Inspired by the comment from @NickCox  I found this note on Wikipedia. We have that the `n-1` term does correct for the `variance` but the non-linear aspect of the square root allows for only partial correction on the `standard deviation`.  There is not a general way to get an unbiased estimate of the latter.

[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/5twF2.png