Is it appropriate to use months abbreviations in academic writing? For example:
As of Aug. 2015, the program usage hit 50%
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This is strictly a question of style. Your best bet is to look at published literature in your field and see what others do. For what it's worth, if you search the Corpus of Contemporary American English (COCA) for "as of Aug", "as of Jan", etc., you find very few hits overall that include just the month but not the day; and among those, only one hit comes from an academic source (from Boston College Environmental Affairs Law Review). The Chicago Manual of Style simply says that abbreviations for the month names can be used 'where space restrictions require' it.
In short: try to avoid it, but if you are really pressed for space, it's probably OK to use it. In any case, check the practices of other writers in your field.
And if this is for a course, the only thing that matters is what your teacher/instructor says, so you should ask them.