Americans offer a lot of rationalizations for not getting vaccinated against the coronavirus. Last month, the Kaiser Family Foundation conducted a poll in which those who hadn’t yet received a vaccine dose were invited to explain why. The rationalizations ranged from outright misinformation to incorrect assumptions (including that the vaccine costs money).

About an eighth of those who said they hadn’t been vaccinated, though, pointed to the idea that the vaccines themselves didn’t do much: 8 percent said that the vaccines weren’t needed and 4 percent that they weren’t effective at protecting against the virus.

Those particular misapprehensions should be easily addressed with a simple experiment.

The Centers for Disease Control and Prevention gathers data on the rate of coronavirus infections and covid-19 deaths relative to a patient’s vaccination status. In other words, over the course of the year, the CDC has collected information about whether someone who contracted the virus or died from it was vaccinated before that happened. It updates the figures regularly, but they haven’t changed much of late. Those who are vaccinated are about six times less likely to contract the virus and 14 times less likely to die from it.

Those rates vary by age, but, helpfully, the CDC provides data by age group. Among those 80 and older, the group most at risk, the effects were not as sharp. The CDC’s most recent data (from early October) suggests that the unvaccinated were nearly three times as likely to contract the virus and more than five times as likely to die from it. But if you’re 85, that’s still a significant difference.

We can visualize this. We took the CDC’s rates of infection or death for two weeks this year, at the peak of the delta variant surge in August and in its most recent data, and created a tool that generates random theoretical patients. Each rate of infection or death is relative to pools of 100,000 people, so the interactive below generates a random number between zero and 100,000. If that number is below the rate of infection or death for the group (you can choose which), it’s marked with a colored dot. If it’s above the rate of infection or death, it’s a gray dot.

The difference should quickly be obvious. At the bottom of the graphic, the relative likelihood of infection or death between the two groups, calculated based on the dots that had already been displayed.

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There are certainly questions about the extent to which vaccines will continue to provide this additional layer of effectiveness. Research about the omicron variant hasn’t yet determined whether it is more likely to escape the protection of immunization. (New analysis from Pfizer and partner BioNTech suggests that a three-dose regimen of their vaccine is effective.) But it’s clear from the data — and the visual above — that there is an appreciable benefit to the vaccine regardless.

Much of the opposition to vaccination isn’t rooted in rational consideration of the effects, of course. But if you encounter someone who indicates that they aren’t confident the vaccines do anything, show them the interactive above. Everyone wants to be a gray dot in this simulation, and the odds are much better that you will be if you’re in that right-hand column.