The well-known Loewner order expresses that one self-adjoint operator's expectation value is less than or equal to the other's with respect to all quantum states. The Loewner order has been extensively examined in the literature, and many of its properties are well understood. In particular, Molnar's theorem describes the form of all Loewner order-automorphisms of the set of all self-adjoint operators. In this talk we propose a similar order relation based on the variance, and prove two results. First, we show that one self-adjoint operator's variance is less than or equal to the other's with respect to all quantum states if and only if the former is a $1$-Lipschitz function ot the latter. Second, we characterise all order-automorphisms with respect to this newly proposed order relation, and show that in some sense they have a more rigid form than in the case of Loewner order-automorphisms. This is a joint work with Nazar Miheisi (King's College London, UK).