A2: The Denominator Demand

The number you already believed

Imagine a post from a group whose cause you share. It says that incidents of the thing you have long worried about have risen by 60 percent in a year, and that 400 people were affected. You feel the small, grim satisfaction of being proved right, and your thumb is already on the share button.

Now suppose the same post had come from the other side, about the thing they worry about. You would probably ask questions at once. Sixty percent of what? Four hundred out of how many? Rose from what, and is that a lot or a little for a country this size? Those questions were just as good when the number flattered you. It is easy not to ask them then, which is the habit A4: The Mirror Check exists to catch.

The move

The Denominator Demand is a tool, which on this site means a mental operation you can run on a claim in real time, before you believe it or pass it on. Its whole content fits in the tagline: when you meet a frightening number, ask "out of how many?"

A count of things, on its own, tells you very little. Four hundred cases is an emergency in a village and a rounding error in a nation. A rise of 60 percent is alarming if it means thousands more people harmed and close to meaningless if it means five cases became eight. The number underneath, the one the count should be divided by, is called the denominator. A frightening statistic often arrives without it. The most everyday form of the demand is asking for a per-person, or "per capita," figure instead of a total, since a large state will have more of almost anything than a small one.

Running it takes four steps.

1. Notice the bare number: a count, a percentage rise, a "doubled" or "tripled," a list of cases.
2. Ask what it is out of. How many people, events or opportunities could the thing have happened to?
3. Ask what it rose from, and what the usual background rate is. Is this year unusual, or is it a normal year that happens to be in the news?
4. Rewrite the claim with the denominator in it, and see whether it still frightens you, and how much.

The fourth step matters most. The tool does not decide whether a harm is real. It turns a feeling into a quantity you can argue about.

Why it works

The idea can be put bluntly: a number without a base rate is a mood, not a fact. A base rate is how often something happens in the relevant population over a normal stretch of time. Without it, a count tells you that something happened, not whether it is common, rare, growing or shrinking.

This tool belongs to the family the site calls Bias Literacy, which deals with errors that happen before reasoning starts. Research on a closely related error is long established. In a 1974 paper in Science, the psychologists Amos Tversky and Daniel Kahneman summarized an experiment they had reported the year before, in which people told that a group held 70 engineers and 30 lawyers, or the reverse, used those proportions correctly when told nothing else, but called it an even chance when given a description of one member that said nothing about his job. "When worthless evidence is given," the authors wrote, "prior probabilities are ignored." The habit is now known as base-rate neglect. Their experiment concerned judging one person from a description, not reading a news statistic, but the failure is of the same kind: a background rate was dropped.

How fixed the habit is remains disputed. Gerd Gigerenzer and Ulrich Hoffrage reported in 1995 that when the same kind of problem was posed as plain counts rather than probabilities, the share of correctly reasoned answers in their first study rose from 16 percent to between 46 and 50 percent. In the 2007 review, Gigerenzer and his colleagues call such counts "natural frequencies": "ten out of every 1,000 women," for example, rather than a probability of 1 percent. The tool's fourth step, rewriting a claim as a count out of a stated whole, applies the same idea to the numbers in the news. That application is this site's, not the researchers' claim.

In health news, one common way a denominator goes missing is the gap between relative risk, the kind of figure behind a headline saying a drug doubles your risk, and absolute risk, the kind that says the risk went from one in 7,000 to two in 7,000. As Gigerenzer and four colleagues put it in a 2007 review of how doctors and patients read health statistics, "Absolute risks are typically small numbers while the corresponding relative changes tend to look big—particularly when the base rate is low."

The site's editorial standards count this among five tools that carry disproportionate weight and get priority for long treatment, because, in the standards' words, most manufactured fear dies on contact with a real base rate. That is the site's working judgment, not a research finding, and the site's articles will test it.

Using it well

The legitimate use is asking what a rising statistic rose from, and against what background rate.

In October 1995 Britain's Committee on Safety of Medicines warned that newer contraceptive pills doubled the risk of dangerous blood clots. As the 2007 review tells it, the doubling was what the studies showed, but the denominator was missing: "of every 7,000 women who took the earlier, second-generation oral contraceptive pills, about 1 had a thrombosis; this number increased to 2 among women who took third-generation pills." Citing a 1999 paper by Ann Furedi, whose listed affiliation was the British Pregnancy Advisory Service, a charity that provides abortions and today describes itself as "advocating and caring for women and couples who decide to end a pregnancy," the review reports "an estimated 13,000 additional abortions" in England and Wales the following year as women stopped taking the pill. A figure from an interested source is worth checking, and the government's own statistics bear out its size: abortions to residents of England and Wales rose from 154,315 in 1995 to 167,916 in 1996, about 13,600 more. A rise in one year does not by itself show how much of it the scare caused. The review adds that pregnancy and abortion themselves carry a higher clot risk than the newer pill did.

