The β″ phase is the major hardening precipitate in Al–Mg–Si alloys. The present study aims to improve understanding of the industrially important hardening process by systematically combining advanced microscopy, ab initio atomistic calculations and strength measurements of Al–Mg–Si alloys containing β″ precipitates. The microscopy identified Mg₄Al₃Si₄ as the most likely precipitate composition, with possibilities for compositional variation within a single precipitate. The atomistic calculations enabled quantification of precipitate formation energies and strain fields inside and around the precipitates. Associated measurements of precipitate size, hardness and yield strength in samples only containing β″ precipitates gave new, empirical relations between these parameters. This demonstrated that the particle sizes were effectively larger than directly observed, owing to coherency strain fields. The study demonstrates that atomistic insight now can be directly linked to the bulk elastic properties of advanced structural materials.