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CHAPTER 5

Numbers and Nonsense

O UR WORLD IS THOROUGHLY QUANTIFIED. Everything is counted, measured, analyzed, and
assessed. Internet companies track us around the Web and use algorithms to predict what we
will . Smartphones count our steps, measure our calls, and trace our movements
throughout the day. “Smart appliances” monitor how we use them and learn more about our
daily routines than we might care to realize. Implanted medical devices collect a continuous
stream of data from patients and watch for danger signs in real time. During service visits, our
cars upload data about their performance and about our driving habits. Arrays of sensors and
cameras laid out across our cities monitor everything from traffic to air quality to the
identities of passersby.

Instead of collecting data about what people do through costly studies and surveys,
companies let people come to them—and then record what those consumers do. Facebook
knows whom we know; Google knows what we want to know. Uber knows where we want to
go; Amazon knows what we want to . Match knows whom we want to marry; Tinder
knows whom we want to be swiped by.

Data can help us understand the world based upon hard evidence, but hard numbers are a
lot softer than one might think. It’s like the old joke: A mathematician, an engineer, and an
accountant are applying for a job. They are led to the interview room and given a math quiz.
The first problem is a warm-up: What is 2 + 2? The mathematician rolls her eyes, writes the
numeral 4, and moves on. The engineer pauses for a moment, then writes “Approximately 4.”
The accountant looks around nervously, then gets out of his chair and walks over to the fellow
administering the test. “Before I put anything in writing,” he says in a low whisper, “what do
you want it to be?”

Numbers are ideal vehicles for promulgating bullshit. They feel objective, but are easily
manipulated to tell whatever story one desires. Words are clearly constructs of human minds,
but numbers? Numbers seem to come directly from Nature herself. We know words are
subjective. We know they are used to bend and blur the truth. Words suggest intuition,
feeling, and expressivity. But not numbers. Numbers suggest precision and imply a scientific
approach. Numbers appear to have an existence separate from the humans reporting them.

People are so convinced of the primacy of numbers that skeptics claim they “just want to
see the data,” or demand to be shown “the raw numbers,” or insist that we “let the
measurements speak for themselves.” We are told that “the data never lie.” But this
perspective can be dangerous. Even if a figure or measurement is correct, it can still be used
to bullshit, as we’ll demonstrate later in the chapter. For numbers to be transparent, they

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must be placed in an appropriate context. Numbers must presented in a way that allows for
fair comparisons.

Let’s start by thinking about where people find their numbers. Some numbers are
obtained directly, through an exact tally or an immediate measurement. There are 50 states in
the United States of America. There are 25 prime numbers less than 100. The Empire State
Building has 102 floors. Baseball legend Tony Gwynn had 3,141 hits in 9,288 at-bats, for a
lifetime Major League batting average of .338. In principle, exact counts should be fairly
straightforward; there is a definitive answer and usually a definitive procedure for counting or
measuring that we can use to find it. Not that this process is always trivial—one can miscount,
mismeasure, or make mistakes about what one is counting. Take the number of planets in the
solar system. From the time that Neptune was recognized as a planet in 1846 through to 1930
when Pluto was discovered, we believed there were eight planets in our solar system. Once
Pluto was found, we said there were nine planets—until the unfortunate orb was demoted to a
“dwarf planet” in 2006 and the tally within our solar system returned to eight.

More often, however, exact counts or exhaustive measurements are impossible. We cannot
individually count every star in the measurable universe to arrive at the current estimate of
about a trillion trillion.

Likewise we rely on estimates when we consider quantities such as the average height of
adults by country. Men from the Netherlands are supposedly the tallest in the world, at an
average height of 183 cm (six feet), but this was not determined by measuring every Dutch
man and taking an average across the entire population. Rather, researchers took random
samples of men from the country, measured the members of that sample, and extrapolated
from there.

If one measured only a half dozen men and took their average height, it would be easy to
get a misleading estimate simply by chance. Perhaps you sampled a few unusually tall guys.
This is known as sampling error. Fortunately, with large samples things tend to average out,
and sampling error will have a minimal effect on the outcome.

There can also be problems with measurement procedures. For example, researchers
might ask subjects to report their own heights, but men commonly exaggerate their heights—
and shorter men exaggerate more than taller men.

Other sources of error, such as bias in the way a sample is selected, are more pernicious.
Suppose you decide to estimate people’s heights by going to the local basketball court and
measuring the players. Basketball players are probably taller than average, so your sample
will not be representative of the population as a whole, and as a result your estimate of
average height will be too high. Most mistakes of this sort aren’t so obvious. We devote the
rest of the chapter to considering the subtle ways in which a sample can turn out to be
uncharacteristic of the population.

