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I am submitting my first academic paper (in computer science). It is a short paper for a conference. In my abstract, I state what my empirical research uncovers. Is it appropriate to then explain why that empirical result is important? Or does that argument belong in a different part of the paper?

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Yes it is generally better when the abstract indicates why the experiment is important, since it's key to hook the readers (otherwise why bother reading the paper if it brings no "important" results?), but succinctly as anything else in the abstract. There are typically 2 parts in the abstract where you can explain why the experiment is important:

  • when explaining why the problem it addresses is important (somewhere in the first half of the abstract)
  • when summarizing the implication of your results (at the end of the abstract).

I love the following abstract for a paper on how to write an abstract written by Steve Easterbrook; I italicized the two parts I mentioned above:

The first sentence of an abstract should clearly introduce the topic of the paper so that readers can relate it to other work they are familiar with. However, an analysis of abstracts across a range of fields show that few follow this advice, nor do they take the opportunity to summarize previous work in their second sentence. A central issue is the lack of structure in standard advice on abstract writing, so most authors don’t realize the third sentence should point out the deficiencies of this existing research. To solve this problem, we describe a technique that structures the entire abstract around a set of six sentences, each of which has a specific role, so that by the end of the first four sentences you have introduced the idea fully. This structure then allows you to use the fifth sentence to elaborate a little on the research, explain how it works, and talk about the various ways that you have applied it, for example to teach generations of new graduate students how to write clearly. This technique is helpful because it clarifies your thinking and leads to a final sentence that summarizes why your research matters.

That said, to make sure your in the same tone as the conference, reading abstracts of papers published in the same conference in the last years is the ultimate reference.

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