Dictionary definitions
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at variance
From The Collaborative International Dictionary of English v.0.48:
Variance \Va"ri*ance\, n. [L. variantia.]
1. The quality or state of being variant; change of
condition; variation.
[1913 Webster]
2. Difference that produces dispute or controversy;
disagreement; dissension; discord; dispute; quarrel.
[1913 Webster]
That which is the strength of their amity shall
prove the immediate author of their variance.
--Shak.
[1913 Webster]
3. (Law) A disagreement or difference between two parts of
the same legal proceeding, which, to be effectual, ought
to agree, -- as between the writ and the declaration, or
between the allegation and the proof. --Bouvier.
[1913 Webster]
4. (Statistics) The expected value of the square of the
deviation from the mean of a randomly distributed
variable; the second moment about the mean. This is also
the square of the standard deviation.
[PJC]
At variance, in disagreement; in a state of dissension or
controversy; at enmity. "What cause brought him so soon at
variance with himself?" --Milton.
[1913 Webster]

