The Small Punch (SP) test serves the screening of mechanical material properties and their degradation in a virtually non-invasive way. It requires robust frameworks for the derivation of mechanical properties and microstructure-mechanical property correlation. The tensile yield stress sigma(y) is commonly associated with an elastic-plastic transition force F-e via sigma(y) = alpha F-e/h(2) with h denoting the SP disc thickness and a dimensionless coefficient alpha considered constant. Here it is shown that alpha cannot be taken as a constant. Instead a new self-consistent data reduction scheme is proposed for the determination of sigma(y) which is based on the curvature of the force-displacement curve rather than a single F-e force level. The scheme derives from finite element simulations of a wide range of strength coefficients C and hardening exponents n of power law flow sigma = C epsilon(n). To a good approximation the scheme depends only on the hardening exponent n, which depends on the curvature, whereas C and the elastic modulus barely matter. The method is validated by comparing the yield stress predictions with the actually implemented yield stresses in the simulations, using various types of hardening rules, as well as experimental data. The uncertainty of yield stress determination by SP tests is thereby largely reduced as compared to the traditional scheme. (C) 2019 The Authors. Published by Elsevier Ltd.