The tool did not say the risk was zero or the warning false. It found the missing quantity and pointed to the comparison that bore on the decision: the risk of the pill against the risk of what replaces it. When the base rate confirms the alarm, the tool has done its job just as well.

It is harder to run on a number that confirms what you already think. In November 2018 a report on the website of a Jacksonville television station opened: "Hate crimes across the United States spiked 17 percent in 2017 - marking a rise for the third straight year - with a 37 percent increase in anti-Semitic hate crimes, according to an FBI report released Tuesday." The FBI's own release supplied the missing base in its second paragraph: "Although the numbers increased last year, so did the number of law enforcement agencies reporting hate crime data—with approximately 1,000 additional agencies contributing information." The count was out of the agencies that reported, and that number had grown. This is a change in who was counting rather than in the population, but the effect is the same: some of the rise could be new counters, not new crimes. (The station did add, in its fourth paragraph, that some of the rise "may be the result of better reporting by police departments.") None of this shows hate crimes did not rise; it shows this figure could not say by how much. A reader who already believed hatred was rising had every reason to share the headline, and the same reason to ask the question they would have asked at once of a number pointing the other way.

A number can flatter the other side of politics just as easily. At a rally in Evansville, Indiana, on 30 August 2018, President Donald Trump said of wind turbines, as transcribed by FactCheck.org, a fact-checking project of the University of Pennsylvania's Annenberg Public Policy Center: "When there's thousands of birds laying at the base of the windmill, what do we do? ... What about the thousands of birds they're killing?" FactCheck.org ran the demand. Dividing the federal estimate of birds killed by turbines each year by the number of turbines in the country, it wrote, "basic math would suggest that each turbine is responsible for about 12 birds a year," and it cited a 2013 study putting the average at "just over five." A reader who distrusts wind power had every reason to repeat the rally line and the same reason to ask how many turbines there were.

But the right denominator is not simply the biggest one available, and the best statement of that on this page comes from the same case. In 2007 a committee of the National Research Council, part of the US National Academies, compared bird deaths from turbines with all bird deaths caused by human activity in the United States. It put turbine deaths at an estimated 20,000 to 37,000 birds a year in 2003, against 97 million to 976 million a year from collisions with buildings alone and a total from all human causes that may approach a billion. Turbines, it concluded, caused "less than 0.003% in 2003." The industry welcomed the comparison. In May 2007 Randall Swisher, then executive director of the American Wind Energy Association, an industry trade group, told one trade publication: "More than a thousand times as many birds are killed flying into buildings, for example, than wind turbines." For birds as a whole, that is fair. But the committee itself said it approached the comparison "with great hesitation." "The ecological consequences and conservation implications of the deaths of 10,000 starlings," it wrote, "are far different from those of the deaths of 10,000 bald eagles," and a national total "is not the appropriate spatial scale to address the question of interest." Six years later, in November 2013, Duke Energy Renewables pleaded guilty to federal charges over "14 golden eagles and 149 other protected birds" found at two of its Wyoming wind projects between 2009 and 2013. Fourteen eagles are nothing against every bird in the country, and may be a great deal against the eagles living near those turbines. The honest use asks which of those questions is being answered.

When it becomes a weapon

The weaponized form is what the site calls denominator-flooding: drowning a real, concentrated harm in an irrelevantly huge base rate to make it disappear. In the case below, as with the national bird figure above, the broad denominator is not absurd. It is plausible and somewhat too large for the question being asked, and that is what would make it persuasive. The example below shows a contest over which denominator to use; it does not establish that a harm was hidden, because whether there was a harm is the question in dispute.

The Denka Performance Elastomer plant in St. John the Baptist Parish, Louisiana, makes synthetic rubber. According to a 2019 ProPublica report, it "is the only one in the country that emits chloroprene, which was classified as a likely carcinogen by the U.S. Environmental Protection Agency in 2010," and in 2015 an EPA analysis said the airborne cancer risk in the census tract nearest the plant was the highest in the nation. (A census tract is a small area the Census Bureau uses to report statistics, generally home to between 1,200 and 8,000 people. The 2015 figure was an estimate of risk from what was in the air, not a count of cancers that had occurred.)

