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Birnbaum's 1999 study serves as a classic demonstration of the problems associated with a lack of context in between-subjects designs. When participants rated the number $$9$$ or $$221$$ on a size scale in separate groups, the group rating $$9$$ judged it as mathematically 'larger' than the group rating $$221$$. Without the context of a within-subjects comparison, participants spontaneously compared $$9$$ to other single-digit numbers and $$221$$ to other three-digit numbers, skewing their relative size judgments.

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