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... but there remains an inconsistency that should be rather addressed now than postponed.

I did this pull request to show the changes (and the additional confusion) that would be necessary in my eyes to resolve the inconsistent default value for corrected in mean_and_var(x) [default corrected=true consistent with Statistics.std(x)] and mean_and_var(x,w) [default corrected=false consistent with StatsBase.std(x,w) which is however deprecated].

Also, I hope that the code update is adequate because I did not figure out how to test the docstring properly [thankful for easy to reproduce advice].

I personally would not accept the pull request, but I would rather stimulate discussion about whether it is the right time to make a breaking release and resolve the deprecated behavior in StatsBase. But this is obviously not my decision to make.

…ins an inconsistency that should be rather addressed now than postponed
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