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Bhutan is a Buddhist country famous for its Gross National Happiness metric. So far, poverty is a drag on that metric: Bhutan Happiness Index: Buddhist Country Fails On Its 'Gross National Happiness'
Other economists have thought about happiness and how Buddhism might figure in to it, ref. Happiness and economics: a Buddhist perspective (too much material to summarize quickly).
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Buddhism is not about happiness. Its about letting go.
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You need to be careful with how you use statistics to demonstrate things because there can be a lot of confounding variables. Many Buddhist countries have political instability, poverty, various public health problems, and much of the population aren't full blown practitioners of the teachings, but are mostly nominal Buddhists who might occasionally participate in a festival or ceremonies, but don't know much about the teachings and how to practice them.
To draw scientific conclusions you would need to be able to compare groups of people who are similarly situated in things except for practicing Buddhists and everyone else, and that would be very hard.
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This article describes a study from 2001, involving monks with EEGs and functional MRIs.
INDEPTH: MEDITATION
The Pursuit of Happiness
CBC News Online | April 23, 2004
His prize subjects β and collaborators β are the Dalai Lama's lamas, the monks.
"The monks, we believe, are the Olympic athletes of certain kinds of mental training," Davidson says. "These are individuals who have spent years in practice. To recruit individuals who have undergone more than 10,000 hours of training of their mind is not an easy task and there aren't that many of these individuals on the planet."
The latter part of the article describes meditation's also being useful to non-Buddhists.
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On the point of happiness as a 'slippery notion', I think there's a very helpful distinction to make between hedonic and eudaemonic happiness. Roughly speaking (and this is just my take - I welcome refinements or corrections): through the first 2 noble truths, a Buddhist practitioner stops seeking ultimate refuge in hedonic happiness, turning away from the sorrow and disappointment that inevitably follow from putting all of one's eggs in this basket; and through the 3rd and 4th noble truths a practitioner begins to perfect his or her eudaemonic happiness.
As to how successful Buddhist practitioners are in this endeavor as a whole... honestly, it's a fascinating question but I have no idea how to answer it empirically on a large scale. For myself and the practitioners I've met, simply turning toward eudaemonic happiness has been a profound and lasting source of joy - a path good in the beginning, good in the middle, and good at the end.