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Precision is about variability while accuracy (in contrast to precision) is about bias. This tag pertains to measurement or estimation; use [precision-recall] when talking about classifiers.

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What do confidence intervals say about precision (if anything)?

I think the precision fallacy is a true fallacy, but not necessarily one we should care about. It isn't even that hard to show it's a fallacy. … Whenever we do data analysis we can only estimate precision, so of course the two individuals will reach different conclusions. …
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