The aim of the paper is to derive essential elements of quantum mechanics from a parametric structure extending that of traditional mathematical statistics. The main extensions relate to symmetry, the choice between complementary experiments and hence complementary parametric models, and use of the fact that there is a limited experimental basis that is common to all potential experiments. Concepts related to transformation groups together with the statistical concept of sufficiency are used in the construction of the quantum mechanical Hilbert space. The Born formula is motivated through recent analysis by Deutsch and Gill, and is shown to imply the formulae of elementary quantum probability/quantum inference theory in the simple case. Planck's constant, and the Schrödinger equation are also derived from this conceptual framework. The theory is illustrated by one and by two spin 1/2 particles; in particular, a statistical discussion of Bell's inequality is given.