When people say "do analysis with and without outliers", what do they mean? When people say that outliers ($\neq$ experimental error) shouldn't be removed, but that analysis should be done both with and without them, do they mean that I should use the same model for both cases, and then report on potential differences? Even if the model with outliers is not satisfying the assumptions of e.g. normality?
Is that what people usually do? Or do people usually use different models?
 A: Researchers may want to know if a result is being driven by a few extreme observations.


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*If no, then what you do about outliers is arguably a moot point.

*If yes, then it matters that you take a reasonable/correct approach for your situation because it affects the result!


Some extreme examples:


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*If the outliers are clear measurement error (eg. negative weight), then it would be crazy/insane to publish a result that's driven by these bogus observations.

*If you're estimating average returns and the outliers are negative -100% returns (eg. firm bankruptcy), then excluding them would be dishonest/insane.

A: You should not generally remove outliers from your raw data -- that is dishonest.  The proper thing to do is to present the raw data, then describe how you chose the outliers to be deleted and the analysis with those outliers deleted.
The reason for this is that, sometimes, the 'outliers' are actually telling you something. As somebody else said in a similar discussion sometimes 'outliers' result in new patents.
A: This question Is it OK to remove outliers from data? discusses the outlier detection in detail. As discussed there, I usually don't recommend removing outliers from data. 
However, in certain cases where you are sure that it is true outlier, e.g. negative age and weight, you could remove them. 
Coming back to the question, you could be best served by asking the person who asked you analyze data with all the knowledge you get from the linked 
question above. If I have to answer that question, I'd say I will analyze both sets of data with same model and report the differences but would fight for 
including outliers in the analysis till my death.