In these examples, we are observing a population with a range of values—a range of
heights, for instance—and then summarizing that information with a single number that we
call a summary statistic. For example, when describing the tall Dutch, we reported a mean
height. Summary statistics can be a nice way to condense information, but if you choose an
inappropriate summary statistic you can easily mislead your audience. Politicians use this
trick when they propose a tax cut that will save the richest 1 percent of the population
hundreds of thousands of dollars but offer no tax reduction whatsoever to the rest of us.
Taking a mean of the tax cuts, they report that their tax plan will save families an average
$4,000 per year. Maybe so, but the average family—if by that we mean one in the middle of
the income range—will save nothing. The majority of us would be better off knowing the tax
cut for a family with the median income. The median is the “middle” income: Half of US

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families earn more and half earn less. In this particular case, the median family would get no
tax cut at all, because the tax cut benefits only the top 1 percent.

Sometimes one cannot directly observe the quantity one is trying to measure. Recently
Carl blew through a speed trap on a straight flat highway in the Utah desert that was
inexplicably encumbered with a 50 mph speed limit. He pulled over to the side of the road
with the familiar red and blue lights glaring in the rearview mirror. “Do you know how fast
you were going?” the state patrolman asked.

“No, I’m afraid not, Officer.”

“Eighty-three miles per hour.”

Eighty-three: a hard number with the potential to create serious problems. But where did
this number come from? Some traffic cameras calculate speed by measuring distance you
travel over a known interval of time, but that’s not how the state patrol does it. The patrolman
measured something different—the Doppler shift in radio waves emitted from his portable
radar gun when they reflected off of Carl’s speeding vehicle. The software built into his radar
gun used a mathematical model grounded in wave mechanics to infer the car’s speed from its
measurements. Because the trooper was not directly measuring Carl’s speed, his radar gun
needed to be regularly calibrated. A standard method for getting out of a speeding ticket is to
challenge the officer to produce timely calibration records. None of that in Carl’s case, though.
He knew he had been speeding and was grateful to face only a stiff fine for his haste.

Radar guns rely on highly regular principles of physics, but models used to infer other
quantities can be more complicated and involve more guesswork. The International Whaling
Commission publishes population estimates for several whale species. When they state that
there are 2,300 blue whales left in the waters of the Southern Hemisphere, they have not
arrived at this number by going through and counting each animal. Nor have they even
sampled a patch of ocean exhaustively. Whales don’t hold still, and most of the time you can’t
see them from the surface. So researchers need indirect ways to estimate population sizes. For
example, they use sightings of unique individuals as identified by markings on the flukes and
tail. To whatever degree these procedures are inaccurate, their estimate of population size
may be off.

There are many ways for error to creep into facts and figures that seem entirely
straightforward. Quantities can be miscounted. Small samples can fail to accurately reflect the
properties of the whole population. Procedures used to infer quantities from other
information can be faulty. And then, of course, numbers can be total bullshit, fabricated out of
whole cloth in an effort to confer credibility on an otherwise flimsy argument. We need to
keep all of these things in mind when we look at quantitative claims. They say the data never
lie—but we need to remember that the data often mislead.

DISTILLING NUMBERS

T hough unaged whiskey has become trendy of late,*1 freshly distilled whiskey can often be
harsh and laden with undesirable by-products from the distillation process. A couple of years
in a freshly charred oak barrel (for bourbon) or a longer period in a previously used barrel (for
scotch) brings about a remarkable transformation. Flavors from the wood penetrate the liquor
and some of the undesirable chemicals in the liquor are extracted through the wood.

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This alchemy does not transpire for free. As the liquor ages in the barrel, a portion seeps
out and evaporates into the air. A barrel that begins full will hold only a fraction of its initial
volume by the time the aging process is complete. The portion of spirits lost to evaporation is
known as the “angels’ share.” Romantic imagery aside, the angels’ share represents a
substantial cost in the production of bourbon and scotch.

How can we best describe this cost? We could start with the total loss: Approximately
440,000 barrels of whiskey are lost to evaporation in Scotland each year. Most people don’t
know how big a whiskey barrel is (about 66 gallons), so we might do better to say that in
Scotland, about 29 million gallons are lost every year to the angels. We usually encounter
whiskey in 750 ml bottles rather than gallons, so perhaps we would want to report this as a
loss of 150 million bottles a year.

