What variable is being studied.
• What groups are being compared.
• Which group has the higher value.
• Better: “The infant mortality rate (IMR) for black
infants is more than twice as high as for white infants.”
• See podcast on writing about ratios.
The Chicago Guide to Writing about Numbers, 2nd edition.
Percentage difference
• Percentage difference = difference (from subtraction)
divided by the initial value multiplied by 100.
percentage difference = [(Y-X)/X] * 100
• Example: The infant mortality rate (IMR) for blacks =
11.5 deaths per 1000 live births; the IMR for whites =
5.1.
– Calculate: [(11.5 - 5.1)/5.1] * 100 = 125%
– Write: “Black infants are 125% more likely than their white
counterparts to die before their first birthday.”
The Chicago Guide to Writing about Numbers, 2nd edition.
Percentage change
• Calculated the same way as percentage difference.
– Compares values from two points in time.
• Subtract the earlier value (T1) from the later value
(T2)
• Divide by the value for the earlier time point and
multiply by 100:
Percentage change = [(T2 - T1)/T1] * 100
• Direction of percentage change
– A positive percentage change means an increase over time.
– A negative percentage change means a decrease over time.
The Chicago Guide to Writing about Numbers, 2nd edition.
Reporting numeric examples
• Once you have interpreted the result of the
comparison, report the numbers to document that
pattern.
• Sometimes the numbers can go in the same
sentence with the description.
– “Mortality is nearly 3 times as high among persons aged
85 and older (15.3 deaths per 1000 persons) as among
those aged 75-84 (5.7 deaths per 1000).” OR
– “Mortality is nearly 3 times as high among persons aged
85 and older than among those aged 75-84: 15.3 deaths
per 1000 persons and 5.7 deaths per 1000, respectively.”
The Chicago Guide to Writing about Numbers, 2nd edition.
Reporting numeric examples, cont.
• Sometimes it is simpler to report the numbers
in a separate sentence.
– “Michael Phelps’ margin of victory in the 200
meter individual medley at the 2012 Olympic
Games was 0.63 seconds. {Phelps finished in
1:54.27, followed by Ryan Lochte in 1:54.90.”
The Chicago Guide to Writing about Numbers, 2nd edition.
Choosing which types of quantitative
comparisons to use in your writing
• The choice of which calculations to include depends
on your topic and discipline.
– Report results of a race, election, or marketing study in
terms of rank and difference.
– Describe time trends using difference or percentage change.
• To express large changes such as tripling, or halving using a ratio.
– Describe variations in risk or probability in terms of ratios.
– Report only one measure of rank (e.g., position, percentile,
decile, or quartile).
Type of variable constrains
your choice of quantitative comparison
• The level of measurement (a.k.a. type of variable)
determines which types of quantitative comparisons
make sense for that variable.
• For each of the variables in your analysis, need to
know its level of measurement.
– See suggested resources at the end of this lecture for
readings and podcasts on that topic.
Categorical variables
• Categorical variables are grouped into
categories or ranges.
– Nominal variables have named categories with no
inherent numeric order.
• e.g., gender, race, religion
– Ordinal variables have ordered categories. They
can, but do not have to, have numeric units. E.g.,
• Age group (in years) and income range (in $) have
numeric units
• Likert scale and letter grade do not have numeric units
The Chicago Guide to Writing about Numbers, 2nd edition.
Continuous variables
• Continuous variables are measured in numeric
units but are not grouped.
– Interval variables can take on negative values.
• e.g., temperature in °Fahrenheit
– Ratio variables cannot have negative values.
• e.g., temperature on the Kelvin scale, weight in pounds,
income in Euros
Quantitative comparisons for
continuous variables
Can do calculations with their values.
• Ratio variables
– Rank, e.g., tallest person
– Difference, e.g., 7 inches taller
– Ratio, e.g., twice as tall
• Interval variables
– Rank, e.g., coldest day in history
– Difference, e.g., 10 degrees colder
– CANNOT interpret a ratio of values
• e.g., if yesterday was -2 °F and today is 2 °F, what would a
ratio of -1 mean?
The Chicago Guide to Writing about Numbers, 2nd edition.
Quantitative comparisons for
categorical variables
More limited comparisons
• Ordinal variables
– Rank, e.g., “excellent health” is better than “poor health.”
– CANNOT interpret difference or ratio of values
• e.g. what would “excellent health” minus “poor health” mean?
• Nominal variables
– Can only ask “same or different” values
– CANNOT rank, or calculate difference or ratio
• e.g., what would it mean to say Jewish is higher than Catholic?
The Chicago Guide to Writing about Numbers, 2nd edition.
Review: Which type(s) of comparison
fits which level(s) of measurement?
• Can assess “same” or “different” values across cases
for all types of variables.
– Nominal, ordinal, interval, or ratio variables.
• Can rank values for any of the quantitative variables.
– Ordinal, interval, or ratio variables.
• Can subtract values for either type of continuous
variable.
– Interval or ratio variables.
• Can divide values only for ratio variables.
Summary
• How to choose which type(s) of quantitative
comparison best suit to the question at hand.
– Each type of calculation answers some questions well but
leaves other questions unanswered.
– Pick types of quantitative comparison(s) that suit the levels
of measurement of the variables in your analysis.
• How to report and interpret the results of those
calculations clearly and correctly.
– Convey the topic, not just the numbers.
– Report direction and magnitude.
– Avoid using jargon about the calculation.
The 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 4, section on levels of measurement
– Chapter 5, on types of quantitative comparisons
• Chambliss, D.F., and R.K. Schutt. 2012. Making Sense
of the Social World: Methods of Investigation. 4th ed.
Thousand Oaks, CA: Sage Publications.
– Chapter 4, on levels of measurement
The Chicago Guide to Writing about Numbers, 2nd edition.
Suggested online resources
• Podcasts on
– Reporting one number
– Comparing two numbers or series of numbers
– Getting to know your variables
– Choosing a comparison group or value
– Writing about ratios
The Chicago Guide to Writing about Numbers, 2nd edition.
Suggested practice exercises
• Study guide to The Chicago Guide to Writing
about Numbers, 2nd Edition.
– Exercises #1-6 and #8 in the problem set for
chapter 5
– Suggested course extensions for chapter 5
• “Reviewing” exercise #1
• “Applying statistics” exercise #1
• “Writing and revising” exercises #1 and 2
The Chicago Guide to Writing about Numbers, 2nd edition.
Contact information
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.