We investigate the regulators of elliptic curves with rank 1 in some families of quadratic twists of a xed elliptic curve. In particular, we formulate some conjectures on the average size of these regulators. We also describe an ecient algorithm to compute explicitly some of the invariants of a rank one quadratic twist of an elliptic curve (regulator, order of the Tate-Shafarevich group, etc.) and we discuss the numerical data that we obtain and compare it with our predictions.