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The website Brainstats lists IQs for different countries and lists 92 for Ireland while it lists 100 for the United Kingdom. Those numbers are surprising to me. Are they backed by good evidence?

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I cannot answer about the UK and Ireland specifically, but the data presented there is not reliable. It comes from books by Lynn and Vanhanen, that have been heavily criticized by other scientists.

The summary given by Brainstats is subtly wrong; Lynn and Vanhanen did not "conduct" the IQ studies in different countries, they reviewed studies done by other people. This is quite problematic, and their methods did not make things better. From a book review published in the American Psychological Associations review of books:

Building a representative IQ estimate for any single country is clearly a tremendously difficult task, requiring extensive preliminary work to understand the distribution of individuals within the population by age, education, socioeconomic status, and so on, as well as extensive testing for each subgroup. Establishing representative IQ data for all 81 countries in the primary group discussed here is, thus, a Herculean challenge, and extrapolating to the total world group of 185 countries, based on neighboring countries and racial mix, is more daunting still. These authors have made a valiant effort to accomplish this task, gathering estimates from numerous individual studies from around the globe. Unfortunately, when the data available are of such questionable representativeness, it is impossible to draw meaningful conclusions without adjusting for geographical sample bias, age group, sample selection criteria, and so on. The authors do this in selected cases but, to be meaningful, such an adjustment would need to be done systematically across the board and based on evidence rather than conjecture. This has not been done, rendering the resulting comparisons virtually meaningless.

Wikipedia's article on the book summarizes quite a few scholarly criticisms of the book, and each criticism is traceable back to its published source.

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