Aggregate totals are hard to grasp unless one knows the total amount of scotch being
produced. We could break these numbers down and describe the amount of liquid lost by a
single distillery during the process of barrel aging. Running at full capacity, the large Speyside
distillery Macallan loses about 220,000 LPA—liters of pure alcohol—per year. (Notice yet
another type of measurement; a distillery’s capacity is often reported by tallying only the
alcohol produced, not the total volume including water.) By contrast, the smaller Islay
distillery Ardbeg loses about 26,000 LPA per year.

Because distilleries vary widely in size, perhaps we should report loss per barrel or, better
yet, as a percentage of the starting volume. During the aging process for the legendary Pappy
Van Winkle twenty-three-year-old bourbon, 58 percent of its initial volume is lost to
evaporation. But instead of describing the loss as a percentage of the starting volume, I could
describe it as a percentage of the final volume. For this bourbon, 1.38 liters are lost to
evaporation for every liter bottled, so we could report the loss as 138 percent of the final
volume. It is the exact same figure as the 58 percent of starting volume described above, but
this way of presenting the data makes the loss appear larger.

Of course, different whiskies are aged for different amounts of time. Maybe instead of
describing the total loss, it would make more sense to describe the annual loss. Scotch
whiskies lose about 2 percent in volume per year of aging, or roughly 0.005 percent per day.
Bourbons typically are aged under higher temperatures than scotch, and thus experience
higher rates of evaporation; some may lose upward of 10 percent per year. Moreover, the rate
of loss is not constant. The aforementioned Pappy Van Winkle loses about 10 percent of its
volume over its first year in the barrel, but this drops to about 3 percent per year later in the
aging process.

There are other decisions to make as well. For example, alcohol and water leave the barrel
at different rates; we could report changes in alcohol volume, water volume, or the total. And
then there is the issue of units: Metric or imperial? Liters or milliliters? Gallons or ounces?

To tell an honest story, it is not enough for numbers to be correct. They need to be placed
in an appropriate context so that a reader or listener can properly interpret them. One thing
that people often overlook is that presenting the numbers by themselves doesn’t mean that
the numbers have been separated from any context. The choices one makes about how to
represent a numerical value sets a context for that value.

So what does it mean to tell an honest story? Numbers should be presented in ways that
allow meaningful comparisons.

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As one of us (Carl) is writing this chapter, the supply of malted milk ball candies in the box
of Hershey’s Whoppers on his desk is gradually dwindling—but there is no guilt, because a
prominent splash of color on the box announces “25% Less Fat* Than the Average of the
Leading Chocolate Candy Brands.” The asterisk directs us to small print reading “5 grams of
fat per 30 gram serving compared to 7 grams of fat in the average of the leading chocolate
candy brands.” Here’s an example of a number provided without enough context to be
meaningful. What brands are chosen as comparators? Is this an apples-to-apples comparison
or are we comparing chocolate-covered malt balls to straight chocolate bars? How about
sugar? Refined sugar may be a bigger health concern than fat; are Whoppers higher or lower
in sugar? Are there other bad ingredients we should be concerned about? And so on, and so
forth. The 25 percent figure sounds like an important nutritional metric, but it is really just
meaningless numerosity.

PERNICIOUS PERCENTAGES

T he twelfth chapter of Carl Sagan’s 1996 book, The Demon-Haunted World, is called “The
Fine Art of Baloney Detection.” In that chapter, Sagan rips into the advertising world for
bombarding us with dazzling but irrelevant facts and figures. Sagan highlights the same
problem we address in this chapter: people are easily dazzled by numbers, and advertisers
have known for decades how to use numbers to persuade. “You’re not supposed to ask,” Sagan
writes. “Don’t think. .”

Sagan focuses on the marketing tactics used in drug sales, a problem that expanded in
scope a year after Sagan wrote his essay. In 1997, the United States legalized direct-to-
consumer marketing of prescription drugs. But rather than take on that troubling and
complex issue, let’s instead consider an amusing and largely harmless example.

Arriving in Washington, DC, late one evening, Carl was searching for something to drink
before bed. He picked up a packet of instant hot cocoa in the hotel lobby. “99.9% caffeine-
free,” the packaging boasted. Given that he was already dealing with jet lag, a 99.9 percent
caffeine-free drink seemed like a prudent alternative to a cup of coffee. But pause and think
about it for a minute. Even though there’s a lot of water in a cup of cocoa, caffeine is a
remarkably powerful drug. So is a 99.9 percent caffeine-free drink really something you want
to drink right before bed?