In a letter to the agency dated 1 March 2021, the company's lawyer argued from a larger denominator: state tumor-registry reports "continue to show that St. John the Baptist Parish regularly exhibits average or below-average rates of the cancer incidence compared with the state average." The company did not reach only for the parish. In 2019 its spokesman had told ProPublica that registry data showed "the number of all cancer cases combined in St. John the Baptist Parish and in the census tracts close to the facility were not statistically different than the numbers for the state." ProPublica added that "census tracts are still relatively large," and that those around Denka run from areas close to the plant to areas "at least a couple of miles away." On 21 July 2021 the Johns Hopkins epidemiologist Miranda Jones, in a reply to the company's letter that the agency published, objected that the parish comparison "assumes that all residents in St. John the Baptist Parish are exposed to the same level of chloroprene and that no individuals residing outside the parish have been exposed. This is likely not true."

This page takes no position on whether the plant's emissions have caused cancer. As of October 2026, no study this page has read settles it: the federal lawsuit filed over the emissions in 2023 was dropped by the Justice Department in March 2025 before it came to trial, and in May 2025 the company suspended production at the plant, citing financial losses. The point here is narrower: an average over a whole parish, or even over tracts that reach miles from the fence, answers a different question from the one the plant's nearest neighbors were asking.

This page's one documented case of a disputed, too-broad denominator comes from a company defending itself, and a reader who dislikes chemical companies should notice how comfortable that feels. The move is open to anyone minimizing a harm their own side would rather not count. The national bird figure above, fair for the question it answered, would become flooding if offered as the answer to whether the Wyoming eagles mattered. This page has not yet found a dated quotation of anyone using it that way, and will add a documented case from any side when it finds one.

The weaponized form sounds exactly like the legitimate tool. The difference is in which denominator gets chosen, and why. The honest use picks the population that was actually exposed to the risk, and asks what rate would matter for the decision at hand; flooding picks the largest population available and treats any small fraction as proof that nothing matters. The honest use stays open to the answer that a harm is concentrated, rare overall and severe where it lands; flooding stops the moment the fraction looks small. And the honest user demands denominators for numbers that help their side as readily as for numbers that hurt it, and states benefits and harms in the same terms. The 2007 review warns against the opposite habit in drug advertising, which is to "report benefits in relative risks (big numbers) and harms in absolute risks (small numbers)."

Run it on your own side first

Every tool on this site comes with this instruction, and it is the real test of whether the tool has been understood. Try these:

• Think of the statistic you quote most often on an issue you care about. Do you know what it is out of, and what it was the year before last?
• Think of a harm your side tends to wave away as rare. Have you divided it by everyone, or by the people actually exposed to it?
• When a number on your side went up recently, did you check what it rose from before you repeated it?

If you cannot answer the first question, you have been passing on a mood.

Grammars it works against

A grammar, on this site, is a recurring pattern of argument that turns up across many different subjects. This tool is one of the first ones to reach for against two of them.

• G1: Purity and Contamination. This grammar treats contact, not quantity, as what matters, so that a trace of the contaminant is held to spoil the whole. The Denominator Demand puts quantity back: how much, out of how much, and compared with what background level. It checks both poles, the person who treats any trace as poison and the person who denies that any real hazard exists, because an honest denominator can show a genuine risk as easily as it can dissolve a false one.
• G6: The Hidden Hand. Accounts of secret coordination often rest on a list of striking connections or coincidences. The tool asks how many such connections you would expect by chance in a network that size, and how many conspiracies of this kind have actually turned out to be real, bearing in mind that the known ones are a skewed sample: they are only the conspiracies that were exposed. That question cuts both ways: it deflates coincidence-hunting, and it also challenges the reflex that dismisses every claim of coordination, since real coordination has a base rate too.

Related tools

• A1: The Invisible-Cases Question asks which cases were filtered out before you ever saw them. A2 asks what the cases you did see are out of. The two are often run together.
• B1: The Effect-Size Reflex picks up where A2 leaves off: once you have a real rate, how big is the effect, for whom, and compared with what?
• B4: The Reference-Class Question asks what a case should be compared with. Choosing a denominator is choosing a reference class, and both can be shopped for.
• A3: The Newsfeed Inversion explains why a number can feel newly huge after a week of coverage: how often something appears in your feed is not how often it happens.
• A4: The Mirror Check asks whether you scrutinized a claim or just liked its conclusion. It is the reason to run this tool hardest on your own side's numbers.