  The purpose of this paper is to exhibit a quantitative stability result for
the class of M\"obius transformations of $\mathbb{S}^{n-1}$ when $n\geq 3$. The
main estimate is of local nature and asserts that for a Lipschitz map that is
apriori close to a M\"obius transformation, an average conformal-isoperimetric
type of deficit controls the deviation (in an average sense) of the map in
question from a particular M\"obius map. The optimality of the result together
with its link with the geometric rigidity of the special orthogonal group are
also discussed.
