I created a solution  for ENH: Computing std/var and mean at the same time, 
issue #23741. The solution can be found here: 
https://github.com/soundappraisal/numpy/tree/stdmean-dev-001 

I still need to add tests and the solution does touch the implementation of 
var. But before starting a pull request I like to check whether mean_std is a 
welcome addition.

Also I was struggling with the internally needed format of the arrays 
containing mean and std and the format produced as a result. Which makes me 
uncertain whether the chosen solution is correct for all cases.
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