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It means that they can analyze problems very quickly and solve problems very quickly.

The above definition is incorrect. There are dozens of different kinds of problems each requiring a different kind of thinking to handle them. For example a problem in human relationships requires different kinds of cognition than a math problem, or a budgeting problem, or an exercise regimen problem, or a marketing problem etc. So the above definition is essentially meaningless.

Does anyone here have a real definition?

I was wondering the same thing as I was reading my econ book.

dictionary.com states that being analytical means " to be skilled or habitually using analysis" and an analysis is "a philosophical method of exhibiting complex concepts or propositions as compounds or functions of more basic ones."

so...This is just my definition: Being analytical means that one is able to break something down (a problem or machine for example) into smaller components to draw a conclusion. This inhances one's ability to reason rationally.

freedictionary.com states:

an·a·lyt·ic

(n-ltk) or an·a·lyt·i·cal (--kl)adj.1. Of or relating to analysis or analytics.

2. Dividing into elemental parts or basic principles.

3. Reasoning or acting from a perception of the parts and interrelations of a subject: "Many of the most serious pianists have turned toward more analytic playing, with a renewed focus on the architecture and ideas of music" (Annalyn Swan).

4. Expert in or using analysis, especially in thinking: an analytic mind; an analytic approach. See Synonyms at logical.

5. Logic Following necessarily; tautologous: an analytic truth.

6. Mathematicsa. Using, subjected to, or capable of being subjected to a methodology involving algebra or other methods of mathematical analysis.

b. Proving a known truth by reasoning from that which is to be proved.

7. Linguistics Expressing a grammatical category by using two or more words instead of an inflected form.

8. Psychoanalytic.

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