"Cuts your risk in half" can mean one person in a hundred
Two in a hundred becomes one in a hundred. That is a 50% reduction and a 1-point one, and both numbers describe the same result.
A drug halves your risk of a disease. How much does it help you?
The honest answer is that the question cannot be answered from that sentence, because "halves" is a ratio and a ratio has thrown away the thing you need.
If the risk was 50% and is now 25%, one person in four is saved from it. If the risk was 2% and is now 1%, one person in a hundred is. Both are a 50% relative reduction. The first is a transformation of your life; the second means that if a hundred people take the drug, ninety-nine of them get exactly what they would have got anyway.
Three numbers from the same fact
Take the 2% → 1% case and write it three ways.
- Relative risk reduction: 50%. The largest number available, and the one that appears in the headline.
- Absolute risk reduction: 1 percentage point. What actually changes for the population.
- Number needed to treat: 100. The reciprocal of the absolute reduction — a hundred people take it for one to benefit.
None of these is wrong. They are the same fact with different denominators, and only one of them is a fifty. Which one gets printed is a choice, and it is made overwhelmingly in one direction: benefits reported in relative terms, harms in absolute ones, so a treatment reads as a large gain and a small cost.
The reciprocal in the third line is worth pausing on, because it behaves badly in a familiar way. As the absolute reduction shrinks, the number needed to treat rises without limit: a 0.1-point reduction needs a thousand people, a 0.01-point one needs ten thousand.
Base rates again, in a different disguise
Underneath, this is the same failure the Bayes tool makes visible with diagnostic tests: a number that sounds like it describes you is really a conditional, and the condition is a base rate you were not shown.
Set that tool to a prevalence of 0.10% with a 99% accurate test and look at what a positive result is worth. The test's accuracy is a property of the test; what a positive means for you depends on how common the disease is, and at low prevalence most positives are false. Relative risk has exactly the same structure: the ratio is a property of the treatment, and what it means for you depends on your starting risk, which the ratio does not contain.
Gigerenzer and Edwards showed in 2003 that this is not a public-understanding problem: doctors given the percentage version of these problems get them wrong at similar rates, and the same doctors get them right when the identical facts are stated as natural frequencies — "10 people in 1,000" rather than "1%". The arithmetic is unchanged; the format does the work. Converting to counts out of a fixed group is error correction, not dumbing down.
Two more places the same gap opens
Precision. A relative reduction quoted without an interval hides how well it is known. The confidence interval tool's point applies directly: a 50% reduction with an interval from 5% to 75% is a different claim from 50% ± 3%, and the point estimate is identical in both. Any risk number without a range is half a result.
What it cost to find. Rare outcomes need enormous studies, and the sample size tool shows why: precision improves as √n, so pinning down a 1-point difference takes vastly more people than a 20-point one. That is also why small absolute effects are so often reported in relative terms: the relative number is stable and impressive while the absolute one is small and uncertain, and the choice of framing quietly follows the incentive.
The test to apply is one subtraction. When you read that something halves, doubles or cuts by a third, ask: from what, to what? If the two numbers are not there, the sentence has told you about a ratio and nothing about your life. And if they are there, the subtraction takes a second and gives you the only figure that describes what happens to actual people.
References (1)
- natural frequencies instead of percentages, and why doctors get this wrong too Gigerenzer & Edwards (2003). Simple tools for understanding risks: from innumeracy to insight. BMJ 327(7417).