Let’s figure it out. How much caffeine is in a cup of coffee? According to the Center for
Science in the Public Interest, there are 415 milligrams of caffeine in a 20-ounce Starbucks
coffee. That corresponds to about 21 mg of caffeine per ounce. A fluid ounce of water weighs
about 28 grams. Thus, a Starbucks drip coffee is about 0.075 percent caffeine by weight. In
other words, strong coffee is also 99.9 percent caffeine free!*2

So while there’s nothing inaccurate or dangerous about the 99.9 percent assertion, it’s a
pointless claim. Most regular coffees could be labeled in the exact same way. Nestlé has
provided us with an excellent example of how something can be true and still bullshit of a
sort. It is bullshit because it doesn’t allow us to make meaningful comparisons the way a claim
such as “only 1% as much caffeine as a cup of coffee” would.

A notorious Breitbart headline similarly denied readers the opportunity to make
meaningful comparisons. This particular bit of scaremongering proclaimed that 2,139 of the
DREAM Act recipients—undocumented adults who came to the United States as children—
had been convicted or accused of crimes against Americans.*3 That sounds like a big number,
a scary number. But of course, the DREAM Act pertains to a huge number of people—nearly
700,000 held DACA status concurrently and nearly 800,000 were granted DACA status at
some point before the program was eliminated. This means that only about 0.3 percent of all

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DACA recipients—fewer than 1 in 300—have been accused of crimes against Americans. That
sounds better, but how does this number compare to similar rates for American citizens?
With 0.75 percent of Americans behind bars, US citizens are twice as likely to be presently
incarcerated as DACA recipients are to have been accused of a crime. About 8.6 percent of
American citizens have been convicted of a felony at some point in their lives, making the
DACA population look better still.

Of course, DACA recipients are younger and had generally not been convicted of a crime
prior to being awarded DACA status,*4 so they have had less time to commit crimes than your
average American. But it turns out that 30 percent of Americans have been arrested for
something other than a traffic violation by age twenty-three. Even assuming that Breitbart’s
figures were correct, the news outlet presented them without the appropriate information a
reader would need to put them into context.

Listing a raw total like this can make a small quantity, relatively speaking, appear to be
large. We put this number into context by expressing it as a percentage. Indeed, percentages
can be valuable tools for facilitating comparisons. But percentages can also obscure relevant
comparisons in a number of ways. For starters, percentages can make large values look small.

In a blog post, Google VP of engineering Ben Gomes acknowledged the problem that their
company faces from fake news, disinformation, and other inappropriate content:

Our algorithms help identify reliable sources from the hundreds of billions of pages
in our index. However, it’s become very apparent that a small set of queries in our
daily traffic (around 0.25 percent), have been returning offensive or clearly misleading
content, which is not what people are looking for.

Two things are going on here. First, a large and largely irrelevant number is presented as if
it helps set the context: “hundreds of billions of pages in our index.” Juxtaposed against this
vast number is a tiny one: “0.25 percent.” But the hundreds-of-billions figure is largely
irrelevant; it is the number of pages indexed, not anything about the number of search
queries. It doesn’t matter whether ten thousand or a hundred billion pages are indexed; if
0.25 percent of Google searches are misleading, you have a one-in-four-hundred chance of
getting bullshit in your search results.*5

We are not told how many search queries Google handles per day, but estimates place this
figure at about 5.5 billion queries a day. So while 0.25 percent sounds small, it corresponds to
well over thirteen million queries a day. These two ways of saying the same thing have very
different connotations. If we tell you that Google is returning inappropriate search results
only one time in four hundred, the system seems pretty sound. But if we tell you that more
than thirteen million queries return inappropriate and inaccurate content every day, it sounds
as though we are facing a serious crisis in information delivery.

Percentages can be particularly slippery when we use them to compare two quantities. We
typically talk about percentage differences: “a 40 percent increase,” “22 percent less fat,” etc.
But what is this a percentage of? The lower value? The higher value? This distinction matters.
In the month of December 2017, the value of the bitcoin digital currency first surged to
$19,211 per unit on the seventeenth of the month, and then plummeted to a low of $12,609
per unit thirteen days later. This is a decrease of $6,602 per unit. But what was the percentage

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change? Should we say it was 34 percent (because $6,602 is 34.3 percent of $19,221), or is it
52 percent (because $6,602 is 52.4 percent of $12,609)?

