statistics.variance also has the same problem. 

>>> import statistics
>>> statistics.stdev([1,2])
0.7071067811865476
>>> statistics.stdev([1,2], 3)
0.7071067811865476
>>> statistics.stdev([1,2], 1.5)
0.7071067811865476

should be 
0.7071067811865476
2.23606797749979
0.5

The relevant code is in the _ss() helper function:

    # The following sum should mathematically equal zero, but due to rounding
    # error may not.
    U, total2, count2 = _sum((x-c) for x in data)
    assert T == U and count == count2
    total -=  total2**2/len(data)

The intent was to correct for small rounding errors, but the effect is to undo any xbar value that differs from the true mean.

From a user point-of-view the xbar parameter should have two effects, saving the computation time for the mean and also giving the ability to recenter the stdev/variance around a different point.   It does save a call to mean; however, that effort is mostly throw-away by the rounding adjustment code which does even more work than computing the mean.

Likely, the fix for this is skip the rounding adjustment code if the user supplies an xbar value.

Perhaps this would work:

diff --git a/Lib/statistics.py b/Lib/statistics.py
index c76a6ca519..93a4633464 100644
--- a/Lib/statistics.py
+++ b/Lib/statistics.py
@@ -682,8 +682,10 @@ def _ss(data, c=None):
     calculated from ``c`` as given. Use the second case with care, as it can
     lead to garbage results.
     """
-    if c is None:
-        c = mean(data)
+    if c is not None:
+        T, total, count = _sum((x-c)**2 for x in data)
+        return (T, total)
+    c = mean(data)
     T, total, count = _sum((x-c)**2 for x in data)
     # The following sum should mathematically equal zero, but due to rounding
     # error may not.


Matti, where do you get 0.5 as the expected outcome for the third example?   The actual mean is 1.5, so I would expect the third case to give sqrt(2)/2 or 0.707.

Thanks Raymond, that is the intended effect, and your analysis seems 
plausible.

If we estimate the mean using a sample we loose one degree of freedom so it will be divided by N-1, while if we have the mean independent of the sample it should be divided by N to be unbiased. 

i.e. 
example 1 
sqrt(((1-1.5)²+(2-1.5)²)/(2-1)) = 0.7...
example 3
sqrt(((1-1.5)²+(2-1.5)²)/(2)) = 0.5

Hi Raymond and Steven!

I'm happy that you are solving this issue but do you have any comment on my previous answer?

> do you have any comment on my previous answer?

I see what you're trying to do but think that interpretation is surprising
and is at odds with the existing and intended uses of the *xbar* argument.  

The goals were to allow the mean to be precomputed (common case) or to be recentered (uncommon).  Neither case should have the effect of changing the divisor.  

We can't break existing code that assumes that stdev(data) is equal to stdev(data, xbar=mean(data)).

>>> data = [1, 2]
>>> stdev(data)
0.7071067811865476
>>> stdev(data, xbar=mean(data))
0.7071067811865476

New changeset d71ab4f73887a6e2b380ddbbfe35b600d236fd4a by Raymond Hettinger in branch 'master':
bpo-40855: Fix ignored mu and xbar parameters (GH-20835)
https://github.com/python/cpython/commit/d71ab4f73887a6e2b380ddbbfe35b600d236fd4a

New changeset 55c1d21761e2e5feda5665065ea9e2280fa76113 by Miss Islington (bot) in branch '3.9':
bpo-40855: Fix ignored mu and xbar parameters (GH-20835) (#GH-20862)
https://github.com/python/cpython/commit/55c1d21761e2e5feda5665065ea9e2280fa76113

New changeset 811e040b6e0241339545c2f055db8259b408802f by Miss Islington (bot) in branch '3.8':
bpo-40855: Fix ignored mu and xbar parameters (GH-20835) (GH-20863)
https://github.com/python/cpython/commit/811e040b6e0241339545c2f055db8259b408802f

Thanks for the bug report 😊

>I see what you're trying to do but think that interpretation is surprising
>and is at odds with the existing and intended uses of the *xbar* argument.  
>
>The goals were to allow the mean to be precomputed (common case) or to be recentered (uncommon).  Neither case should have the effect of changing the divisor.  
>
>We can't break existing code that assumes that stdev(data) is equal to stdev(data, xbar=mean(data)).

Maybe the requirement are buged? It seems to me that recalculating the mean is a very niche use case. You will very little time on a call you do once. 

But what good is it to supply a re-centered mean if you get a wrong estimation of the standard deviation? If the mean is not the mean of the sample it was not calculated using the sample so there is no loos of degrees of freedom.

I meant to write "pre-calculate".