In concentrated solid solutions, the random distribution of elements of different sizes induces characteristic displacement and stress fields at the root of solid solution strengthening. The aim of this two-part article is to derive the statistical properties of these elastic fields. The present Part I focuses on the variance of the elastic fields, while Part II addresses their spatial correlations. In this first part, we develop two elastic models of random solid solutions, based respectively on realspace and spectral methods, to derive the mean square displacement and shear stress. Both approaches hold advantages and drawbacks that are discussed here. We show in particular that both the mean square displacement and shear stress are directly proportional to the variance of the atomic eigenstrains, which embodies the atomic size differences and simplifies to the classical lattice mismatch parameter if the alloy satisfies Vegard's law. This allows to clarify the scaling relations between various quantities (mismatch parameter, mean square displacement and yield stress) and to bridge energy-based and stress-based models of concentrated solid solution strengthening proposed in the literature. The elastic predictions for the mean-square displacement and stresses are also successfully compared with atomistic results obtained for a model Lennard-Jones system and with a more complex Al-Mg interatomic potential.