Quantum measurement bounds beyond the uncertainty relations

In quantum mechanics, the Heisenberg uncertainty relations and the Cramer-Rao inequalities typically limit the precision in the estimation of a parameter through the {\it standard deviation} of a conjugate observable. Here we extend these relations by giving a bound to the precision of a parameter in terms of the expectation value of the conjugate observable. This has both foundational and practical consequences: in quantum optics it resolves a controversy on which is the ultimate precision in interferometry.