Watch Eric Schmidt, chairman of China Entrepreneurs in Beijing, talks about China's growing economic might.
So what is behind these feelings about China's growing economic weight?
The survey does not tell us for sure, but there are some obvious candidate explanations.
In the period since the earlier poll - in 2005 - the world has been through an episode called the great recession, a result of the financial crisis.
The developed world was hard hit. The rebound now underway in the global economy is led by developing countries, notably China.
The recovery in the rich nations by contrast is more sluggish. The rise in unemployment caused by the recession is likely to take years to reverse.
Tom Friedman, the influential New York Times columnist and Pullitzer Prize winner, told the BBC: "there's no question that China's rise, coinciding with a sense of stagnation and paralysis among many of the leading western democracies, is psychologically unsettling".
End QuoteTom FriedmanNew York Times
There is also a very specific economic issue, and that does emerge in the BBC survey.
People were asked if they think China trades fairly with other countries.
Those saying China is unfair were above 50% in Japan, South Korea, Germany and Italy. In the US, the figure was 45%, compared with 24% saying that it was fair.
The particular policy that has attracted so much attention, in the media and in business, is China's approach to its currency, holding its value down by intervening in the foreign exchange market.
Critics, and there are many of them, say that gives Chinese industry an unfair competitive advantage.
Tom Friedman is particularly caustic about this policy: "That's part of a broader concern of people which is that China is still in many ways a freeloader on the international system. It's not a stakeholder."