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Vaccinated people are getting sick — but some counterintuitive math puts it in context

It’s a mistake to measure vaccine effectiveness by the percentage of hospitalized patients.

The spelling error isn't the only thing wrong with this slide cited by White House press secretary Jen Psaki.Anna Moneymaker/Getty

As infectious as a new strain of the coronavirus, a new kind of misleading statistic has cropped up in COVID-19 conversations over the past week.

It has emerged in response to a question being asked now with greater urgency: Will vaccines really keep us safe from the virus? An increasingly common way of answering the question is to cite the percentage of vaccinated people among those contracting COVID.

For example, officials in Provincetown were alarmed to find that the majority of cases in a recent outbreak were among vaccinated people. Reports out of Texas that 1 in 20 people coming to the hospital for COVID-19 had been vaccinated also prompted a discussion about vaccine effectiveness. Meanwhile, at the White House on Tuesday, press secretary Jen Psaki made the case for people to get their shots by pointing out that “virtually all” hospitalizations and deaths were among unvaccinated people.

The problem is, no matter the number, that percentage is the wrong stat to look at; it gets things backwards. What matters for gauging vaccine effectiveness is not the proportion of hospitalized people who were vaccinated but the proportion of vaccinated people who wind up in the hospital.


In fact, a high share of vaccinated people among the very sick could just be a sign that a lot of people have been vaccinated. Let’s say that vaccines are 95 percent effective, reducing someone’s chance of being hospitalized from 1 percent to 0.05 percent. And imagine that in a group of 1 million people, 90 percent were vaccinated. We’d expect 0.05 percent of the 900,000 vaccinated people — that is, 450 people — to go to the hospital. By comparison we’d have 1 percent of the 100,000 unvaccinated people — 1,000 people — in the hospital. The vaccinated would account for about one out of every three hospitalizations.


By analogy, among people who go to the dentist for a filling, almost everyone will have been brushing their teeth regularly, but that shouldn’t be taken as evidence that toothbrushing is ineffective at preventing cavities. It’s precisely because it is so important that it’s nearly universal. So should it be with the COVID vaccines. The higher the percent of the vaccinated in the population, the higher we’d expect the proportion of vaccinated sick people to be, all other things being equal. The correct measure of whether we’re on the right track, then, is whether the absolute number of new cases is going down.

The evidence has been compelling since the first clinical trials and remains clear today that vaccines are miraculously effective at preventing serious illness, even against new variants. No vaccine is perfect, and extra measures, such as the CDC’s re-endorsement of indoor masking for people in high-risk communities, may continue to be warranted. But the shortest path to lessening the numbers of sick and dead is to get the vaccine into as many arms as possible as quickly as possible. One hopes that someday soon, nearly everyone admitted to the hospital for COVID-19 will have been vaccinated because nearly everyone will be vaccinated, period.

Aubrey Clayton, a mathematician in Boston, is the author of “Bernoulli’s Fallacy: Statistical Illogic and the Crisis of Modern Science.” Follow him on Twitter @aubreyclayton.