One can make a legitimate case for either number. In general, we advocate that a
percentage change be reported with respect to the starting value. In this case the starting
value was $19,211, so we say bitcoin lost 34 percent of its value over those thirteen days. This
can be a subtle issue, however. One would say that bitcoin lost 34 percent of its value over this
period, because when we talk about a loss in value, the starting value is the appropriate
comparison. But we would also say that bitcoin was apparently overvalued by 52 percent at
the start of December 2017, because when we talk about something being overvalued, the
appropriate baseline for comparison is our current best estimate of value. Different ways of
reporting give different impressions.

Instead of listing percentage changes, studies in health and medicine often report relative
risks. New drivers aged sixteen and seventeen years have some of the highest accident rates
on the road. But their accident rates depend on whether they are carrying passengers, and on
who those passengers are. Compared to teen drivers without passengers, the relative risk of a
teen driver dying in a crash, per mile driven, is 1.44 for teens carrying a single passenger
under the age of twenty-one. This relative risk value simply tells us how likely something is
compared to an alternative. Here we see that teen drivers with a young passenger are 1.44
times as likely to be killed as teen drivers without a passenger. This is easily converted to a
percentage value. A teen driver with a passenger has a 44 percent higher chance of being
killed than a teen driver without a passenger. Carrying an older passenger has the opposite
effect on the risk of fatal crashes. Compared to teen drivers without passengers, the relative
risk of a teen driver dying in a crash while carrying a passenger over the age of thirty-five is
0.36. This means that the rate of fatal crashes is only 36 percent as high when carrying an
older passenger as it is when driving alone.

Relative risks can help conceptualize the impact of various conditions, behaviors, or
health treatments. But they sometimes fail to provide adequate context. A worldwide study of
alcohol-related disease was reported with stark headlines such as “There’s ‘No Safe Level of
Alcohol,’ Major New Study Concludes.” In particular, even very modest drinking—a single
drink a day—was found to have negative health consequences. That sounds bad for those of us
who enjoy a beer or glass of wine with dinner. But let’s look more closely.

The press release from The Lancet, where the study was published, reports that:

They estimate that, for one year, in people aged 15−95 years, drinking one alcoholic
drink a day increases the risk of developing one of the 23 alcohol-related health
problems by 0.5%, compared with not drinking at all.

Scary stuff? To evaluate whether this is a substantial increase or not, we need to know how
common are “alcohol-related health problems”—liver cirrhosis, various cancers, some forms
of heart disease, self-harm, auto accidents, and other maladies—among nondrinkers. It turns
out that these problems are rare among nondrinkers, occurring in less than 1 percent of the
nondrinking population in a year. And while a drink per day increases this risk by 0.5 percent,
that is 0.5 percent of the very small baseline rate. In other words, the relative risk of having
one drink a day is 1.005. People who have a drink a day are 1.005 times as likely to suffer
“alcohol-related disease” as those who do not drink.

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The authors of the study calculated that having a single daily drink would lead to four
additional cases of alcohol-related illness per 100,000 people. You would have to have 25,000
people consume one drink a day for a year in to cause a single additional case of illness.
Now the risk of low-level drinking doesn’t sound as severe. To provide further perspective,
David Spiegelhalter computed the amount of gin that those 25,000 people would drink over
the course of the year: 400,000 bottles. Based on this number, he quipped that it would take
400,000 bottles of gin shared across 25,000 people to cause a single additional case of illness.

To be fair, this is the risk from drinking one drink a day; the risk rises substantially for
those who consume larger amounts. People who drink two drinks a day have a relative risk of
1.07 (7 percent higher than nondrinkers) and those who drink five drinks a day have a relative
risk of 1.37. The main point is that simply reporting a relative risk of a disease isn’t enough to
assess the effect unless we also know the baseline rate of the disease.

Percentages get even more slippery when we are comparing one percentage figure to
another. We can look at the numerical difference between two percentages, but we can also
create a new percentage, reflecting the percentage difference between our percentage values.
Even professional scientists sometimes mix up a subtle issue in this regard: the difference
between percentages and percentage points. An example is by far the easiest way to illustrate
the difference. Suppose that on January 1, the sales tax increases from 4 percent to 6 percent
of the purchase price. This is an increase of 2 percentage points: 6% − 4% = 2%. But it is also
an increase of 50 percent: The 6 cents that I now pay on the dollar is 50 percent more than
the 4 cents I paid previously.

So the same change can be expressed in very different ways that give substantially
different impressions. If I want to make the tax increase sound small, I can say that there has
been only a 2 percentage point increase. If I want to make it sound large, I can say that taxes
have gone up 50 percent. Whether accomplished by accident …

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