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An error and a mistake, although share similar denotation, truly do vary in connotative meaning. An error is something wrong that is done when not knowing any different, versus a "mistake" implies something wrong that is done when knowing better. I'm not strong in math, and I typically would make errors on exams since I didn't have control of higher level math rules. However, I make careless mistakes when balancing my check book because I really despise doing it. There is a definite difference in connotation.
As far as error analysis, this is the formal study or examination of the types and causes of errors. In what ways do non-native speakers make errors in language use: pronunciation, writing, listening, and choice of vocabulary? This is, of course, a much different question when it comes to discussing the common mistakes made by native speakers. Or a bigger question may be, in what ways do native speakers make errors in language use? Are these "errors" or are they truly "mistakes"? When a student says or writes, "I ain't got none", is it a statement made due to a lack of knowledge or is it knowlingly a statement which breaks the rules due to laziness and habit? I think we know the answer.
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