The missing denominator

What it looks like

A percentage is a ratio with the bottom half deleted. Once it is gone, the reader cannot recover it, and — this is the part that makes the tactic work — does not usually try.

The same shape appears wherever a number arrives without its base: a risk that doubles, a crime that surges, a test that comes back positive. In each case the figure is arithmetically correct and nearly uninformative on its own.

Why it works

Reyna and Brainerd named the mechanism denominator neglect: people encode the top of a ratio as a salient quantity and the bottom as background, so larger raw counts drive judgments of probability even when the probability is unchanged.1

Its close relative is base-rate neglect, and the classic demonstration is uncomfortable. Casscells and colleagues asked 60 physicians and medical staff a screening question with a disease prevalence of 1 in 1,000 and a 5 percent false-positive rate. Only 11 of the 60 gave the roughly correct answer. Twenty-seven answered 95 percent — they used the test’s error rate and discarded the base rate entirely.2

The applied version is relative versus absolute risk. A Cochrane review of 35 studies concluded that people judge a risk as larger, and are more willing to act on it, when it is stated in relative terms — and that this holds for clinicians as much as for patients.3 "Halves your risk" and "takes it from 2 in 1,000 to 1 in 1,000" are the same finding.

Documented cases

The same 18 percent, with and without its base

In October 2015 the International Agency for Research on Cancer reported an 18 percent increase in colorectal cancer risk for each 50 grams of processed meat eaten daily. On the same day, at the UK Science Media Centre, the statistician David Spiegelhalter supplied what the percentage had left out: around 6 in every 100 people would be expected to develop bowel cancer in their lifetime, and the finding moves that to about 7 in 100.5

Both numbers describe the same result. One of them is checkable against a reader’s own life and the other is not. Cancer Research UK’s 2024 update gives the modern equivalent in the same form: about 45 cases per 10,000 at low intake, and about 14 more per 10,000 at high intake.6

A small base makes ordinary noise look like a trend

The Florida Museum’s International Shark Attack File recorded 69 unprovoked bites worldwide in 2023, 47 in 2024 and 65 in 2025, against a ten-year average of around 70.7

Reported as percentages, that is a 32 percent collapse followed by a 38 percent surge. Reported as counts, it is a small number moving around by a handful. Nothing happened. The File publishes the raw counts alongside the changes, which is exactly the practice that makes the tactic impossible to run on it.

What to do about it

• Ask for the two raw numbers. "Up 300 percent" cannot be checked. "Three to twelve" can be, in seconds.
• Ask "out of how many, and over what period?" A count without a population and a window is not a rate.
• Convert relative to absolute yourself. An 18 percent rise on a base of 6 in 100 is 6 to 7, not 6 to 24. This is arithmetic you can do in your head and almost nobody does.
• Compare against several years, not last year. On a small base, one-year swings are usually noise wearing a percentage.
• For any positive test result, ask what the condition’s prevalence is. Without it, the accuracy of the test tells you much less than it appears to.

Sources

Every figure on this page comes from one of these. Each entry names the publisher, the date and the sample size, so you can check it rather than take our word for it.

1. Reyna, V. F. & Brainerd, C. J. — "Numeracy, ratio bias, and denominator neglect in judgments of risk and probability"Learning and Individual Differences 18(1):89–107 · 2008 · review articlehttps://eric.ed.gov/?id=EJ786722
2. Casscells, W., Schoenberger, A. & Graboys, T. B. — "Interpretation by Physicians of Clinical Laboratory Results"New England Journal of Medicine 299(18):999–1001 · 1978 · n=60https://jamanetwork.com/journals/jamainternalmedicine/fullarticle/1861033
3. Akl, E. A. et al. — "Using alternative statistical formats for presenting risks and risk reductions"Cochrane Database of Systematic Reviews, CD006776.pub2 · 2011 · 35 studies includedhttps://www.cochranelibrary.com/cdsr/doi/10.1002/14651858.CD006776.pub2/full
4. Schwitzer, G. — "How Do US Journalists Cover Treatments, Tests, Products, and Procedures?"PLoS Medicine 5(5):e95 · 27 May 2008 · n=500 storieshttps://journals.plos.org/plosmedicine/article?id=10.1371%2Fjournal.pmed.0050095
5. Science Media Centre — "Expert reaction to IARC classification of processed meat"UK Science Media Centre · 26 October 2015 · IARC drew on 10 studieshttps://www.sciencemediacentre.org/xpert-reaction-to-iarc-classification-of-processed-meat-as-carcinogenic-to-humans-group-1-and-red-meat-as-probably-carcinogenic-to-humans-group-2a/
6. Patrick, G. & Johnson, J. — "How does processed meat cause cancer and how much matters?"Cancer Research UK · August 2024 · drawing on a cohort of ~500,000 adultshttps://news.cancerresearchuk.org/2024/08/01/bacon-ham-hot-dogs-salami-how-does-processed-meat-cause-cancer-and-how-much-matters/
7. International Shark Attack File — annual reports for 2024 and 2025Florida Museum of Natural History, University of Florida · 11 Feb 2025 and 18 Feb 2026 · n=47 and n=65 biteshttps://news.ufl.edu/2026/02/shark-bites/
8. Sumner, P. et al. — "Exaggerations and Caveats in Press Releases and Health-Related Science News"PLoS ONE · 15 December 2016 · 996 press releases, 1,250 news articleshttps://journals.plos.org/plosone/article?id=10.1371%2Fjournal.pone.0168217