How borders are drawn and enforced has far-reaching consequences, whether we live on either side of them or halfway across the world.

Do you think there are too many immigrants in your country? How people answer that question can depend on whether they know how many immigrants there actually are.

That’s the eye-catching finding of a new survey by the German Marshall Fund, the latest in its annual Transatlantic Trends poll about how Americans and Europeans feel about foreign policy. The pollster split respondents into two groups and asked them subtly different questions about the number of immigrants in their countries. In one, people were simply asked whether they thought there were “too many” immigrants; in the other, the question was the same but prefaced with the actual share of migrants in their country:

As you may know, according to official estimates, around [XX]% of the [COUNTRY] population was born in another country. In your opinion, is this too many, a lot but not too many, or not many?

Nearly 40% of Americans said that they thought there are too many immigrants, but this fell to 21% for those who were first given an estimate. It followed a similar pattern in Europe, with 32% saying there are too many, but also only 21% when respondents were told the actual figure first.

The gaps were particularly pronounced in Greece, Italy, and the UK:

This makes for a ”hall of mirrors” quality to the debate about immigration policy, with perceptions of a country overrun with migrants often running ahead of the reality, writes Peter Sutherland, the UN’s special representative for migration. “Anti-immigrant sentiment stems largely from misinformation, not entrenched animus,” he says.