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Lumifer comments on Dancing room for you and applied problems in foreign fields - Less Wrong
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If the statistician is competent, he has a specialized toolset and the necessary skills to use it -- something that a generalist usually lacks.
The question is similar to "what's the difference between complaining about some pain to a doctor and to a broadly educated rationalist".
Moreover, stats is one of those fields where "A little knowledge is a dangerous thing" applies in spades...