the pvariance and variance functions take the argument mu and xbar to pass the population and sample mean to avoid some recomputation.

I assume the keyword arguments are different because the two means accepted are different, but the docstring does not indicate this directly.
It just says mu or xbar is the mean of the data. The module documentation is a little clearer but only in the grey box right at the end.

I would propose to change the docstring and module documentation to explicitly state that mu is the population mean and xbar is the population mean.
E.g.

- The optional argument mu, if given, should be the mean of
the data.
+ The optional argument mu, if given, should be the population mean of
the data.

etc.

xbar is the *sample* mean of course

maybe with proper docstrings the two functions could also use the same keyword argument?

I agree that the current wording in the doc-strings is ambiguous. It should be more careful to distinguish between

mu : true/population mean
xbar : estimated/sample mean

I disagree that the keyword arguments should be made the same. There is an important conceptual difference between these two things that the user needs to be aware of and mu, xbar - as symbols rather than ascii characters - are widely used for this. See e.g. this Wikipedia entry (although it uses ybar instead of xbar):
http://en.wikipedia.org/wiki/Variance#Population_variance

Steven included some draft docs updates in a separate patch for issue 20481 - I have split those out and uploaded them here (since this discussion refers to the docstrings as well).

I've closed issue36099 as a duplicate of this.