At variance

the state, quality, or fact of being variable, divergent, different, or anomalous.
an instance of varying; difference; discrepancy.
Also called mean square deviation. Statistics. the square of the standard deviation.
Physics, Chemistry. the number of degrees of freedom of a system.
Law.

a difference or discrepancy, as between two statements or documents in law that should agree.
a departure from the cause of action originally stated in the complaint.

an official permit to do something normally forbidden by regulations, especially by building in a way or for a purpose normally forbidden by a zoning law or a building code.
a disagreement, dispute, or quarrel.
at variance,

(of things) in a state of difference or disagreement.
(of persons) in a state of controversy or dissension:
at variance with one’s superiors.

noun
the act of varying or the quality, state, or degree of being divergent; discrepancy
an instance of diverging; dissension: our variance on this matter should not affect our friendship
at variance

(often foll by with) (of facts, etc) not in accord; conflicting
(of persons) in a state of dissension

(statistics) a measure of dispersion obtained by taking the mean of the squared deviations of the observed values from their mean in a frequency distribution
a difference or discrepancy between two steps in a legal proceeding, esp between a statement in a pleading and the evidence given to support it
(in the US and Canada) a licence or authority issued by the board of variance to contravene the usual rule, esp to build contrary to the provision of a zoning code
(chem) the number of degrees of freedom of a system, used in the phase rule
(accounting) the difference between actual and standard costs of production
n.

mid-14c., “fact of undergoing change,” from Old French variance, from Latin variantia, from variare “to change” (see vary). Meaning “state of disagreement” is recorded from early 15c. The U.S. zoning sense of “official dispensation from a building regulation” is recorded from 1925.

variance var·i·ance (vâr’ē-əns, vār’-)
n.

The state or quality of being variant or variable; a variation.

The state or fact of differing or of being in conflict.

The square of the standard deviation.

Differing, discrepant; also, in a state of conflict. For example, John’s and Mary’s answers are at variance means that their answers do not agree, or John was at variance with his in-laws means that he strongly disagreed or quarreled with them. [ Early 1500s ]